AVU-PARTNER INSTITUTION MODULE DEVELOPMENT TEMPLATE Prepared by Thierry Karsenti, M.A., M.Ed., Ph.D.
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AVU-PARTNER INSTITUTION
MODULE DEVELOPMENT TEMPLATE
Prepared by Thierry Karsenti, M.A., M.Ed., Ph.D.
With : Salomon Tchaméni Ngamo, M.A. & Toby Harper, M.A.
Version 19.0, Thursday December 21st, 2006
I.
INTRODUCTION
1. TITLE OF MODULE
Mechanics
2. PREREQUISITE COURSES OR KNOWLEDGE
Linear Algebra 1 and Calculus 3 is prerequisite.
3. TIME
The total time for this module is 120 study hours.
4. MATERIAL
Students should have access to the core readings specified later. Also, they will
need a computer to gain full access to the core readings and web links in the
materials.
5. MODULE RATIONALE
This topic of mechanics has been treated as that branch of mathematics that
mathematically attempts to explain our physical environment. It is that branch of
mathematics that spans the divide between our natural sciences and
mathematics.
The module has infused as its basis, both modern mathematics and traditional
mathematics as it explores the basic concepts of Kinematics – which is the
study of motion without referrence to forces which cause motion of particles;
Kinetics which relates the action of forces on particles and bodies to their
resulting motions; Dynamics – the study of the general causes of motion; and
Statics – the equillibrium mechanics of stationary bodies
Practical examples and their implications for classroom practice have been
suggested whenever and wherever appropriate in the module in order to help the
teacher to impart the knowledge of mechanics to pupils.
II.
CONTENT
6. Overview
Overview: Prose description
This module is a first degree course in Mechanics. The module starts with a
treatment of vectors and vectors calculus and attempts to explain all the topics in
mechanics from this basis. It is hoped that the students enrolling for this course
will acquaint themselves with the basic notions of force and the resultant motion
of its application. Four areas of mechanics: Statics, Dynamics, Kinetics and
Kinematics of particles and rigid bodies have been treated in this module.
The student is strongly advised to consult Physics sources on Mechanics in
conjunction with this module for practical examples which are mathematically
modelled in the module.
Outline: Syllabus
Unit 1: Force, Energy and Motion
Level 1. Priority A. Linear Algebra 1 and Calculus 3 is prerequisite. (Note overlap
with Vector Analysis)
Vectors, Velocity and Acceleration: Dot product, cross product, triple products.
Derivatives of vectors, Integrals of vectors. Relative velocity and acceleration.
Tangential and normal acceleration, Circular motion. Gradient, divergence and
curl. Line integrals and independent of path.
Newton's Law of Motion - Work, Energy and Momentum: Work, power, kinetic
energy. Conservative force fields, potential, conservation of energy. Impulse,
Torque and angular momentum, conservation of momentum.
Motion in a Uniform Force Field. Falling bodies and projectiles. Potential and
potential energy in a uniform force field. Projectiles, motion in a resisting medium.
Constrained motion. Friction. Because of the volume of content in this unit, we
have seen it prudent to divide this unit into two parts – 1a and 1b.
Unit 2: Oscillations
Level 1. Priority A. Mechanics 1 is prerequisite.
The Simple Harmonic Oscillator and the Simple Pendulum: Energy of a simple
harmonic oscillator. Over-damped, critically-damped and under-dumped motion.
Simple pendulum. The two and three dimensional harmonic oscillator.
Unit 3: Dynamics
Level 2. Priority B. Mechanics 2 is prerequisite.
Central Forces and Planetary Motion: Equations of motion for a particle in a
central field; Potential energy of a particle in a central field; Conservation of
energy. Kepler's laws of planetary motion.
Moving Coordinate Systems: Rotating coordinate systems, Derivative
Operations, Velocity, and Acceleration in a moving system. Coriolis and
centripetal acceleration (and force). Motion of a particle relative to the earth.
Systems of Particles: Conservation of momentum, angular momentum, external
torque. Kinetic energy, work, potential energy. Principle of virtual work;
D'ALembert's principle.
Rockets and Collisions: Problems involving changing mass. Rockets, Collisions
(direct and Oblique).
Unit 4: Rigid Bodies and Energy
Level 3. Priority C. Mechanics 3 is prerequisite.
Plane Motion of Rigid Bodies: Euler's Theorem. Chasle's Theorem. Moment of
inertia. Radius of gyration. Parallel axis theorem. Perpendicular axis theorem.
Couples. Kinetic energy and angular momentum about a fixed axis. Principle of
angular momentum. Principle of conservation of energy. Principle of virtual work
and D'Alembert's principle. Principle of minimum potential energy.
Graphic Organiser
RIGID
BODIES
DYNAMICS
OSCILLATIONS
NEWTON’S
LAWS OF
MOTION
VECTOR
ALGEBRA
PLANE AND
CURVILINEAR
MOTION
7. General Objective(s): for the whole module
By the end of the module, the trainee should be able to:
o Relate mathematical notions to physical quantities such as force and
motion.
o Model some physical phenomena mathematically as required for effective
teaching of mathematics in the secondary school.
o Relate traditional mechanics operations to vector calculus and vice-versa.
8. Specific Learning Objectives (Instructional Objectives): separate
objectives for each unit
You should be able to:
1. Be equipped as a student and as a teacher with vector operations.
2. Be infused with the basic tools of analysis into vector quantities.
3. Infuse the basic tools of analysis into various types of motion e.g. Simple
Harmonic Motion.
III.
TEACHING AND LEARNING ACTIVITIES
9. PRE-ASSESSMENT
Title of Pre-assessment: Basic Algebraic Ideas Test
Rationale: To check learner’s familiarity with some concepts assumed in the
module.
QUESTIONS
1. Velocity is the
a. rate of change of displacement
b. rate of change of speed
c. rate of change of distance
d. rate of change of time
2. Which of the following is group of vectors?
a. speed, acceleration and time
b. displacement, velocity and acceleration
c. direction, displacement and velocity
d. force, velocity and time
3. The resultant of velocities 8ms-1 and 6ms-1 at right angles is
a. 12ms-1
b. 10ms-1
c. 7ms-1
d. 9ms-1
4. The momentum of a body is the
a. mass of a body times its speed
b. weight of a body times its velocity
c. mass of a body times its velocity
d. weight of a body times its speed
5. Newton’s first law of motion states that
a.
If a body is at rest it might remain at rest or if it is in motion it moves with a
speed until it stops.
b.
c.
d.
If a body is at rest it remains at rest or remains in motion until it is acted on
by a resultant force
If a body is at rest it remains at rest or if it is in motion it moves with uniform
velocity until it is acted on by a resultant force
If a body a body is at rest it remains at rest
6.
A car of mass 1.0 103 kg travelling at 72 km h-1 on a horizontal road is
brought to rest in a distance of 40 m by the action of the brakes and
frictional forces. Find the average stopping force.
a.
b.
5.0 102 N
c.
5.0 104 N
d.
5.0 101 N
5.0 103 N
7. A scalar quantity has
a. direction only
b. magnitude only
c. direction and magnitude
d. none of the above
8.
A train which is moving with uniform acceleration is observed to take 20 s
and 30 s to travel successive 400 metres. How much farther will it travel
before coming to rest if the acceleration remains uniform?
a.
b.
c.
d.
163.3 m
963.3 m
800 m
663.3 m
9. The following is NOT an equation of motion in a straight line.
a. v u at
b. x ut 12 at 2
c. v 2 u 2 2ax
d v u at 2
10. Power
a. is the ability to have energy
b. is the ability to run
c. is the ability to have speed
d. is the ability to do work
11. A mass of 5 kg moves on smooth horizontal plane with a speed of 8 m/s,
being attached to a fixed point on the plane by a string of length 4 m. find
the tension of the string.
a.
b.
c.
d.
16 N
40 N
80 N
20 N
12. A particle of mass 3 kg, resting on a smooth table and attached to a fixed
point on the table by a rope 1.2 m long, is making 300 rev/min. Find the
angular velocity.
a.
b.
c.
d.
10 rev/s
10π rad/s
5rev/s
5 π rad/s
13. An engine, of mass 80 Mg, is moving in arc of a circle of radius 240 m, with
a speed of 48 km/h. what force must be exerted by the rails towards the
center of the circle.
a.
b.
c.
d.
0.59×105 N
0.59×104 N
0.59×103 N
0.59×102 N
14. Impulse is defined as
a.
b.
c.
d.
the product of force and distance
the product of force and mass
the product of force and time
the product of force and velocity
15. The rate of change of angular momentum of a rigid body rotating about a
fixed axis equals
a.
b.
c.
d.
the moment about that axis of the internal forces acting on the body.
the moment about that axis of the external forces acting on the body.
the moment about that axis of the speed of the body.
the moment about that axis of the acceleration of the body.
16. Angular velocity is
a.
b.
c.
d.
radius of a circle over the speed of a particle
radius of a circle over the velocity of a particle
speed of a particle over the radius of a circle
velocity of a particle over the radius of a circle
17. A particle is said to move with simple harmonic motion if:
a.
the particle moves so that its acceleration along its path is directed towards
a fixed point in that path, and varies inversely as its distance from this fixed
point
b.
the particle moves so that its acceleration along its path is directed towards
a fixed point in that path, and varies directly as its distance from this fixed
point.
c.
the particle moves so that its speed along its path is directed towards a
fixed point in that path, and varies directly as its distance from this fixed
point
d.
the particle moves so that its acceleration along its path is directed towards
a fixed point in that path, and varies directly as its velocity from this fixed
point
18. A simple pendulum:
a.
consists of a heavy paticle or bob attached to a fixed point by a heavy string
and swinging in a vertical plane
b.
consists of a heavy paticle or bob attached to a fixed point by a weightless
string and swinging in all directions
c.
consists of a heavy paticle or bob attached to a fixed point by a heavy string
and swinging in all directions
d.
consists of a heavy paticle or bob attached to a fixed point by a weightless
string and swinging in a vertical plane
19. Which of the following does not typically denote a vector:
4
(a) -5 (b) (1, 2, 3) (c) A
(d) 8
3
20. A Subspace of vector space:
(a)
(b)
(c)
(d)
Is also a vector space
Is not a vector space
Is not a linear space
Is half of a vector space
ANSWER KEY
1.
a ((b),(c),(d) have scalar quantities speed, distance and time respectively.
So since velocity is a vector quantity a is correct)
2.
b (for (a) speed and time are not vectors, for (c) direction is not a vector, for
(d) time is not a vector )
3.
b (using the pythagorus theorem take 8ms-1 to be on the opposite side and
6ms-1 to be on the adjacent side then the resultant side is 10ms-1 )
4.
c (momentum is the product of mass and velocity since the particle is
moving in a particular direction. So (a), (b) and (d) are not correct)
5.
c ((c) is correct because the body is moving in a particular direction and
only stops when a force e.g. friction is applied on it)
6.
b (The initial velocity is 72 km h-1 or 20 m/s and final velocity 0 m/s and so
acceleration is 5 ms-2. thus force is mass times acceleration giving the
answer in (b))
7.
b (a vector quantity is the one with magnitude and direction so (b) is correct)
8.
a
9.
d ((d) is not the equation of a sraight line because of the t 2 in the equation.
The other equations are correct)
10. d (power is force times velocity or the rate of doing work so (d) is correct)
11. c (here we are looking at motion in a circle so acceleartion towards the fixed
point is (velocity)2/radius=64/4=16 ms-2. therefore tension is mass times
acceleration= 5×16= 80 N)
12
b ( motion is in a circle so we multiply the revolutions made per second by
2π since each revolution is made up of 2π )
13. a (48 km/h= 40/3 ms-1 and force exerted is mv2/r= 80 000×(40/3)2/240)
14. c
15. b
16. d
17. b
18. d
19. a ((c) normally denotes a vector or matrix, (b) and (d) denote vectors )
20. a
PEDAGOGICAL COMMENT FOR LEARNERS
The pre-assessment has been designed in such a way that it introduces the
student to basic motions kinetics and kinematics. It covers such concepts as
identifying the equations of motion in a straihgt line,basic notions of force and so
on.
Familiarity with basic algrebraic process is a fundamental skill and it is assumed
that all students enrolling for this course possess the skill.
A score of 50 % or less should be a cause for concern and will require the
student to revisit his “O” level work on Algebra and its processes. It is essential
that the student reads widely on that content they are not familiar with as it is
important to have this prior knowledge before embarking on the following units,
10. LEARNING ACTIVITIES
Unit 1a : Vectors, Vector Calculus, Velocity and Acceleration
Specific learning objectives
At the end of these activities you should be able to:
Define vectors and perform operations on vectors.
Differentiate and Integrate vector valued functions.
Define velocity and acceleration in terms of vectors and describe the
relationships between velocity and acceleration.
Given appropriate situations, define and calculate the relative velocities
and accelerations of bodies in motion.
Describe circular motion and calculate tangential and normal acceleration
of particles moving in circular motion.
Define and apply the concepts of gradient, divergence and curl.
Define and evaluate line integrals and independence of paths.
Summary of the learning activity
In this activity you will familiarise yourself with elementary vector calculus and its
application to motion in two and three dimensions.
Compulsory Reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Relevant Links and Resources
Vectors
http://en.wikipedia.org/wiki/Vector_(spatial)
Vector-valued function
http://en.wikipedia.org/wiki/Vector-valued_function
Acceleration
http://en.wikipedia.org/wiki/Acceleration
Velocity
http://en.wikipedia.org/wiki/Velocity
Divergence
http://en.wikipedia.org/wiki/DIVERGENCE
Curl
http://en.wikipedia.org/wiki/CURL
Gradient
http://en.wikipedia.org/wiki/Gradient
Gradient
http://hyperphysics.phy-astr.gsu.edu/hbase/gradi.html
Key Words (for description/definitions see glossary)
Scalar
Vector
Velocity
Acceleration
Function
Bridging The Gap
An army sergeant gives the following order to one group of new army recruits on
parade.
“Walk for five hours.”
To another group he orders:
“Run for five kilometres”
To a third he barks:
“On- the- double at ten kilometres per hour!”
As you may well be aware, army recruits are not allowed to question orders
given by a senior.
Describe the situation at the parade ground immediately after these orders and
the possible questions each recruit might be asking himself/herself.
What would happen if the sergeant had given similar instructions to a group of
pilots with the appropriate terminology and figures?
Detailed description of the activities
In this activity, we revisit the ideas of vectors and vector valued functions with the
intention of exploring vector calculus. We then use the results of vector calculus
to define velocity acceleration force and in Activity 2 finally discuss the motion of
particles and bodies in various settings in Activity 3. You will have opportunity to
examine various problem and solution situations as well as opportunities to solve
problems on your own..
1a.1
Vectors and Scalars
http://en.wikipedia.org/wiki/Vector_(spatial)
1a.1.1 Examples of Scalar Quantities
Vectors are quantities that require not only a magnitude, but a direction to specify
them completely. Let us illustrate by first citing some examples of quantities that
are not vectors. The number of litres of petrol in the fuel tank of your car is an
example of a quantity that can be specified by a single number---it makes no
sense to talk about a "direction" associated with the amount of petrol in a tank.
Such quantities, which can be specified by giving a single number (in appropriate
units), are called scalars. Other examples of scalar quantities include the
temperature, your weight, or the population of a country; these are scalars
because they are completely defined by a single number (with appropriate units).
1a.1.2 Examples of Vector Quantities
However, consider a velocity. If we say that a car is
going 70 km/hour, we have not completely specified its
motion, because we have not specified the direction
that it is going. Thus, velocity is an example of a vector
quantity. A vector generally requires more than one
number to specify it; in this example we could give the
magnitude of the velocity (70km/hour), a compass
heading to specify the direction (say 30 degrees from
North), and an number giving the vertical angle with
respect to the Earth's surface (zero degrees except in
chase scenes from action movies!). The adjacent figure
shows a typical coordinate system for specifying a vector in terms of a length r
and two angles, and
1a.1.3 Vectors in in 2-d and 3-d
Definition: The component forms of a vector v in 2-d and 3-d whose initial
point is the origin and whose terminal points are x1 , x2 and x1 , x2 , x3
respectively are given by
v = x1 , x2
v = x1 , x2 , x3
Definition: The length of a vector v will be defined as
v =
x12 x 22 for 2-d and
v =
x12 x 22 x32 for 3-d
Definition: If v is a non-zero vector in the 2-d or 3-d space, then the vector
u=
v
1
v
v
v
has length 1 in the direction of v.
Definition: The standard unit vectors (1, 0) and (0, 1) in 2-d and
1,0,0, 0,1,0and 0,0,1 in 3-d are
i = (1, 0) and j = (0, 1) and
i = 1,0,0 , j = 0,1,0 and k = 0,0,1
It is the assumption here that you have done work on vector addition and scalar
multiplication and the only operations we will discuss here are the dot product
and cross product Your core text, however, has sections that treat these
operations as indicated below. .
Compulsory Reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP pp 34-38
N.B. You will have recognized that the vector addition described is the
component wise addition that you are familiar with.
Vector magnitude P. 35
Pythagoras Theorem (3.6) P. 35
N.B. Note that the vector magnitude is also called the modules of the vector.
Equation (3.6 and (3.7)
Scalar Multiplication P. 35
Component wise scalar multiplication 3.8 P. 36
Diagonals of a parallelogram P. 36 – 38
N.B. Equations 3.9 – 3.13
Teaching Tip
This gives a geometrical interpretation of vector addition and scalar multiplication
which is a very useful tool when teaching secondary school students.
1a.1.4 The Dot Product
The Dot product of
u = v1 ,v2 and v = v1 ,v2 is
u v u1v1 u2 v2
The Dot product of u = u1 , u 2 , u and v = v1 , v 2 , v3 is
u v u1v1 u 2 v2 u3 v3
vv v
N.B.
2
Theorem: If is the angle between two non-zero vectors u and v, then
cos
uv
u v
Definition: The work done, W, by a force acting along the line of motion of an
object is given by
W= Force x distance = F PQ
Compulsory Reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP pp 40
N.B. The scalar product is also referred to as the “DOT” product. The
geometrical figure given in Figure 15 explains this vector operation very well.
1a.1.5
The cross product
Definition: The cross product of vectors u= v1i v2 j v3 k and v= v1i v2 j v3 k
is
u x v = (u 2 v3 u3 v2 )i u1v3 u3 v1 ) j (u1v2 u 2 v1 )k
A more convenient way is to write it as
i
j
u x v = u1 u 2
v1
v2
k
u3
v3
which is the determinant of a 3x3 matrix. Remember to refer to your module on
linear algebra to refresh yourself on properties of determinants.
DO THIS:
Verify that the two definitions are indeed the same
N.B. The cross product is not defined for vectors in 2-d
Theorem: Algebraic properties of the cross product
Let u and v be vectors and c be a scalar:
1.
2.
3.
4.
5.
6.
uxv=vxu
u x (v + w) = (u x v) + (u x w)
c(u x v)= cu x v= u x cv
ux0=0xu=0
u x u =0
u (v w) (u v) w
Theorem: Geometric properties of the cross product
Let u and v be non-zero vectors and be the angle between them.
1. u x v is orthogonal to both u and v.
2. u v u v sin
3. u x v = 0 if and only if one is a scalar multiple of the other
u v area of parallelogram of sides u and v
4.
N.B. The cross product can be used torque- the moment M of a force F about a
point P.
Example: If the point of application of the force is Q, the moment of F about P is
M= PQ x F
The magnitude of the moment F measures the tendency of the vector PQ to
rotate counter clockwise about an axis directed along the vector M
DO THIS:
Exercise: Prove that u x v = u v sin , where u and v are vectors, the angle
between them and x is the cross product.
(ANSWER: Hint. Remember that sin 1 cos and that cos
(u v) 2
u
1a.1.6
2
v
2
The Triple Scalar Product
Definition: The triple scalar product is the dot product of u and v + w
u (v w)
Theorem: For u = u1i u 2 j u3 k , v= v1i v2 j v3 k and w= w1i w2 j w3 k
u1
u (v w) = v1
u2
v2
u3
v3
w1
w2
w3
DO THIS:
Prove the theorem
)
1a.1.7
VECTOR VALUED FUNCTIONS
Definition: A vector valued function is a function where the domain is a subset
of the real numbers and the range is a vector In other words, vector valued
functions assign a vector to a number.
More specifically,
In 2-d
r(t) = x(t)i + y(t)j or r(t) = (x(t) , y(t)), or
In 3-d
r(t) = x(t)i + y(t)j + z(t)k or r(t) = (x(t) ,y(t) ,z(t))
You will notice the strong resemblance to parametric equations. In fact there is
an equivalence between vector valued functions and parametric equations.
Example
Sketch the graph of
r(t) = (t - 1)i + t2 j
Solution (Hide)
We draw vectors for several values of t and connect the dots. Notice that the
graph is the same as
y = (x + 1)2
1a.1.8
Calculus of Vector Valued Functions
The formal definition of the derivative of a vector valued function is very similar to
the definition of the derivative of a real valued function.
1a.1.9
The Derivative of a Vector Valued Function
Let r(t) be a vector valued function, then
= x'(t)i + y'(t)j
Because the derivative of a sum is the sum of the derivative, we can find the
derivative of each of the components of the vector valued function to find its
derivative.
Go to this link: http://en.wikipedia.org/wiki/Vector-valued_function
Examples
d/dt (3i + sintj) = costj
d/dt (3t2 i + cos(4t) j + tet k) = 6t i -4sin(t)j + (et + tet) k
1a.1.10
Properties of differentiation of Vector Valued Functions
All of the properties of differentiation still hold for vector values
functions. Moreover because there are a variety of ways of defining
multiplication, there is an abundance of product rules.
Suppose that v(t) and w(t) are vector valued functions, f(t) is a scalar function,
and c is a real number then
1. d/dt(v(t) + w(t)) = d/dt(v(t)) + d/dt(w(t))
2. d/dt(cv(t)) = c d/dt(v(t))
3. d/dt(f(t) v(t)) = f '(t) v(t) + f(t) v'(t)
4. (v(t) . w(t))' = v'(t) . w(t)+ v(t) . w'(t)
5. (v(t) x w(t))' = v'(t) x w(t)+ v(t) x w'(t)
6. d/dt(v(f(t))) = v'(f(t)) f '(t)
1a.1.11
Integration of vector valued functions
Definition: We define the integral of a vector valued function as the integral of
each component. This definition holds for both definite and indefinite integrals.
1.If r(t) = x(t)i +y(t)j, where x and y are continuous on [a , b] then
r (t )dt x(t )dt i y(t )dt j
and
b
b
r
(
t
)
dt
x
(
t
)
dt
i y (t )dt j
a
a
a
b
2. If r(t) = x(t)i + y(t)j + z(t)k, and x, y, and z are continuous on [a , b] then
r (t ) x(t )dt i y(t )dt j z (t )dt k
b
b
b
r
(
t
)
dt
x
(
t
)
dt
i
y
(
t
)
dt
j
z
(
t
)
dt
k
b
a
a
a
b
Example:
Evaluate
(sin t)i + 2t j - 8t3 k dt
Solution
Just take the integral of each component
(
(sin t)dt i) + (
2t dt j) - (
8t3 dt k)
= (-cost + c1)i + (t2 + c2)j + (2t4 + c3)k
Notice that we have introduce three different constants, one for each component
and that the three scalar constants produce one vector constant
Activity 1a.2
VELOCITY AND ACCELERATION
Define velocity and acceleration in terms of vectors and describe the
relationships between velocity and acceleration.
Given appropriate situations, define and calculate the relative
velocities and accelerations of bodies in motion
1a.2.1 Velocity
Go to this link: http://en.wikipedia.org/wiki/Velocity
Definition: Velocity and Speed
In single variable calculus the velocity is defined as the derivative of the position
function. For vector calculus, we make the same definition for both 2-2 and 3-d
space.
Let r(t) be a differentiable vector valued function representing the position vector
of a particle at time t. Then the velocity vector is the derivative of the position
vector.
In 2-space:
r(t) = x(t)i + y(t)j and Velocity = v(t) = r'(t) = x'(t)i + y'(t)j
Speed = v(t ) r ' (t )
x' (t )2 y' (t )2
In 3-space: r(t) = x(t)i + y(t)j + z(t)k and velocity = v(t) = r'(t) = x'(t)i + y'(t)j +
z'(t)k
Speed = v(t ) r ' (t )
x' (t )2 y' (t )2 z' (t )2
Example:
Find the velocity vector v(t) if the position vector is
r(t) = 3ti + 2t2j - sin t k
We just take the derivative
v(t) = 3i + 4tj + cos t k
N.B. When we think of speed, we think of how fast we are going. Speed should not be
negative. In one variable calculus, speed was the absolute value of the velocity. For
vector calculus, it is the magnitude of the velocity.
1a.2.2
Motion in one dimension.
Compulsory reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP pp18-31
This section introduces the concepts of displacement, velocity and acceleration
and the motion of a particle varying velocities such as constant, uniform and so
on
You should be able to relate the ideas of the position vector to the important
concept of displacement
Discussion: Are the two concepts of displacement and position vector the same?
1a. 2.3
Motion in three dimensions
Compulsory reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP p 33-52
N.B. You should note hear the introduction of the three dimensional Cartesian
plane to provide a suitable frame of reference to describe motion in three
dimensions.
For practical examples you can use the idea of an aircraft taking off from an
airport. At any given time, its position with reference to the airport can be
described by asking the questions:
How far north of the airport is the plane?
How far east?
1a.2.4
Acceleration
Go to this link: http://en.wikipedia.org/wiki/Acceleration
In one variable calculus, we defined the acceleration of a particle as the second
derivative of the position function. Nothing changes for vector calculus.
1a.2.5
Definition of Acceleration
Let r(t) be a twice differentiable vector valued function representing the position
vector of a particle at time t. Then the acceleration vector is the second
derivative of the position vector.
In 2-space, r(t) = x(t)i + y(t)j and acceleration = a(t) = r''(t) = x''(t)
In 3- Space r(t) = x(t)i + y(t)j + z(t)k
and acceleration =a(t) = r''(t) = x''(t)i + y''(t)j + z''(t)k
Example:
Find the velocity and acceleration of the position function
r(t) = 4t i + t2 j
when t = -1. Then sketch the vectors.
Solution
The velocity vector is
v(t) = r'(t) = 4 i + 2t j
Plugging in -1 for t gives
v(-1) = 4 i - 2j
Take another derivative to find the acceleration.
a(t) = v'(t) = 2j
Below is a picture of the vectors.
DO THIS:
Sketch the path of motion of an object whose position vector is
r(t) = (t 2 4)i j
DO THIS:
An object starting from rest at P(1, 2, 0) has acceleration
a(t) = j + 2k
where a(t ) is measured in m s 2 Find the location of the object after 2seconds
Do not look over the page until you are finished!
Answer
We have initial conditions as v(0) = 0 and r(0) = x(0)i + y(0)j + z(0 k
i.e. r(0) = 1i + 2j + 0k = i + 2j
v(t ) a(t )dt ( j 2k )dt tj 2tk CwhereC C1i C 2 j C3 k
When t = 0, v(0) C1i C 2 j j C3 k 0 C1 C 2 C3 0
Thus the velocity at any time t is
v(t) = tj + 2tk
Now r (t ) v(t )dt ( j 2k )dt
1 2 2
t t k CwhereC C 4 i C j C 6 k
2
Also r (0) C4 i C5 j C6 k i 2 j C4 1, C5 2, C6 0
1
Thus r (t ) i t 2 2 j t 2 k
2
Compulsory Reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Motion with constant velocity.p23
Motion with constant acceleration p24
Equations of motion in a straight line p26
1a.2.6
MOTION OF FALLING BODIES
Aristotelian Physics tell us that the speed of falling of a
body depends completely on its weight, so a one
Kilogramme stone will fall faster than a half-kilogramme
stone. Galileo denied this, arguing that everybody falls
down with the same speed and acceleration, for
example, if you have got a stone in your hand, and you
suddenly stop holding it, it will fall to the ground with
speed V. And now, you have a paper, and you throw it to
the ground, its falling speed will be now v (a smaller
speed), but if you make a little "ball" with the paper, its
speed will be V. (the same than the stone). From here he
postulated that the speed does not depend on the
weight, every time the acceleration is the same, but in the case of the plain paper
the air has more resistance and that's the cause of the smaller speed. This
experiment was done by Galileo in Pisa's Tower.
1a.2.7
EXPERIMENT
Purpose
In this experiment you'll be able to see the acceleration of different objects and
compare them such as Galileo (the precursor of Einstein) did.
Materials
A tennis ball
A football
A notebook
A sheet
Procedure
1.
2.
3.
4.
5.
6.
7.
8.
9.
Take both balls.
Hold them at the same level, as high as you can (shoulder, face, etc.)
Drop them to the ground at the same time
Both reach the soil at the same time.
Do you think this only happen because they have got he same shape?.
Try with the notebook and a ball then.
Both reach the soil at the same time!
Now, try with the notebook and the sheet of paper. What happens? Why
do you think this happens?
Now make a little "ball" with the sheet of paper and repeat the experiment.
They should reach the soil at the same time because the air resistance
has been reduced.
Compulsory Reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Free fall under gravity p 26-28
Examples p28-31
DO THIS
Exercise
A child leans out of a window of a building from a height 10m above the ground.
She throws a ball vertically upwards with an initial velocity of 12m/s. What is the
maximum height above the ground reached by the ball and the total amount of
time it takes to strike the ground.
Answer (hide)
Maximum height = 17.4m and total time elapsed = 3.11s
1a.2.8
PROJECTILE MOTION
N.B. This should be read together with the section on Newton’s laws in
Activity 2.
As mentioned earlier it will be assumed that the only force acting on the projectile
after its launching is the force of gravity. Thus the motion occurs in a vertical
plane. For a projectile of mass m, the foce due to gravity is
F = - mgj
Comparing this with
F = ma (from Newton’s second law of motion) we have
a = -gj which becomes our acceleration vector.
v0
r0
Now, as shown in the diagram above, if the projectile is launched with initial
velocity v0 from a position r0 then
v(t) = a(t )dt gjdt gtj C1
r(t) = v(t )dt ( gtj C1 )dt
1 2
gt j C1t C 2
2
Now, v(0) v0 ands(0) s0 and this gives rise to
C1 v0 andC2 r0
And therefore
r(t)=
1 2
gt j tv0 r0 which gives our position vector.
2
Recall that
v0 xi yj = ( v0 cos )i ( v0 sin ) j = v0 cos i v0 sin j
Substituting into the above, we have
r(t) =
1 2
gt j tv0 cos i tv0 sin j hj h being the initial height above the ground
2
Rearranging , we have The position Function of a Projectile as
1
r(t) = (v0 cos )ti h (v0 sin )t gt 2 j
2
DO THIS: A catapult throws a stone from 3m above the ground at an
angle of 45 o from the horizontal at 100m per second. Find the stone’s maximum
height. Will the stone go over a 10m wall located 300m from the point of
projection
Do not look over the page until you are finished!
Answer
We are given h=3, v0 100, and 45 0
4
. Using g = 9.8 m s 2
r(t) = (100 cos )ri 3 (100 sin )t 4.9t 2 j
4
4
= (50 2t )i (3 50 2t 4.9t 2 ) j .
The maximum height occurs when the vertical component of v is 0.
That is:
y ' (t ) 50 2 9.8t 0 which means that t=
25 2
seconds.
4 .9
25 2
25 2
Maximum height is y 3 50 2 (
) 4.9
4.9
4
.
9
2
DO THIS:
Simplify the above equation and find the actual value of y
For the wall, x(t ) 300 50 2t
Which means t 3 2 and y 3 50 2 (3 2) 4.9(3 2 ) 2 3 300 - 88.2 214.8
This means that the stone clears the wall.
Software Activity
Work through the activity in the compulsory readings entitled: Software Activity
Projectiles. Notice that this gives a treatment of motion using a traditional
approach without vector notation.
Compulsory Reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Projectile motion P41-44
1a.2.9
CIRCULAR MOTION
In general, circular motion is rotation along a circle, a circular path or a circular
orbit. The rotation around a fixed axis of a three-dimensional body involves
circular motion of its parts. We can talk about circular motion of an object if we
ignore its size, so that we have the motion of a point mass in a plane.
Examples of circular motion are: an artificial satellite orbiting the Earth in
geosynchronous orbit, a stone which is tied to a rope and is being swung in
circles (cf. hammer throw), a racing car turning through a curve in a racetrack, an
electron moving perpendicular to a uniform magnetic field, a gear turning inside
the gearbox of a car.
A special kind of circular motion is when an object rotates around its own center
of mass. This can be called spinning motion, or rotational motion and will be
discussed in later module. I am sure you can come up with your own examples of
motion in a circle.
Circular motion involves acceleration of the moving object by a centripetal force
which pulls the moving object towards the center of the circular orbit. Without this
acceleration, the object would move inertially in a straight line the tangent the
circle, according to Newton's first law of motion. Circular motion is accelerated
even though the speed is constant, because the object's velocity vector is
constantly changing direction. You should deduce from this that the acceleration
vector and the velocity vector are orthogonal
Compulsory readings
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Introduction P. 136
Uniform Circular Motion P. 136 – 138
N.B. Equations 7.1 – 7.11
Centripetal Acceleration P. 138 – 141
N.B. Equations 7.12 – 7.17
Definition : Centripetal Acceleration (7.15) p139
The Conical Pendulum p 141 – 142
N.B. Equations 7.18 – 7.25
Non uniform circular motion P. 143 – 147
Explanations: Radial unit vector P. 143
Tangential unit vector P. 143
Radial velocity and tangential velocity P. 144
Radial acceleration and tangential acceleration P. 144
N.B. Equations to note(7.26) – (7.45)
The Vertical Pendulum P. 148 – 150
N.B. Equations to note 7.46 – 7.53
Animated picture)
N.B. Equations 7.54 – 7.64
You need to note the different approach that is not based on vector algebra
and the traditional equations that are employed in your compulsory reading
text
Example:
Find the velocity vector, speed, acceleration vector of the circle
r (t ) 2 sin
t
t
i 2 cos j
2
2
And sketch the circle.
Solution.
The velocity vector is
t
t
v(t ) r / (t ) 2 cos i 2 sin j
2
2
The speed at any time is:
r / (t ) cos 2
t
t
sin 2 1
2
2
The acceleration vector is
1
t
1
t
a (t ) r // (t ) sin i cos j
2
2
2
2
N.B. The parametric equations for the curve are
t
and
2
t
y 2 cos
2
x 2 sin
DO THIS:
Exercise
Verify that the rectangular equation of the circle is
x2 y2 4
Exercise
DO THIS:
A particle starts from rest at point P(1,2,0) with acceleration
a (t ) j 2k
in the usual units. Find the position of the particle after 2 seconds.
Do not look over the page until you are finished!
Answer
You should be able to deduce that
v ( 0) 0
and
r (0) x(0)i y (0) j z (0)k
1i 2 j 0k
i2j
To find the position function, you have to integrate twice, each time using one of
the initial conditions to find the constants of integration. Thus
v (t ) a(t )dt ( j 2k )dt tj 2tk C
Where
C C1i C2 j C3k
When t 0 and v (0) 0 you get
v (0) C1i C2 j C3k o C1 C 2 C3 0
Thus the velocity at any time t is
v (t ) t ( j ) 2t (k )
When we integrate once more we produce
t2
r (t ) v (t )dt (tj 2tk )dt j t 2k C
2
Where C C4 i C5 j C6k
When t 0 and r (0) i 2 j we have
r (0) C4 i C5 j C6k i 2 j C4 1,C5 2,C6 0
Thus the position vector is
t 2
r (t ) i 2 j t 2k
2
The position of the particle after 2 seconds is
r (2) i 4 j 4 k given by the coordinates (1,4,4)
Compulsory Reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP p 136-160
Note that in our compulsory reading we introduce notions of angular velocity,
centripetal acceleration and so on. This section should be read in conjunction
with the sections on curvilinear motion below and you should be able to draw
parallels between circular motion and curvilinear motion which is an extension of
secular motion.
1a.2.10
Relative Velocity
We have been able to describe the position of a body and its velocity with
reference to the origin of a given co-ordinate system. Ordinarily, this origin is
taken to be fixed in some other body may be in motion relative to a third and so
on. For example, when we speak of the velocity of a car, we usually mean the
velocity of the car relative to the earth… But the earth is in motion relative to the
sun… the sun is in motion relative to some other star…. And so on…..
Suppose a long train is moving towards the right along a straight level track and
an athlete is running on the train towards the right.
Diagram of a flat car train with man running on the train. In the diagram,
uTE represents the velocity of the train T relative to the earth E
u AT represents the velocity of the athlete A relative to the trainT
The velocity of the athlete relative to the earth, u AE ,
is evidently equal to the sum of u AT and uTE :
u AE u AT uTE
N.B. The velocity u AE is the algebraic sum of u AT and uTE and when combining
relative velocities
Write each velocity with a double subscript in the proper order meaning
“velocity of (first subscript) relative to (second subscript)
When velocities are added, the first letter of any subscript is to be the
same as the last letter of the second subscript
The first letter of the subscript of the first velocity in the sum, and the
second letter of the subscript of the last velocity, are the subscripts, in that
order, of the relative velocity represented by the sum.
These three lengthy and cumbersome statements are very important to consider
when teaching vector addition in general where the subscripts actually represent
the vectors themselves.
Compulsory Reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP pp 44-48 (Relative velocity)
Worked examples
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP pp 48
1a.2.11 Curvilinear Motion
1a.2.12
Tangent Vectors and Normal Vectors
We have in the last section shown that the velocity vector always points\in the
direction of motion. In this section we use this observation to extend this to a
concept which applies to any smooth curve which is not necessarily described in
terms of time
1a.2.13
The Unit Tangent Vector
The derivative of a vector valued function gives a new vector valued function that
is a tangent to the defined curve. The analogue to the slope of the tangent line is
the direction of the tangent line. Since a vector contains a magnitude and a
direction, the velocity vector contains more information than we need. We can
strip a vector of its magnitude by dividing by its magnitude.
1a.2.14
Definition of the Unit Tangent Vector
Let r(t) be a differentiable vector valued function and v(t) = r'(t) be the velocity
vector. Then we define the unit tangent vector as the unit vector in the direction
of the velocity vector.
v(t)
T(t) =
, v(t)
0
||v(t)||
Example
Let
r(t) = t i + et j - 3t2 k
Find the T(t) and T(0).
Solution
We have
v(t) = r'(t) = i + et j - 6t k
and
To find the unit tangent vector, we just divide
To find T(0) plug in 0 to get
DO THIS
Show that the unit tangent vector to the curve given by
r (t ) ti t 2
j when t 1 is given by
1
(i 2 j )
5
N.B. In this exercise the direction of the unit tangent vector is determined by the
orientation of the curve. Sketch the curve and verify that the unit tangent vector
to the curve
r (t ) (t 2)i (t 2)2 j at point (1,1)
would still be the same but will point in the opposite direction.
DO.THIS
FindT (t ) to the curve given by
r (t ) 2 cos ti 2 sintj tk at the pointwhen t
and
4
show that the parametric equations for the tangent line are:
x x1 as 2 2s
y y1 bs 2 2s
z z1 cs
4
s
using the point (x1, y1, z1) ( 2, 2, )
4
1a.2.15
The Principal Unit Normal Vector
A normal vector is a perpendicular vector. Given a vector v in space, there are
infinitely many perpendicular vectors. Our goal is to select a special vector that
is normal to the unit tangent vector. Geometrically, for a non straight curve, this
vector is the unique vector that points into the curve or, in other words, the one
that points towards the concave side of the curve. Algebraically we can compute
the vector using the following definition.
Definition: Let r(t) be a differentiable vector valued function and let T(t) be the
unit tangent vector. Then the principal unit normal vector N(t) is defined by
T'(t)
N(t) =
||T'(t)||
Comparing this with the formula for the unit tangent vector, if we think of the unit
tangent vector as a vector valued function, then the principal unit normal vector is
the unit tangent vector of the unit tangent vector function. You will find that
finding the principal unit normal vector is almost always cumbersome. The
quotient rule of differentiation usually appears to complicate this process!.
Example
Find the unit normal vector for the vector valued function
r(t) = ti + t2 j
and sketch the curve, the unit tangent and the principal unit normal vectors when
t = 1.
Solution
First we find the unit tangent vector
Now use the quotient rule to find T'(t)
Since the unit vector in the direction of a given vector will be the same after
multiplying the vector by a positive scalar, we can simplify by multiplying by the
factor:
The first factor gets rid of the denominator and the second factor gets rid of the
fractional power. We have
Now we divide by the magnitude (after first dividing by 2) to get
Now plug in 1 for both the unit tangent vector to get
The picture below shows the graph
and the two vectors.
DO THIS (with a colleague)
Given the curve
r (t ) 3ti 2tj
Find N (t ).
What is the value of N (t ) when t 1
Do not turn the page until you have finished the problem!
Solution
r / (t ) 3i 4 j and r / (t ) 9 16t 2
T (t )
r / (t )
1
(3i 4tj )
/
r (t )
9 16t 2
1
T / (t )
9 16t
2
12
3
2 2
16t
(4 j )
3
2 2
(3i 4tj )
(9 16t )
(4ti 3 j )
(9 16t )
9 16t 2
12
2 3
(9 16t )
9 16t 2
T / (t ) 12
N (t )
T / (t )
/
T (t )
1
9 16t 2
(4ti 3 j )
Substituting for t 1
1
N (1) (4 i 3 j )
5
1a.2.16
Tangent and Normal Components of Acceleration
Imagine yourself driving down the hill along a curving road towards and having
your brakes fail. As you are riding you will experience two forces (other than the
force of terror) that will change the velocity. The force of gravity will cause the
car to increase in speed. A second change in velocity will be caused by the car
going around the curve. The first component of acceleration is called the
tangential component of acceleration and the second is called the normal
component of acceleration. As you may guess the tangential component of
acceleration is in the direction of the unit tangent vector and the normal
component of acceleration is in the direction of the principal unit normal
vector. Once we have T and N, it is straightforward to find the two components.
Definition: The tangential component of acceleration is
aT a T
va
v
and the normal component of acceleration is
aN a N
va
v
and
a = aNN + aTT
N.B. The normal component of the acceleration is also called the centripetal
component of the acceleration
Example
Prove that the acceleration vector a(t) is in the plane that contains T(t) and N(t)
Proof
First notice that
v = ||v|| T
and
T' = ||T'|| N
(N.B.We have simplified notation here and used, for example, T for T (t ) and so
on)
Taking the derivative of both sides gives
a = v' = ||v||' T + ||v|| T' = ||v||' T + ||v|| ||T' || N
This tells us that the acceleration vector is in the plane that contains the unit
tangent vector and the unit normal vector.
Example
Find the tangential and normal components of acceleration for the prior example
r(t) = ti + t2
j
Solution
Taking two derivatives, we have
a(t) = r''(t) = 2j
We dot the acceleration vector with the unit tangent and normal vectors to get
DO THIS
Exercise: Show that
If r (t ) 3ti tj t 2k then aT
4t
10 4t 2
and aN
2 10
10 4t 2
Exercise:
The position function for a projectile is given by:
r (t ) (50 2t )i (50 2t 16t 2 ) j
Find the tangent component of acceleration when:
t = 0,1, and
25 2
16
Do not turn the page until you have finished the problem!
Solution
v(t ) 50 2i (50 2 32t ) j
v(t ) 2 502 16(50) 2t 162
Activity 1a.3
GRADIENT, DIVERGENCE AND CURL
This section is fully covered in the analysis one module of this course. Students
can refresh their understanding of this work by referring to the compulsory
readings given in the analysis one module.
1a.3.1DIVERGENCE
In vector calculus, the divergence is an operator that measures a vector field's
tendency to originate from or converge upon a given point. For instance, for a
vector field that denotes the velocity of air expanding as it is heated, the
divergence of the velocity field would have a positive value because the air is
expanding. Conversely, if the air is cooling and contracting, the divergence would
be negative.
A vector field which has zero divergence everywhere is called solenoidal.
Let x, y, z be a system of Cartesian coordinates on a 3-dimensional Euclidean
space, and let i, j, k be the corresponding basis of unit vectors.
The divergence of a continuously differentiable vector field F = F1 i + F2 j + F3 k is
defined to be the scalar-valued function:
Although expressed in terms of coordinates, the result is invariant under
orthogonal transformations, as the physical interpretation suggests.
The common notation for the divergence ∇·F is a convenient mnemonic, and an
abuse of notation, where the dot denotes something just reminiscent of the dot
product: take the components of ∇, apply them to the components of F, and sum
the results.
Physical interpretation
In physical terms, the divergence of a three dimensional vector field is the extent
to which the vector field flow behaves like a source or a sink at a given point.
Indeed, an alternative, but logically equivalent definition, gives the divergence as
the derivative of the net flow of the vector field across the surface of a small
sphere relative to the volume of the sphere. (Note that we are imagining the
vector field to be like the velocity vector field of a fluid (in motion) when we use
the terms flow, sink and so on.) Formally,
where S(r) denotes the sphere of radius r about a point p in R3, and the integral
is a surface integral taken with respect to n, the normal to that sphere.
In light of the physical interpretation, a vector field with constant zero divergence
is called incompressible – in this case, no net flow can occur across any closed
surface.
The intuition that the sum of all sources minus the sum of all sinks should give
the net flow outwards of a region is made precise by the divergence theorem.
Go to this link
http://en.wikipedia.org/wiki/DIVERGENCE
1a.3. 2 CURL
http://en.wikipedia.org/wiki/CURL
1a.3.3 GRADIENT
In vector calculus, the gradient of a scalar field is a vector field which points in
the direction of the greatest rate of increase of the scalar field, and whose
magnitude is the greatest rate of change.
Consider a room in which the temperature is given by a scalar field φ, so at each
point (x,y,z) the temperature is φ(x,y,z) (we will assume that the temperature
does not change in time). Then, at each point in the room, the gradient at that
point will show the direction in which the temperature rises most quickly. The
magnitude of the gradient will determine how fast the temperature rises in that
direction.
Consider a hill whose height above sea level at a point (x,y) is H(x,y). The
gradient of H at a point is a vector pointing in the direction of the steepest slope
or grade at that point. The steepness of the slope at that point is given by the
magnitude of the gradient vector.
The gradient can also be used to measure how a scalar field changes in other
directions, rather than just the direction of greatest change, by taking a dot
product. Consider again the example with the hill and suppose that the steepest
slope on the hill is 40%. If a road goes directly up the hill, then the steepest slope
on the road will also be 40%. If instead, the road goes around the hill at an angle
with the uphill direction (the gradient vector), then it will have a shallower slope.
For example, if the angle between the road and the uphill direction, projected
onto the horizontal plane, is 60°, then the steepest slope along the road will be
20% which is 40% times the cosine of 60°.
This observation can be mathematically stated as follows. If the hill height
function H is differentiable, then the gradient of H dotted with a unit vector gives
the slope of the hill in the direction of the vector. More precisely, when H is
differentiable the dot product of the gradient of H with a given unit vector is equal
to the directional derivative of H in the direction of that unit vector.
Go to the links below
http://en.wikipedia.org/wiki/Gradient
Worked Examples
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP pp154 – 159
Exercise
DO THIS
Find the magnitude of the
acceleration a s from the diagram
below. What will be the
magnitude and direction of a s if
and
r 100 m
v 15.0 m/s
(about 34 mph)?
as
Do not turn the page until you have finished the problem!
Solution
Since the car moving on a properly banked road is equivalent to the bob of a
conical pendulum, the conditions governing the vector a s , are the same as those
governing the equivalent vector a s for the conical pendulum. That is, the
v2
and a sy g .
r
To find the magnitude of the vector a s , you use the Pythagorean theorem:
horizontal and vertical components of a s must be given by a sx
a s (a sx 2 a sy 2 )1/ 2
1/ 2
v 2 2
g 2
r
1/ 2
v4
2 g2
r
For the numbers given, you have the magnitude
1/ 2
(15.0 m / s) 4
as
(9.80 m / s 2 ) 2
2
(100 m)
10.1 m / s 2
or about 3 percent more than the acceleration of gravity. Using the equation
v2
you can solve for to calculate the ideal banking angle. You have
tan
rg
tan 1
v2
rg
tan 1
(15.0 m / s) 2
12.90
2
100 m 9.80 m / s
Thus the direction of a s is about 130 from the vertical. This is a relatively steep
angle of bank under ordinary roadway conditions.
UNIT 1B: NEWTON’S LAWS OF MOTION
Specific learning objectives
By the end of this activity you should be able to:
State Newton’s Laws of motion and apply them.
Define the concepts Work, Energy, Power and Momentum, and use them
to solve related problems.
State the Laws of conservation of energy and momentum
Define the terms impulse and torque.
Describe the motion of falling bodies and projectiles in a uniform force
field.
Describe the motion of bodies in a resting medium.
Define Friction and describe the motion of a body subject to given
constraints.
Describe potential energy as energy due to position and derive potential
energy as mgh
Describe kinetic energy as energy due to motion and derive kinetic energy
as mv2/2
State conservation of energy laws and solve problems where energy is
conserved
Define power as rate of energy transfer
Define couple, torque and calculate work done by a variable force or
torque
Solve problems where energy is lost due to friction
Summary of the learning activity
In this activity you will acquaint yourself with Newton’s three laws of motion
and their application in the related fields of energy and momentum
Compulsory Reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Relevant Links and Resources
Friction
http://en.wikipedia.org/wiki/Friction
Newton’s Laws of Motion
http://en.wikipedia.org/wiki/Newton's_laws_of_motion
Newton’s Laws of Motion
http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html
http://www.waukesha.k12.wi.us/South/physics1/1.4/notes.html
Work
http://en.wikipedia.org/wiki/Work
Work
http://hyperphysics.phy-astr.gsu.edu/hbase/work.html
Kinetic Energy
http://en.wikipedia.org/wiki/Kinetic_energy
Kinetic Energy
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html
Power
http://en.wikipedia.org/wiki/Power
Potential Energy
http://en.wikipedia.org/wiki/Potential_energy
Potential Energy
http://hyperphysics.phy-astr.gsu.edu/hbase/pegrav.html
List of relevant MULTIMEDIA resources
Hyper physics Link
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Date
This link has introductory content very good diagrams on Newton’s laws
and their applications which will provide you with the required insights for
your progression through the activity.
Simulations and Experiments on Newton’s Laws
www.compadre.org/precollege/static/unit.cfm?sb=3
http://www.glenbrook.k12.il.us/gbssci/phys/Class/newtlaws/newtltoc.html
Key Words with description/definitions
Impulse
Torque
Momentum
Energy
Power
Work
1b Learning Activity: Newton’s Laws of Motion
Introduction
In this activity, you will familiarise yourself with three laws that generally govern
the motion of particles and bodies in space- Newton’s three laws of motion and
be able to apply them to real life situations. Through related questions, exercises
and experiments, you will then be taken through the central theme that runs
through all the concepts in this activity- the fundamental notion of force. The
relationships between force, motion, energy, momentum and other related
notions will be dealt with through your involvement with everyday examples and
their mathematical models.
Bridging The Gap
Bedside Wallet example;
You go to sleep at night and place your wallet on the bed side table. All
things being equal, you should be able to find your wallet on the same
bedside table. If on the other hand, you find the wallet in the bathroom
what would have happened? Obviously the wallet cannot move on its own!
What would happen if, when approaching a bus stop, on a straight road,
the brakes of the bus failed?
o What effect do the brakes of the bus have on its motion?
A driver wakes up in the morning to find his car battery flat – he requires a
push:
o One person gives him a push and the car won’t move.
o Two persons give him a push, the car moves, coughs but won’t
start
o Three persons give him a push and the vehicle starts.
Describe the effect of the three situations on the movement of the car with the
help of a colleague.
Here is an interesting scenario: you and your friend want to collect fruit from
a tree but you are too short and need something to stand on. You get a stool
and succeed in obtaining the fruit. Your friend tries the same thing but is
unsuccessful as the stool breaks.
What could be the possible reasons for your success and your friend’s failure?
The stool does not break in your case but does so in your friend’s.
You need to feed in order to do work. Why?
You place a book on a table. What makes the book to stay on the table and
not fall through the table top?
Two vehicles collide and there is minimal damage. What happens
immediately after the collisions? What would an investigating police officer
ask about the movement of the cars?
If you have not been able to answer or explain these scenarios then go through
the activity and attempt to answer them.
1b.1 Newton’s Laws of Motion
Visit the links below
http://en.wikipedia.org/wiki/Newton's_laws_of_motion
http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html
1b.1.1 Newton's First Law:
Consider the section on bridging the gap above and this time the motion, of a
cart driven by a span of two oxen. If the cart is detached from the oxen that were
pulling it would quickly come to rest because there is no tractive force pulling it
along. However, other forces also act on the cart such as the force of friction,
which helps slow down the cart.
Newton's first law summarizes these ideas and is stated thus.
Every object continues in a state of rest, or of uniform motion in a straight line,
unless it is compelled to change that state by forces acting on it.
The implications of this law are that.
an object at rest remains at rest, and
an object in motion continues to move with constant velocity.
Inertia - the tendency of an object to maintain its current state of motion, whether
at rest or in motion.
Because of inertia, only external forces can alter an object’s state of motion;
internal forces cannot.
Internal force - forces generated by one part of a system of objects on another
part of the same system.
Example. Your left hand pushes on your right hand. The engine of a car
pushes on the transmission which pushes on the wheels.
External force - forces generated by something outside the system of objects.
Example. A tennis racket hits a tennis ball. Your foot kicks a chair.
The key point to remember is that no change of motion can occur unless a force
from outside the object causes it to change.
N.B. Care must be taken when applying this law as it only holds for
measurements made in an absolute frame of reference but are subject to
correction depending on the relative position of the observer.
1b.1.2 Newton’s Second Law of Motion
The acceleration of a particle is proportional to the resultant force acting on it and
is in the direction of the force.
Other interpretations are
When there is a net force acting on an object the object undergoes
acceleration in the same direction as the force.
The acceleration and the force are proportional in magnitude.
The amount of inertia an object possesses is indicated by its mass.
Mass - A measure of an object’s resistance to change in motion.
The larger the mass of an object, the more it resists a change in its
motion. Smaller masses have less resistance to change in motion. This is an
operational definition of mass, it requires direct measurement of an object’s
change in motion to determine its value.
If an object experiences an external force, not equal to zero, it will accelerate
proportional to the magnitude of the force, but inversely proportional to its own
mass.
In equation form: FT = Ma
where Force (F) is in Newtons,
mass (M) is in Kg, and acceleration (a) in m/s2
This means that the acceleration of an object depends on the ratio of the total
force to the mass of the object. The direction of the acceleration is the same as
that of the total force which caused it.
Newton’s second law is the work horse of our mechanical world; it connects the
changes in motion of all objects with the forces needed to make that change. Its
a simple equation to work, but its application can become complicated as you will
see in the next section.
Some examples:
a) If a total force of 10 N is placed on 2 Kg ball, what is its acceleration?
b) If you push with a force of 25 N on a 2 Kg box to slide it across the floor,
what is the acceleration of the box if the frictional force is 10 N?
This law led to the equation;
F ma , F Force , a acceleration , m mass of object
Another way of stating this law is:
The rate of change of momentum of a body is proportional to the resultant force
and occurs in the direction of the force.
Mathematically modelling this law we have, if the body changes velocity from u to
v in t seconds
change of momentum mv mu
mv mu
t
m (v u )
t
rate of change of momentum
Thus F ma F kma
and it follows, by defining the unit of Force as the Newton,
F ma
1b.1.3 Newton's Third Law of motion
The forces of action and reaction between interacting bodies are equal in
magnitude, opposite in direction and are co-linear.
Simply put, forces are social beings, they always operate in pairs. It is not
possible to have an isolated, or single force. Isaac Newton expressed it a bit
more formally:
If an object expresses a force upon another object, then the latter of
the two exerts a force of equal magnitude back upon the first.
We call these action and reaction forces. It doesn’t make any difference which
you call the action and which the reaction since they happen concurrently and
are reversible.
Example. Your body pushes on the chair, the chair pushes on your body.
A tennis racket hits the ball, the ball hits the racket.
Point to remember: Although these two forces are equal, but oppositely
directed, they do not cancel each other out. They operate on different objects so
they cannot be added together.
If you kick a chair, your foot exerts a force on the chair. The chair exerts a force
on your foot.
See, one force is on the chair, the other force is on your foot; two different
objects.
The only force which can cancel any applied force is another force on the same
object. The only force which can cancel your foot’s force on the chair is another
force on the chair, like friction between the chair and the floor.
Remember inertia? How an internal force cannot cause a change in
motion? Here’s how it works in the real world.
Stand up and jump straight up into the air. If you think about what you did you
bent your knees, pushed on the floor, and went up into the air. We say “I
jumped”.
Actually, you cannot do anything to yourself to make you go into the air from rest
on the floor. There has to be a force on your from outside of you to make you go
into the air.
Why do you push on the floor? To get the floor to push on you! That’s the
action/reaction part of the operation. The floor’s force on you is what’s needed to
make you go up. Pretty tricky, huh? That’s why it took you a while to learn to
walk and jump.
More simply put, if body A exerts a force on body B, then body B an equal and
opposite force on body A.
Thus forces never occur singly but in pairs as a result of the interaction between
the bodies.
N.B. You will note that in all the systems in which we will study Newton's Second
Law, we will also be implicitly and explicitly studying and using Newton's Third
Law.
Worked Examples
Two shot put balls, one weighing 0.70 kg and the other 7.0 kg both fall towards
the grant with the same downward acceleration of magnitude 9.8m / s 2 . Find the
force, w, exerted by gravity on each ball.
Fig: Two masses with the force of gravity acting on them
By Newton's Second Law,
For the smaller shot
w mg 0.20 9.8
6.9 Newtons
and for the larger shot-put
W Mg 7.0 9.8
69 Newtons
Example:
A Block slides down a long frictionless plane which is at an angle of 690 with
respect to the horizontal. Find the acceleration of the block and the distance the
block has moved 3 seconds from rest.
Solution
The components of W parallel and perpendicular to the plane are
Wx mg sin
and
Wy mg cos
Since the place is frictionless then N x 0
And N y N (when N x and N y are the reactions of the plane to the block parallel
and perpendicular to the plane).
And therefore, the only force that is working on the block is Wx max
mg sin ma
a g sin
g sin 69
Using S ut 12 at 2
and putting u 0 and S x
Then
x 12 ax (30) 2
12 g sin 69 9
92 g sin 69 metres
DO THIS:
EXERCISE
A block of mass m slides down a plane supported
from the earth at an incline the angle between the
plane and the horizontal is adjusted until the block
slides with constant speed. Find the co-efficient of
friction between the block and the slope. If is
350, what is the value of μ.
Do not turn the page until you have completed the solution!
SOLUTION
Since the block is sliding down the plane, then
FR
, where FR is the Frictional force and N the normal reaction
N
We know that N mg cos
and since
FR N
then
FR mg cos
Since motion is down the slope, when the block begins to move
mg sin mg cos
tan
tan 35
0.70
1b.1.4 APPLICATION OF NEWTONS LAWS
The most widely applied Newton’s Law is the Second Law of motion,
F ma
which is applied in the study of motion of bodies in a variety of systems.
N.B. The quantity F in the equation is the Net force of all the forces acting on the
parts of the system and thus F is the vector sum of all the forces acting on the
part. The question then to answer is:
What are the magnitude and directions of all the forces?
1b.2 Motion of Connected bodies
The contraption consists of two bodies of mass m1 and m2
attached to the ends of weightless and inextensible string
which runs over a smooth, weightless pulley. If m1 and m2
are not equal the system will begin to move as soon as it is
released.
If T1 and T2 are the forces exerted by the pulley on the weights and, m1 g and
m2 g are the gravitational forces exerted on the two then, if m1 m2
m1 g T1 m1a1
But surely
T1 T2 and
and
T2 m2 g m2 a2
a1 a2
then
T1 m2 g m2 a1
and
m1 g T1 m1a1
Adding
(m1 m2 ) g (m1 m2 )a1
Thus
a1
m1 m2
g
m1 m2
and this gives us the equation of motion for the system.
DO THIS:
EXERCISE
In the diagram, the string and the pulley have negligible
mass and the friction in the pulley system is negligible. If the
two bodies are initially at rest at the same level, how long
will it be before the vertical separation between them is
1,5m? How fast will they be moving?
Do not turn the page until you have completed the solution!
[Answer: t = 2.5s and v = 0.60s]
1b.3 WORK
Visit the links below:
http://en.wikipedia.org/wiki/Work
http://hyperphysics.phy-astr.gsu.edu/hbase/work.html
Definition
The work done, dU , by a force F during a small displacement ds of its point of
application is given by
dU F ds
which is the dot product between F and ds.
The magnitude of the work is given by
dU F ds cos
From the equation
dU F ds
Then
U F ds
F dx F dy F dz
x
y
z
Ft ds
Example:
The most common example of work done on a body is the action of a body on a
spring to which it is attached.
If the spring constant is K then
F kx
The force exerted on the spring whether it is tension or compression is opposite
to the displacement and so does negative work on the body
Thus
x2
x2
x1
x1
U Fdx kxdx 12 k ( x22 x12 )
Next we consider the work done on a particle of mass m moving along a curved
path under the action of the Force, F. The resultant force is F .
the position of m is described by the position vector r.
the displacement during time dt is given by the change dr in its position
vector.
now
dU F dv
and
x2
U F dv Ft dx
x1
Given that F ma
then
U F dv ma dv
but
a dv at ds
and
at ds v dv
thus
U F dv mv dv
12 m(v22 v12 )
which is the integration carried out between the change v1 to v2 in the velocity.
1b.4 KINETIC ENERGY
Visit the links below:
http://en.wikipedia.org/wiki/Kinetic_energy
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html
The Kinetic Energy of a particle is defined as
K .E. T 12 mv 2
and is the work done on a particle to bring it to a velocity V from rest and is
always a positive quantity.
The unit is the Joule.
thus the equation
U 12 m(v22 v12 )
may be written as
U J
N.B. This is called the Work- Energy equation of a particle and states that
the total work done by all forces acting on a particle during an interval of
its motion i.e. equal to the change in the Kinetic energy of the particle.
Another way of putting it is that the Final K.E., T 2, is the sum of the Initial K.E., T 1
and the work done U.
T2 U T1
The application of the work-energy equations requires an isolation of the particle
or system under consideration.
1b.5 POWER:
Visit the link below:
http://en.wikipedia.org/wiki/Power
Definition: Power is the capacity of a machine to do work and is measured by
the time rate in which it can do the work or derive energy.
P
dU
dx
F F v
dt
dt
and is measured in watts.
Worked Example
In the diagram m 50kg , 0.30 , 15 , u1 4m / s .
Calculate the velocity of the crate at B.
N.B. Can you explain each step of the solution?
U Fs 50(9.81)sin15 14210
152J
Change in K .E. T2 T1 T .
T 12 mv 2 , T 12 50(v 2 42 ) , U T
152 J 12 50(v 2 42 )
v 3.15m / s 2
DO THIS
Exercise
1. A truck, carrying a concrete block of 80kg, starts from rest and attains a
speed of 72km/h after travelling for 75m on a road with constant
acceleration. Find the work done by the Friction working on the block
during this motion if the static and kinetic coefficients of friction between
the block and truck bed are:
(a)
(b)
0.30 and 0.28 respectively, and
0.25 and 0.20 respectively
2. The position vector r, of a particle is given by r 8ti 1.2t 2 j 0.5(t 3 1)k , t
is time in seconds from the start of motion. Determine the power P
developed by the force F 10i 5 j 9k Newtons which acts on the particle
when t 6s .
Do not turn the page until you have completed the solution!
Answers
(a)
(b)
16.0 KJ
8.66 KJ
1b.6 POTENTIAL ENERGY
Visit the links below:
http://en.wikipedia.org/wiki/Potential_energy
http://hyperphysics.phy-astr.gsu.edu/hbase/pegrav.html
Definition: The Potential Energy of a system is the energy the system has
because of the relative positions of its points and, that is, due to its configuration.
It arises when a body experiences a force in a field such as the earth’s
gravitational field and the potential energy is regarded as the joint property of the
body-earth system and not of either body separately.
1b.7 CONSERVATION OF ENERGY
The full definition of conservation of energy is given in the link below:
http://en.wikipedia.org/wiki/Conservation_of_energy
1b.8 IMPULSE and TORQUE
Compulsory Reading:
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP pp 271-272
For information on Torque also see the link below:
http://en.wikipedia.org/wiki/Angular_momentum
1b.9 FRICTION
Definition: Friction is the force that opposes the relative motion or tendency
toward such motion of two surfaces in contact. It is not a fundamental force, as it
is made up of electromagnetic forces between atoms. When contacting surfaces
move relative to each other, the friction between the two objects converts kinetic
energy into thermal energy, or heat. Friction between solid objects and fluids
(gases or liquids) is called drag.
For further reading please refer to the following link:
http://en.wikipedia.org/wiki/Friction
1b.10 MOTION OF FALLING BODIES
Aristotelian Physics tell us that the speed of falling of a body
depends completely on it weight, so a one pound stone will
fall faster than a ½ pound stone. Galileo denied this, arguing
that everybody fells down with the same speed and
acceleration, for example, if you have got a stone in your
hand, and you suddenly stop holding it, it will fall to the
ground with speed V. And now, you have a paper, and you
throw it to the ground, it falling speed will be now v (a smaller
speed), but if you make a little "ball" with the paper, it speed
will be V. (the same than the stone), from here he postulate
that the speed does not depend on the weight, every time the
acceleration is the same, but in the case of the plain paper
the air has more resistance and that's the cause. This
experiment was done by Galileo at the leaning tower in Pisa, Italy.
Compulsory Reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP p 26-28 (Free fall under gravity)
Examples
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP p28-31
DO THIS
Exercise
A child leans out of a window of a building from a height 10m above the ground.
She throws a ball vertically upwards with an initial velocity of 12m/s. What is the
maximum height above the ground reached by the ball and the total amount of
time it takes to strike the ground.
Do not turn the page until you have completed the solution!
Answer
Maximum height = 17.4m and total time elapsed = 3.11s
1b.11 PROJECTILE MOTION
N.B. This should be read together with the section on Newton’s laws in Unit 1b
As mentioned earlier it will be assumed that the only force acting on the projectile
after its launching is the force of gravity. Thus the motion occurs in a vertical
plane. For a projectile of mass m, the foce due to gravity is
F = -mg.
Comparing this with
F = ma (from Newton’s second law of motion) we have
a = -gj, which becomes our acceleration vector.
( Chris A diagram of the parabolic motion of a projectile from an initial height r0
and velocity v0 showing the vertical and horizontal components of v0 in terms of x
and y.)
Now, as shown in the diagram above, if the projectile is launched with initial
velocity v0 from a position r0 then
v(t) =
a(t )dt gjdt gtj C
r(t) = v(t )dt ( gtj C1 )dt
1
1 2
gt j C1t C 2
2
Now, v (0) v0 and s (0) s0 and this gives rise to
C1 v0 andC2 r0
And therefore
r(t)=
1 2
gt j tv0 r0 which gives our position vector.
2
Recall that
v0 xi yj = ( v0 cos )i ( v0 sin ) j = v0 cos i v0 sin j
Substituting into the above, we have
r(t) =
1 2
gt j tv0 cos i tv0 sin j hj h being the initial height
2
above the
ground
Rearranging , we have The position Function of a Projectile as
1 2
r(t) = (v0 cos )ti h (v0 sin )t gt j
2
DO THIS:
A catapult throws a stone from 3m above the ground at an angle of 45 o from the
horizontal at 100m per second. Find the stone’s maximum height. Will the stone
go over a 10m wall located 300m from the point of projection
Do not turn the page until you have completed the solution!
Answer
0
We are given h=3, v0 100, and 45
. Using g = 9.8 m s 2
4
2
r(t) = (100 cos )ri 3 (100 sin )t 4.9t j
4
4
= (50 2t )i (3 50 2t 4.9t 2 ) j . The maximum height occurs when the vertical
component of v is 0. That is
y ' (t ) 50 2 9.8t 0 which means that t=
25 2
seconds.
4 .9
25 2
25 2
Maximum height is y 3 50 2 (
) 4.9
4.9
4
.
9
For the wall,
2
x(t ) 300 50 2t
Which means
t 3 2
and y 3 50 2 (3 2) 4.9(3 2 ) 2 3 300 - 88.2 214.8
This means that the stone clears the wall.
Software Activity
Work through the activity in the compulsory readings entitled: Software Activity
Projectiles. Notice that this gives a treatment of motion using a traditional
approach without vector notation.
Compulsory Reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP pp41-44 (Projectile motion)
Note that in our reading we use the traditional rather than the vector approach in
the treatment of the topic and that we rely very heavily on adaptations of
equations of uniform motion in a straight line to come up with projectile motion
equations. Of particular importance are the derivations of equations such as the
range, the maximum range, the greatest height attained and so-on…
It is advisable that you DO NOT memorize these equations and that you should
be able to work from first principles through given situations
UNIT 2: OSCILLATIONS
Specific learning objectives
By the end of the unit you should be able to:
o Describe Simple Harmonic Motion
o Model motion that is Over-damped, Under-damped and Critically-damped
o Describe the Simple pendulum
o Derive the two and three dimensional oscillator
Summary
In this unit we look at the motion of a body when the resultant force acting on it is
not constant, but varies during the motion. Naturally, there is an infinite number
of ways in which a force may vary; hence no general expressions can be given
for the motion of a body when acted on by a variable force, except that the
acceleration at each instant must equal the force at that instant divided by the
mass of the body. We will look at motion which is oscillatory or periodical using
examples like simple pendulums and masses suspended on a string to illustrate
this motion.
Required Reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Relevant Links and Resources
Simple harmonic motion
http://en.wikipedia.org/wiki/Simple_harmonic_motion
Simple harmonic motion
www.phy.ntnu.edu.tw/java/shm/shm.html
Simple harmonic motion
http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html
Simple harmonic motion
www.kettering.edu/~drussell/Demos/SHO/mass.html
Simple harmonic motion
http://theory.unwinnipeg.ca/physics/shm/nodes2.html
Classical Mechanics Lecture notes
http://farside.ph.utexas.edu/teaching/301/lectures/lectures.html
Classical Mechanics: An Introductory Course
http://www.lulu.com/content/159798
Key Words with description/definitions
Harmonic Oscillator
Simple Pendulum
Natural Circular frequency
Angular Harmonic Motion
Damped Motion
Hooke’s Law
2 Learning Activity: Oscillations
2.1 Bridging the Gap
Two children are playing on a swing- one in the sit and
the other ‘pushing’ And waiting for her turn on the seat.
Have you ever wondered why
After the hardest push, the swing always continues
with its usual motion of coming back to its resting
place and through it until it reaches its highest
backward position and returns. Why does the swing
not coil itself on the horizontal support or, worse still ,
go off its supports ?
If the ‘pusher’ stops and the child on the seat stops
‘kicking’ what happens to the motion of the swing ?
Fig 2.1:
Parents with smiling child on swing
Take a spiral spring, fix one end of the spring, pull the free end slightly and then
release it.
Describe the action of the spring.
Introduction
Consider the motion of the child on the swing above. Before she is pushed or
"kicks" herself in the swing, the swing hangs vertically downwards. In this
position, we say that the swing is in its equilibrium position. When the swing is
pushed, then it is displaced from its equilibrium position but the tendency has
always been to return back to its equilibrium position. What causes this
tendency?
There surely must be some force that causes it to go back-and forth motion past
the equilibrium position and this force has been aptly named the restoring force.
DO THIS
Try it yourself!
Tie a stone on one end of the string
Suspend the stone from the open door frame by fixing one end of the stone to
the top of the door frame
In the equilibrium position the stone hangs straight down
Give it a little displacement from this position by pushing it, say, to the left
What happens to the stone?
The stone does not simply return to the equilibrium position but instead swings
back and forth, past the equilibrium position in a regular repetitive manner.
Can you give other examples of this motion which is defined as periodic or
oscillatory motion?
2.1
THE SIMPLE HARMONIC OSCILLATOR
Fig 2.2:
Simple harmonic Oscillator
Consider the movement of a mass m attached to the end of a spring, the other
end which is held stationary at A. We will describe the position of the body with
the co-ordinate x, where x denotes the displacement from the equilibrium position
O, the position of zero spring deflection. We will restrict our discussion to a linear
spring. Such a spring exerts a restoring force -kx, on the mass which means that
when the mass is displaced to the right, the spring force is to the left and vice
versa.
k is known as the spring constant or modulus or stiffness of the spring
Thus:
F kx
Note here that the magnitude of the restoring force is directly proportional to the
displacement of the system from equilibrium.
That is F x .
From Newton's 2nd Law
F ma
mx
and this implies that
kx mx or
mx kx 0
This differential equation is known as the Simple Harmonic Equation
Read this link:
http://en.wikipedia.org/wiki/Simple_harmonic_motion
DO THIS
EXERCISE
Show that the solution of the equation is
x a cos(t ) where , , a are constants and
k
m
Hint: We are solving a Second Order Differential Equation which is
Homogeneous.
Alternatively, and with the same conditions,
mx kx 0 is normally written as x 2 x 0
is called the natural circular frequency and has units in radians per second
Note that the displacement x oscillates between
x a and x a, where a is the amplitude of the oscillation
the number of complete cycles per minute, the natural frequency f
and the time required for one complete motion’s cycle is given by
n
2
1 2
f
is the period of the motion
Compulsory Reading :
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
pp 237-238
DO THIS
EXERCISE
Fig 2.3:
Model for Periodic motion
A spring is mounted as shown in the diagram above. Through the attachment of
a spring balance to the free end, it is determined that the force is proportional to
the displacement, a force of 4 N causing a displacement of 0.02 m. A body of 2
kg mass is then attached to the free end, pulled for a distance of 0.04 m and
released.
a)
b)
c)
d)
e)
f)
Find the force constant of the spring
Find the period and frequency of vibration
Compute the maximum velocity attained by the oscillating body
Compute the maximum acceleration
Compute the velocity and acceleration when the body has moved halfway
in towards the centre from its initial position
How much time is required for the body to move halfway in to the centre
from its initial position
Do not turn the page until you have completed the solution!
Solution
Before we answer this question, we need to establish the basic equations of
simple harmonic motion.
We know from above that
k
xa x
m
That is, the acceleration at each instant is proportional to the negative of the
instant.
When x has its maximum positive value A, the acceleration has its maximum
A
negative value k
and at the instant the particle passes the equilibrium
m
position x = 0, the acceleration is zero. However, its velocity at this position is
not zero at this point.
Looking back, we can use the Conservation of Energy Principle to analyze some
aspects of simple Harmonic Motion. In a spring, the restoring force is a
conservative force (Remember this from previous activities) and the work done
can be stated in terms of potential energy.
1
P.E. of Conservative Force kx 2
2
1
The Kinetic Energy at any point x is again mv 2 and by the Principle of
2
Conservation of Energy, the total Energy.
1
1
E mv 2 kx 2 is a constant
2
2
Also, when the particle reaches its maximum displacement A it stops and turns
back to its equilibrium position. At the instant of timing, V = 0 and there is no
Kinetic Energy and total Energy
1
E kA2
2
thus we have
1 2 1 2 1 2
mv kx kA
2
2
2
k
v
A2 x 2
m
DO THIS EXERCISE
When does the speed have its maximum value vmax (Discuss with a friend)
Do not turn the page until you have completed the solution!
Answer
The speed has its maximum value at the mid-point because at the mid-point the
energy is all Kinetic and at the end point the energy is all Potential.
Thus:
1
mvmax 2 E
2
2E
vmax
m
1 2 1
kA mvmax 2 E
2
2
k
vmax
A
m
Looking at the equation
v
k
A2 x 2 A2 x 2
m
We cannot tell where the particle is at any given time and to be able to have a
complete description of motion we need to know the position, velocity, and
acceleration at any given time.
We can enliken the simple harmonic motion of the particle at the end of the
spring to a circle of reference given in the Diagram below.
Fig 2.4:
Uniform circular motion
Suppose point Q moves anticlockwise around a circle of radius A with constant
angular velocity (remember this?) W. The vector OQ represents the position of
point Q relative to O and θ is the angle this vector makes with the positive x- axis.
The horizontal component represents the actual movement depicted by the
spring and particle. You will realize that as Q moves, P represents the actual
movement of the particle and spring.
The displacement OP at any time t is x and
x A cos
and if Q is at the extreme right of the diameter at time t = 0, then
t
and hence
x A cos t
Thus, the velocity at any time t
v x A sin t Asin t
And acceleration
a v x 2 A cos t
You will notice that what we have done here is to assume that when t = 0, x = A,
is positive maximum displacement.
At a different initial position, say, if t = 0 OQ makes and angle θ0 while the
positive
x-axis then θ at time t is given by
0 t
And the equations become
x A cos(0 t )
v A sin(0 t )
a 2 A cos(0 t ) 2 x
By using these results you should be able to verify that the answers to the
question above are
(a)
k = 200 Nm-1
(b)
τ = 0.628 s, f = 1.59 Hz and ω = 10 s-1
(c)
vmax = ± 0.4 ms-1
(d)
amax = 4.0 ms-2
(e)
v = -0.346 ms-1, a = -2.0 ms-2
(f)
t = π/30 s
2.2
The Torsion Pendulum (Angular Harmonic Motion)
Fig 2.5: A torsion pendulum
Compulsory Reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
p241-242
The system shown in the compulsory reading is quite self-explanatory. In other
instances Angular Harmonic motion results when a body that is pivoted about an
axis experiences a restoring torque proportional to the angular displacement from
its equilibrium position. The balance wheel from a kinematic watch (obviously
not a battery operated watch) is a very good example of angular harmonic
motion.
Note that in this case the restoring force becomes the restoring torque
The Restoring torque is proportional to angular displacement and is given by
k
where k’ is the torque constant
The moment of inertia I of the pivoted body corresponds to the mass of a body in
linear motion. Thus the period of angular Harmonic motion is
I
2
k
and angular frequency is
k
I
with the period of oscillation being
2
I
2
k
2.3
The Simple Pendulum
Compulsory Reading :
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
p242-244.
Fig 2.6: Simple Pendulum
The simple pendulum is a simplified model of a more complex system. It
consists of a mass suspended by a weightless rod or a weightless and
inextensible string in a uniform gravitation field. When pulled to one side of its
equilibrium position and released, the rod (mass) vibrates about this position.
Does the motion constitute simple Harmonic motion?
We need to establish whether the restoring force is directly proportional to the
coordinate x and oppositely directed. The path of the rod is not a straight line but
an arc of a circle of radius L, the length of the string or rod. The coordinate x
refers to distances measured along this arc.
Then if F = -kx the motion will be simple harmonic.
The tangential components of the net force acting on the pendulum bob is given
by
F mg sin
This means that the restoring force is proportional to sin θ and not to θ and
therefore the motion is not simple harmonic.
How do we get around this problem?
DO THIS
EXERCISE:
Demonstrate that if θ is small, then sin θ is very nearly equal to θ.
If you have demonstrated this then
L x and
x
L
and therefore
F mg mg
x
mg
x
L
L
Thus the restoring force is proportional to the coordinate for small displacements
mg
and the force constant k is represented by
leading to the following results
L
Period 2
2
m
mL
2
k
mg
L
g
The frequency relations are f
1
2T
g
and
L
g
L
Thus the period of oscillation is entirely determined by the length 2 of the
pendulum and only for small oscillations does the motion of the pendulum
approximate Simple Harmonic Motion.
A more analytical way to describe pendulum motion is to obtain an expression for
aT, the tangential component of the acceleration of the pendulum
Fig 2.7: Motion of Simple Pendulum
If we use a small displacement vector ds in a corresponding small interval of time
dt
then
v
ds
dt
But, from the diagram above
ds ld
Therefore
v
ds ld
d
l
dt
dt
dt
The rate of change of speed is
d
d l
dv
dt
dt
dt
d d
l
dt dt
d 2
l 2
dt
Which gives the tangential acceleration aT of the bob
Thus
aT l
d 2
dt 2
Now, from Newton's second law, the tangential component of the force is
FT = maT
but, from above
Ft F mg sin
Thus
mg sin ml
Thus
d 2
dt 2
d 2
g
sin
2
dt
l
[Compare this with l g sin (equation 11.23) Fitzpatrick, R. (2001). Classical
Mechanics: An Introductory Course. Austin, Texas.UTP
on page 244]
Taking
g
l
then the pendulum equation is
d 2
sin
dt 2
The argument goes on to say that, for small angles, sin is the approximately
the same as if is small compared to 1 radian.
Thus
d 2
dt 2
2.4
THE DAMPED OSCILLATOR
In classical mechanics, a harmonic oscillator is a system which, when
displaced from its equilibrium position, experiences a restoring force F
proportional to the displacement x according to Hooke's law:
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic
oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about
the equilibrium point, with a constant amplitude and a constant frequency (which
does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the
harmonic oscillator is described as a damped oscillator. In such situation, the
frequency of the oscillations is smaller than in the non-damped case, and the
amplitude of the oscillations decreases with time.
If an external time-dependent force is present, the harmonic oscillator is
described as a driven oscillator.
Mechanical examples include pendula (with small angles of displacement),
masses connected to springs, and acoustical systems.
Go to this link:
http://hyperphysics.phy-astr.gsu.edu/hbase/oscda2.html
DO THIS
Set up a simple pendulum as in an earlier activity and set it in motion: Does the
pendulum continue oscillating? If it does eventually stop, why does it stop?
A vehicle hits a hump on the road. It is quiet safe to say that the spring will
perform oscillations which would continue if there was no opposing force to stop
then, where, what provides this force?
In the two cases cited above, the amplitude of the oscillations of the pendulum
and the springs of the car gradually decreases to zero due to the sensitive force
that arises from the act, for the pendulum, and that is provided by the shock-
absorber. The motion is therefore not a perfect Simple Harmonic Motion and is
said to be damped by a resistance, friction or the shock absorber.
The behaviour of any mechanical system depends on the extent of damping.
Under-damped oscillations are said to be free and perform near Simple
Harmonic Motion. The graph of this motion is
Fig 2.8: A lightly –Damped or Under-Damped motion
However, every mechanical system possesses some inherent degree of friction,
which acts as a consumer of mechanical energy.
The viscous damper (or dashpot), used as a basis for shock absorbers in
vehicles and related machines is a device in intentionally added to system for the
purposes of limiting or reducing vibratory oscillations. The basic compounds are
a cylinder filled with a viscous fluid and a piston which allows the fluid to flow
from one side of the piston to the other.
Schematically, the diagram below illustrates this system.
Fig 2.9: Viscous Damper
The dashpot exerts a force F whose magnitude is proportional to the velocity of
the mass. The constant of proportionally cv is known as viscous damping coefficient. The direction of the damping force as applied to the mass is opposite to
that of the velocity x . Hence the force on the mass is cv x .
From Newton's Second Law, the equation of motion of the mass is
kx cv x mx
Alternatively
dx
d2x
kx cv
m 2
dt
dt
c
k
and introduce a constant v . The quantity
m
2mwn
(Zeta) is called the Viscous damping factor damping ratio and is a measure of the
severity of the damping.
If we substitute wn
Example
Fig 2.12: Viscous Damping
An 8 kg body is moved 0.2 m to the right of the equilibrium body, from rest at
time t = 0 s. Determine its displacement at time t = 2 s. The viscous damping
co-efficient c is 20 Nsm-1 and the spring co-efficient K is 32 Nm-1
Solution
We must first determine the degree of damping by computing the damping
ratio
n
k
32
2 radians / s
m
8
c
20
0.625
2mn 16 2
Since is less than 1 then the system is underdamped. The damped natural
frequency is
n 1 2 2 1 (0.625) 2 1.561 rad / s
The motion is given by
x Ee nt sin(k t )
Ee 1.25t sin(1.561t )
x 1.25Ee 1.25t sin(1.561t ) 1.561Ee 1.25t cos(1.561t )
When t =0,
x0 E sin 0.2
x0 1.25E sin 1.561cos 0
Solving for
E and we have
E 0.256 m
and
0.896 rad
Therefore Displacement
x 0.256 me1.25t sin(1.561t 0.896)
When t=2
x = -0.0162
DO THIS: EXERCISE
Verify that
is non-dimensional.
Thus the equation becomes
x 2 wn x wn 2 x 0
Alternatively
d 2x
dx
2 wn
wn 2 x 0
2
dt
dt
Hint: This is a Second Order Homogeneous Differential Equation
Do not turn the page until you have completed the solution!
Solution
This is a differential equation with a possible solution of
x Aet
Substituting we have
2 2 wn wn 2 0
giving
1 wn [ 2 1]
2 wn [ 2 1]
or
By inspection, the general solution is
x Ae1t Be2t
1 t Be 1 t
Ae
2
2
n
The discriminant 2 1
the values it can take.
(i)
n
determines the extent of the damping by considering
If 1 :
Remember from your school days that the roots 1 and 2
are real and distinct negative numbers and the motion depicted below
delays so that x approaches zero at t for large value of t. There is no
oscillation and no period associated with the motion. The motion is
OVERDAMPED (HEAVILY DAMPED).
Fig 2.10: Overdamped and Critically Damped Motion
(ii)
If 1 : The roots are equal real negative numbers ie 1 2 n and
the solution to the equation in this case is given by
x ( A Bt )ent
As depicted in the diagram above, the motion decays, x approaches zero
for t
and the motion is non-periodic. This motion is called
CRITICALLY DAMPED and will approach equilibrium factor than will be
an overdamped system.
(iii)
If 1 : The roots are complex for this UNDERDAMPED (LIGHTLY
DAMPED) and transforms the equation to
x Aei
1 2 n t
Be i
1 2 n t
e
n t
where
Setting a new variable k to represent n 1 2
We have
i 1
x Aeik t Be ik t e nt
Employing Euler’s formula
e ix cos x i sin x
We have
x A(cos k t i sin k t ) B(cos k t i sin k t ) ent
( A B)cos k t i( A B)sin k t ent
C cos k t D sin k t e k t
E sin(k t ) e k t
Eek t sin(k t )
DO THIS:
EXERCISE
Show that
C cos k t D sin k t can be written as E sin(k t )
2.5 Underdamped Oscillator
When a damped oscillator is underdamped, it approaches zero faster than in the
case of critical damping, but oscillates about that zero.
Fig 2.11: Underdamped Oscillator
The frequency k n 1 2
d
2
k
and the damped period is given by
2
n 1 2
DO THIS:
Exercise
Which of the motions discussed above best describes the action of shock
absorbers on a vibrating vehicle?
UNIT 3: DYNAMICS
Specific learning objectives
By the end of this unit you should be able to:
o Derive the equation of motion for a particle in a central field.
o Find the potential energy of a particle in a central field.
o State the laws of conservation of energy.
o State, derive and apply Kepler’s laws of planetary motion in solving related
problems.
o Find velocity and acceleration of moving systems.
o Define conservation of momentum and angular momentum.
o Solve problems involving changing mass.
Summary
In this module you will familiarize yourself with historical developments
concerning planetary motion beginning with early attempts to explain this
phenomenon. You will also acquaint yourself with early and modern
mathematical models that exemplified this phenomenon resulting in the famous
Kepler’s Laws.
Required Reading
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Relevant Links and Resources
Classical Mechanics Lecture notes
http://farside.ph.utexas.edu/teaching/301/lectures/lectures.html
Newtonian Physics
http://www.lightandmatter.com/arealbook1.html
Planetary motion
http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/kepler6.html
Planetary motion
http://www-istp.gsfc.nasa.gov/stargaze/kep3laws.htm
Planetary motion
http://en.wikipedia.org/wiki/Kepler’s_laws_of_planetary_motion
Planetary motion
www.windows.ucar.edu/tour/link=/the_universe/uts/planets.html
Coriolis Effect
http://en.wikipedia.org/wiki/Coriolis_effect
Coriolis Force
http://hyperphysics.phy-astr.gsu.edu/hbase/corf.html
Centripetal Force
http://en.wikipedia.org/wiki/Centripetal_force
Angular Momentum
http://en.wikipedia.org/wiki/Angular_momentum
D’Alembert’s Principle
http://en.wikipedia.org/wiki/D%27Alembert%27s_principle
Key Words
Central forces
Planetary motion
Central field
Kepler’s laws
Coriolis
3
Learning Activity Dynamics
3.1
Bridging the Gap
What stories were you told when you were young about
the rising and setting of the sun
the rising and setting of the moon
the general movement of the sun, moon and the earth ?
Consult some of the elders in your local community and hear their views in these
issues.
Have you ever wondered why, when you look at the diagrams of thr planetary
system why
the planets don’t just ‘fall off, into space
The planets and the sun maintain their distance apart and do not collide
Fig 3.1: Solar system
3.2
Central Forces and Planetary Motion
Compulsory Reading:
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Page 262-275
3.2.1 The concept of central force
It is with Kepler that we see the beginning of the concept of central force, that is a
force which acts continuously on the planets to keep them moving in closed,
stable paths from one orbit to the next. It was apparent to Kepler that the force
was directed towards the focus of the ellipse, but he could not describe the
nature of the force.
3.2.2 Motion in a general central force-field
Go to this link
http://farside.ph.utexas.edu/teaching/336k/lectures/node53.html
3.3
Kepler’s Laws Of Planetary Motion
Compulsory Reading:
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Page 260
3.3.1 LAW 1: The orbit of a planet/comet about the Sun is an ellipse with the
Sun's center of mass at one focus
This is the equation for an ellipse:
x2 y 2
1
a 2 b2
3.3.2 LAW 2: A line joining a planet/comet and the Sun sweeps out equal areas
in equal intervals of time
3.3.3 LAW 3: The squares of the periods of the planets are proportional to the
cubes of their semimajor axes:
Ta2 / Tb2 = Ra3 / Rb3
Square of any planet's orbital period (sidereal) is proportional to cube of its
mean distance (semi-major axis) from Sun
Mathematical statement: T = kR3/2 , where T = sideral period, and R =
semi-major axis
Example - If a is measured in astronomical units (AU = semi-major axis of
Earth's orbit) and sidereal period in years (Earth's sidereal period), then
the constant k in mathematical expression for Kepler's third law is equal to
1, and the mathematical relation becomes T2 = R3
3.3.4 Examples of Kepler's Third Law
Planet P (yr) a (AU) T2
Mercury 0.24
R3
0.39 0.06 0.06
Venus
0.62
0.72 0.39 0.37
Earth
1.00
1.00 1.00 1.00
Mars
1.88
1.52 3.53 3.51
Jupiter
11.9
5.20
142 141
Saturn
29.5
9.54
870 868
In simple terms Kepler’s laws are stated as follows:
1. The motion of planets and other bodies subject to the same force is in orbits
that are "conic sections": ellipses or hyperbolae or in very special circumstances
parabolas (all with the sun as a focus), or straight lines.
2. The area swept out per unit time in any orbit is constant.
3. There is a certain specific relation between the period of an elliptical orbit and
a measure of its radius, which relation we will not discuss further.
Gravitational Potential Energy
Definition:
The Gravitational potential energy is the work done by raising a mass of m Kg, h
above a reference point where the work done is taken to be zero.
Potential Energy is then written as
Vg mgh
This work is taken as energy because the work done may be converted to energy
if the body was allowed to work on a supporting body to return to its original
position. Thus change in potential energy for a body moving from initial height
h1 to height h2 is
Vg mg (h2 h1 ) mg h
The attendant work done by the gravitational force is
mg h
which is the negative of the change in potential energy.
Newton's Law of Gravitational governing the attraction between two bodies of
mass m1, m2 is given by
F G
m1m2
r2
where F = force of attraction between two particles
G =Universal Constant of gravitation
6.673 1011 m3 / kg s 2
r =distance between the centres of the particles.
3.3.5 Examples
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Page 275-278
3.3.6 Example
Consider a particle of mass m, moving under the action of the central
gravitational attraction.
mm0
r2
Where m0 is the mass of the attracting assumed to be fixed, G is the universal
gravitational constant, and r is the distance between the centers of the masses.
The particle of mass m could represent the earth moving about the sun, the
moon moving about the earth, or a satellite in its orbital motion about the earth
above the atmosphere. The most convenient coordinate system to use is polar
coordinates in the plane of motion since F will always be in the negative rdirection and there is no force in the direction . This implies that
F G
mm0
m(r r 2 )
2
r
0 m(r 2r )
The second of the two equations when multiplied by r/m is seen to be the same
as d (r 2 ) / dt 0 , which is integrated to give
r 2 h ,
a constant
G
This is reduced to
r h(u / )
And
r h(d 2u / d 2 )
Or
r h2u 2 (d 2u / d 2 )
After substitution we get
Gm0u 2 h 2u 2
d 2u 1 2 4
hu
d 2 u
Or
Gm
d 2u
u 20
2
d
h
Which is a non-homogeneous linear differential equation.
3.3.7 DO THIS
3.3.7.1
Exercise
Verify that the solution of this familiar second-order equation is
Gm
1
u C cos( ) 2 0
r
h
3.3.7.2
Exercise
An artificial satellite is launched from point B on the equator by its carrier rocket
and inserted into an elliptical orbit with a perigee altitude of 2000 km. if the
apogee altitude is to be 4000 km, compute (a) the necessary perigee velocity v p
and the corresponding apogee velocity vA, (b) the velocity at point C where the
altitude of the satellite is 2500 km, and (c) the period for a complete orbit.
Do not turn the page until you have completed the solution!
Solution
(a) The perigee and apogee velocities for specified altitudes are given
by
Where
vP R
g 1 e
g
R
a 1 e
a
rmax
rmin
vA R
g 1 e
g
R
a 1 e
a
rmin
rmax
rmax 6371 4000 10371km
rmin 6371 2000 8371km
a (rmin rmax ) / 2 9371km
Thus
vP R
g rmax
9.825
10371
6371(103 )
3
a rmin
9371(10 ) 8371
7261 m / s or 26140km / h
vA R
g rmin
9.825
8371
6371(103 )
3
a rmax
9371(10 ) 10371
5861 m / s or 21099 km / h
(b) For an altitude of 2500 km the radial distance from the earth’s center is r =
6371 + 2500 = 8871 km. The velocity at point C becomes
1 1
1 1
1
vC2 2 gR 2 2(9.825)[(6371)(103 )]2
3
r 2a
8871 18742 10
47.353(106 )(m / s ) 2
vC 6881 m / s or
24773 km / h
(c) The period of the orbit is given by
2
a 3/ 2
R g
[(9371)(103 )]3/ 2
9026 s
(6371)(103 ) 9.825
or 2.507 h
2
3.4
Moving Co-Ordinate Systems
3.4.1 The Coriolis Effect
The Coriolis effect is the apparent deflection of objects from a straight path if
the objects are viewed from a rotating frame of reference.
See link below
http://en.wikipedia.org/wiki/Coriolis_effect
To get more information on the Coriolis Force visit this site
http://hyperphysics.phy-astr.gsu.edu/hbase/corf.html
3.4.2 Centripetal Acceleration (and Force)
Compulsory Reading:
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Page 138
The rate of change of the object's velocity vector is the centripetal acceleration.
The centripetal force is the external force required to make a body follow a
circular path at constant speed.
See link below
http://en.wikipedia.org/wiki/Centripetal_force
3.4.3 Examples
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Page 155-157
3.4.4 Relative velocity in a moving coordinate system
A man walking at a rate of 4 km per hour toward the forward car of a train when
the latter is traveling at a speed of 50 km per hour is actually traveling at the
speed of 54 km per hour over the round. His relative speed with respect to a
fixed point on the moving train is 4 km per hour. This simple concept of
superposition of speeds can be extended readily to the case of differing
directions making use of the reference system velocity and the relative velocity
by use of vector addition.
3.4.4.1
Example
The compass of an aircraft indicates that it is headed due north, and its air speed
indicator shows that it is moving through the air at 200 km/h. If there is a wind of
80 km/h from west to east, what is the velocity of the aircraft relative to the earth?
Solution
Let A refer to the aircraft, and F to the moving air. E refers to the earth. We have
given
VAF = 240 km/h,
due north
VFE = 100 km/h,
due east
and we wish to find the magnitude and direction of VAE:
vAE vAF 2 vFE 2
v AE 2402 1002
260 km / h ,
22.60 North East
MORE EXAMPLES
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Page 48-52
DO THIS: 3.4.4.2 EXERCISE
Consider the first example above, in what direction should the pilot head in order
to travel due north? What will then be his velocity relative to the earth?
N.B. The magnitude of his airspeed, and the wind velocity, are the same as in
the first example.
Do not turn the page until you have completed the solution!
Solution
We now have given:
VAF = 240 km/h, direction unknown
VFE = 100 km/h, due east,
and we wish to find VAE, whose magnitude is unknown but whose direction is due
north.
We find
v AE 2402 1002
218.2 km / h
sin 1
100
24.60
240
Thus , the pilot should head 24.60 North West, and his ground speed will be
218.2 km/h.
3.5
Systems Of Particles
3.5.1 Angular momentum
In physics, the angular momentum of an object rotating about some reference
point is the measure of the extent to which the object will continue to rotate about
that point unless acted upon by an external torque.
Compulsory Reading:
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Page 204-214
For more information visit the site below
http://en.wikipedia.org/wiki/Angular_momentum
3.5.2 Examples
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Page 214-216
3.5.3 Conservation of angular momentum and Torque
Compulsory Reading:
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Page 271-272
For information on Conservation of angular momentum and Torque also see the
link below:
http://en.wikipedia.org/wiki/Angular_momentum
3.5.4 D’Alembert’s Principle
The principle states that the sum of the differences between the generalized
forces acting on a system and the time derivative of the generalized momenta of
the system itself along an infinitesimal displacement compatible with the
constraints of the system (a virtual displacement), is zero.
That is, at equilibrium,
( F p ) r 0
i
i
i
i
Since Fi pi , where Fi are generalized forces and pi are generalized momenta
See link
http://en.wikipedia.org/wiki/D%27Alembert%27s_principle"
3.5.5 Example
Each of three balls has a mass m and is welded to the rigid equi-angular frame of
negligible mass. The assembly rests on a smooth horizontal surface. If a force F
is suddenly applied to the one bar determine (a) the acceleration of point O and
(b) the angular acceleration of the frame.
Solution
(a) Point O is the mass center of the system of the three balls, so that its
acceleration is given by
F ma
Fi 3ma
a a0
F
i
3m
(b) We determine from the moment principle M G HG . To find H G we note
that the velocity of each ball relative to the mass center O as measured in the
non-rotating axes x-y is r where is a common angular velocity of the spokes.
The angular momentum of the system about O is the sum of the moments of the
relative linear momenta as shown by the expression below.
H 0 H G 3(mr )r 3mr 2
And from
M
Fb
G
HG we have
d
(3mr 2 ) 3mr 2
dt
So
Fb
3mr 2
DO THIS
3.5.6 Exercise
Consider the same conditions as for the example above except that the spokes
are freely hinged at O and so do not constitute a rigid system. Explain the
difference between the two problems.
Do not turn the page until you have completed the solution!
Solution
The generalized Newton’s second law holds for any mass system, so that the
acceleration a of the mass center G is the same as with example above,
namely,
a
F
i
3m
N.B. Although G coincides with O at the instant represented, the motion of the
hinge O is not the same as the motion of G since O will not remain the center of
mass as the angles between the spokes change. Both M G and HG have the
same values fro the two problems at the instant represented. However, the
angular motions of the spokes in this problem are all different and are not easily
determined.
3.6
ROCKETS AND COLLISIONS
Compulsory Reading:
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Page 115-129
Now that people have begun exploration of space, rockets have become a
subject of general interest. The central feature of the rocket, which underlies its
unique capabilities, is its reaction engine. As the rocket engine operates, the
mass of the rocket changes rapidly. Consequently, Newton’s second law of
motion in its general form does not lead to F = ma. We therefore begin the
analysis of the motion of a rocket from the basic principle of momentum
conservation.
3.6.1 Examples
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin,
Texas.UTP
Page129-135
Example
The ram of a pile driver has a mass of 800 kg and is released from rest 2 m
above the top of the 2400 kg pile. If the ram rebounds to a height of 0.1 m after
impact with the pile, calculate (a) the velocity v 'p of the pile immediately after
impact. (b) the coefficient of restitution e, and (c) the percentage loss of energy
due to the impact.
Solution
Conservation of energy during free fall gives the initial and final velocities of the
ram from
v 2 gh . Thus
vr 2(9.81)(2) 6.26 m / s
vr' 2(9.81)(0.1) 1.40 m / s
(a) Conservation of momentum (G y 0) for the system of the ram and pile
gives
800(6.26) 0 800(1.40) 2400v'p
Thus
v'p 2.55 m / s
(b) The coefficient of restitution gives
e
rel. vel. seperation
rel. vel. approach
e
2.55 1.40
0.63
6.26 0
(c) The kinetic energy of the system just before impact is the same as the
potential energy
of the ram above the pile and is
T Vg mgh 800(9.81)(2) 15700 J
The kinetic energy T ' just after impact is
T ' 12 (800)(1.40)2 12 (2400)(2.55)2 8620 J
The percentage loss of energy is therefore,
15700 8620
(100) 45.1%
15700
N.B. The impulses of the weights of the ram and pile are very small compared
with the impulses of the impact forces and thus are neglected during the impact.
DO THIS
3.6.2 Exercise
A cannon is mounted inside a railroad car, which is initially at rest but can move
frictionlessly. It fires a cannonball of mass m = 5 kg with a horizontal velocity of
v = 15 ms-1 relative to the ground at the opposite wall. The total mass of the
cannon and the railroad car is 15 000 kg. (Assume that the mass of the exhaust
gases is negligible.)
(a) What is the velocity V of the car while the cannonball is in flight?
(b) If the cannonball becomes embedded in the wall, what is the velocity of the
car and ball after impact?
Do not turn the page until you have completed the solution!
Solution
When fired, the cannon exerts a force to the right on the ball. The ball exerts an
equal and opposite force on the cannon, so the car and the cannon recoil to the
left. The net momentum is conserved because there is no external frictional
force. The momentum before firing is zero, so after firing the momentum of the
ball to the right must be equal in magnitude to that of the car and the cannon to
the left. Thus mv MV and
V
mv (5 kg )(15 ms 1 )
M
15000 kg
This gives
5 103 ms 1
N.B. The recoil speed of the car and the cannon is very small because of their
large mass.
(b) As the ball becomes embedded in the wall, it exerts a force on the wall to the
right. The wall, in turn, exerts a force to the left on the ball. The ball and car both
stop moving when this happens, since the net momentum is still zero. Meanwhile
the car will have rolled to the left as the ball traveled to the right.
DO THIS
3.6.3 Exercise
A baseball initially at rest is struck with a bat. The velocity of the 0.15-kg ball just
after it is hit is 40 ms-1. If the impact time is 10-3s, what is the average force on
the ball?
Do not turn the page until you have completed the solution!
Answer
The initial momentum p of the ball is zero, since it starts from rest; the final
momentum is p’ = mv’. Thus, from F t p p , the average force F on the ball is
mv (0.15kg )(40ms 1 )
F
6000 N
t
(103 s)
Unit 4: Rigid bodies and Energy
Assumed Knowledge:
Unit 3 of this module is the prerequisite, particularly particle kinematics and
kinetics of systems of particles.
Objectives of the unit:
The student should be able to:
a) Write and explain the equations of rigid body motion and apply them to
general planar motion of a rigid body.
b) Calculate relative velocity and relative acceleration for points on a rigid
body, and also calculate the instantaneous centre of zero velocity for a
rigid body.
c) Apply the equations of planar kinetics to analyze the motion of rigid bodies
undergoing translation, rotation about a fixed and general plane motion.
d) Apply the work-energy principle as well as the conservation of energy
principle to solve problems in planar kinetics.
e) Calculate both the linear and angular momentum of a rigid body and apply
them to solve problems in planar kinetics.
Summary of the learning activity
In this unit students will learn the equations of rigid body motion and apply them
to general planar motion of a rigid body. The students will learn how to calculate
relative velocity and relative acceleration for points on a rigid body, and also
learn how to calculate the instantaneous centre of zero velocity for a rigid body.
The students will learn how to apply the equations of planar kinetics to analyze
the motion of rigid bodies undergoing translation, rotation about a fixed point, and
general plane motion.
The student will learn how to apply the work-energy principle as well as the
conservation of energy principle to solve problems in planar kinetics. The student
will learn how to calculate both the linear and angular momentum of a rigid body
and apply them to solve problems in planar kinetics. Students will also learn
other better methods of solving mechanical problems other than the Newton
laws. They will look at other principles of mechanics like the D’Alembert’s
principles.
Suggested ICT methods
Use of the Internet to get more information on the relevant topic, Software and
Animations. Virtual Science laboratory,
http://www.smartscience.net/SmartScience/SmartScience.html
Content: Plane motion of rigid bodies: Moment of inertia. Radius gyration,
Parallel axis theorem, Perpendicular axis theorem, Couples. Kinetic energy and
angular momentum about a fixed axis, Principle of angular momentum, Principle
of conservation of energy, Principle of virtual work and D’Alembert’s principle,
Principle of minimum potential energy.
Compulsory reading text
Fitzpatrick, Richard, Classical Mechanics: An Introductory Course, Austin, Texas.
Relevant Links and Resources
Our links are from Wikipedia.
Dynamics of Rigid Bodies
http://www.engin.brown.edu/courses/en4/notes/RigidKinematics/rigkin.htm
Key Words
Mechanical system
Inertia
Degrees of freedom
Kinematic constraints
Rectilinear translation
Curvilinear translation
4.
Learning Activity: Rigid bodies and Energy
4.1
PLANE MOTION OF RIGID BODIES
Compulsory Reading:
Mariam J.L. and Kraige L.G. Engineering Mechanics Volume 2: Dynamics
(page 291-459)
Ruina, A. and Pratap, R. Introduction to Statics and Dynamics
Dynamics of Rigid Bodies
http://www.engin.brown.edu/courses/en4/notes/RigidKinematics/rigkin.htm.
http://en.wikipedia.org/wiki/Rigid_body_dynamics
Remark
On the topic plain motion of rigid bodies, I feel guilty not to recommend students
to read Mariam J.L. and Kraige L.G. Engineering Mechanics Volume 2:
Dynamics. All other sources are heavily dependent on this book.
http://www.engin.brown.edu/courses/en4/notes/RigidKinematics/rigkin.htm
Dynamics of Rigid Bodies
http://www.engin.brown.edu/courses/en4/notes/RigidKinematics/rigkin.htm
4.1.1 Introduction
Dynamics is the branch of mechanics which deals with the laws of motion of
material bodies subjected to the action of forces. The motion of bodies from a
purely geometrical point of view is discussed in kinematics. Unlike in kinematics,
in dynamics the motion of material bodies is investigated in connection with the
acting forces and the inertia of the material bodies themselves. Inertia is the
property of material bodies to resist change in their velocity under the action of
an applied force. For example, the velocity of one body changes slower than that
of another body subjected to the same force, the former is said to have a greater
inertia.
A rigid body is an idealization of a body that does not deform or change shape.
Formally it is defined as a collection of particles with the property that the
distance between particles remains unchanged during the course of motions of
the body. Like the approximation of a rigid body as a particle, this is never strictly
true. All bodies deform as they move. However, the approximation remains
acceptable as long as the deformations are negligible relative to the overall
motion of the body.
Schematic showing a planar rigid body.
In the following, we will restrict attention to the planar motion of rigid bodies. In
particular, we will take all rigid bodies to be thin slabs with motion constrained to
lie within the plane of the slab. Unless otherwise indicated, we will assume basis
vectors of the form {i, j, k}, such that i and j lie in the plane, with k is the plane
normal.
DO THIS
Exercise: Kinematics of Rigid Body Motion
1.
2.
What do you understand by kinematics of rigid bodies?
Explain the following types of motion
i)
Rectilinear translation
ii)
curvilinear translation
4.2
DYNAMICS OF A SYSTEM AND RIGID BODY
Compulsory Reading:
Moment of inertia
http://en.wikipedia.org/wiki/Moment_of_inertia
4 .2.1 Mechanical Systems. External and Internal Forces
A mechanical system is defined as such a collection of material points (particles)
or bodies in which the position or motion of each particle or body of the system
depends on the position and motion of all the other particles or bodies. A
classical example of a mechanical system is the solar system, all the component
bodies of which are connected by the forces of their mutual attraction. Other
examples of mechanical systems are machines or any mechanism whose
members are connected with pins, rods, cables, belts, belts, etc. In this case the
bodies of the system are subjected to the reciprocal compressive or tensile
forces transmitted through the constraints.
A collection of bodies not connected by interacting forces does not comprise a
mechanical system (e.g, a group of lying aircraft). In this book we shall consider
only mechanical systems, calling them just "systems" for short. It follows from
the above that the forces acting on the particles or bodies of a system can be
subdivided into external and internal forces.
External forces are defined as the forces exerted on the members of a system
by particles or bodies not belonging to the given system.
Internal forces are defined as the forces of interaction between the members of
the same system. We shall denote external forces by the symbol F e
, and
internal forces by the symbol Fi .
Internal forces possess the following properties:
The geometrical sum (the principal vector) of all the internal forces of a system is
zero. This follows from the third law of dynamics, which states that any two
particles of a system (Fig . 15) act on each other with equal and oppositely
directed forces Fi12
and Fi21, the sum of which is zero. Since the same is
true for any pair of particles of a system,
Fki 0
Fig 15
The sum of the moments (the principal moment) of all the internal forces of a
system with respect to any centre or axis is zero. For if we take an arbitrary
centre
O , it is apparent from Fig. 15 that m0 ( F12i ) m0 ( F21i ) 0 . The same
result holds for the moments about any axis. Hence, for the system as a whole
we have:
m0 ( Fki ) 0 or mx Fki 0
It does not follow from the above, however, that the internal forces are mutually
balanced and do not affect the motion of the system, for they are applied to
different particles or bodies and may cause their mutual displacement. The
internal forces will be balanced only when a given system is a rigid body.
4.2.2 Mass of a System. Centre of Mass
The motion of a system depends, besides the acting forces, on its total mass and
the distribution of this mass. The mass of a system is equal to the arithmetical
sum of the masses of all the particles or bodies comprising it.
M mk
We shall conventionally denote mass by the same letter M as the moment of a
force. Any possibility of confusion is precluded by the fact that when the symbol
M denotes moment of a force, it is always provided with a subscript (e.g.
M c , M x , M t ).
In a homogeneous field of gravity, where g = const., the weight of every particle
of a body is proportional to its mass, hence the distribution of mass can be
judged according to the position of the centre of gravity. The equations defining
the coordinates of the centre of gravity can be written as
xc
m x
k
M
k
, yc
m y
k
M
k
, zc
m z
k k
M
(1)
The equations include only the masses mk of the material points (particles) of
the body and their coordinate ( xk , yk , zk ). Hence, the position of point C ( insert
equation) gives the distribution of mass in the body or in any mechanical system,
where
mk and ( xk , yk , zk ). are the masses and coordinates of the system's
respective points.
The geometric point C whose coordinates are given b Eqs. (1) is called the
centre of mass, or centre of inertia of a mechanical system. If the position of the
centre of mass is defined by is radius vector rc we can obtain from Eqs. (1) the
following expression:
rc
m r
k k
M
where rk is the radius vector of particle k of the system.
Although in a homogenous gravitational field the Centres of mass and gravity
coincide, the two concepts are not identical. The concept of centre of gravity, as
the point through which the resultant of the forces of gravity passes, has meaning
only for a rigid body in a uniform in field of gravity. The concept of centre of
mass, as characteristic of the distribution of mass in a system, on the other hand,
has meaning for any system of particles or bodies, regardless of whether a given
system is subjected to the action of forces or not.
4.2.3
Moment of Inertia of a Body about an Axis: Radius of Gyration
The position of centre of mass does not characterize completely the distribution
of mass in a system. For if in the system in Fig.16a the distance h of each of the
two identical spheres A and B from the axis Oz is increased by the same
quantity, the location of the centre of mass will not change, though the
distribution of mass will change and influence the motion of the system (all other
conditions remaining the same, the rotation about axis Oz will be slower).
Fig 16a
Accordingly, another characteristic of the distribution of mass, called the moment
of inertia, is introduced in mechanics. The moment of inertia of a body (system)
with respect to a given axis Oz (or the axial moment of inertia) is defined as the
quantity equal to the sum of the masses of the particles of the body (system)
each multiplied by the square of its perpendicular distance from the axis:
J z mk hk2
(2)
It follows from the definition that the moment of inertia of a body (or system) with
respect to any axis is always positive.
It will be shown further on that axial moment of inertia plays the same part in the
rotational motion of a body a mass does in translational motion, i.e. moment of
inertia is a measure of a body's inertia in rotational motion.
By Eq. (2), the moment f inertia of a body is equal to the sum of the moments of
inertia of all its parts with respect to the same axis.
For a material point located at distance h from an axis, J z mh 2 . The unit for the
moment of inertia in the SI system is 1 kg-m2, and in the mkg(f) system 1 kgf-ms.
In computing the axial moments of inertia the distances of the points from the
axis can be expressed in terms of their coordinates xk , yk , zk (e.g., the square
yk2 zk2 , etc.
of the distance from axis Ox is
Then the moments of inertia about the axes Oxyz will be given by the following
equations:
(3)
J x mk ( yk2 zk2 ), J y mk ( xk2 zk2 ), J z mk ( yk2 xk2 ).
The concept of the radius of gyration is often employed in calculations. The
radius of gyration of a body with respect to an axis Oz is a linear quantity g
defined by the equation
J z M g2
where M is the mass of the body.
(4)
It follows from the definition that geometrically the radius of gyration is equal to
the distance from the axis Oz to a point, such that if the mass of the whole body
were concentrated in it, the moment of inertia of the point would be equal to the
moment of inertia of the whole body.
Knowing the radius of gyration, we can obtain the moment of inertia of a body
from Eq. (4) and vice versa.
Eqs. (2) and (3) are valid for both rigid bodies and systems of material points. In
the case of a solid body, dividing it into elementary parts, we find that in the limit
the sum in Eq. (2) becomes an integral. Hence, taking into account that
dm dV
, where is the density and V the volume, we obtain:
J z h 2 dm, or
J z h 2 dV
(5)
The integral extends over the whole volume V of the body, and the density and
distance H depend on the coordinates of the points of the body. Similarly, for
solid bodies Equations. (3) take the form:
J x ( y 2 z 2 )dV , etc.
(5’)
Equations.
(5) and (5') are useful in calculating the moments on inertia of
homogeneous bodies of geometric shape. As in that case the density is
constant, it can be taken out of the integral sign.
Let us determine the moments of inertia of some homogeneous bodies.
4.2.4 Thin Homogeneous Rod of Length L and Mass M.
Let us find its moment of inertia with respect to an axis
Az perpendicular to the rod (Fig. 16b). If we lay off a
coordinate axis Ax along AB, for any line element of
length
dx
we
have
h
=
x
and
its
mass dm 1dx, where 1 M / l , is the mass of a
unit length of the rod, and Eq. (5) gives.
Fig. 16b
l
l
l3
J A x dm 1 x dx 1 .
3
0
0
2
2
Substituting the expression for
1 , we obtain finally
1
J A Ml 2
3
(2) Thin Circular Homogeneous Ring of Radius R and Mass M.
(6)
Let us find its moment of inertia with respect to an axis Cz perpendicular to the
plane of the ring through its centre.
Fig 17.
As all the points of the ring are at a distance
from axis Cz, Eq. (2) gives
J c mk R2 ( mk ) R2 MR2
Here and further on J denotes the moment of inertia with respect to an axis
through A perpendicular to the plane of the cross section in the diagram.
Hence, for a ring J c MR 2
(7)
It is evident that the same result is obtained for the moment of inertia of a
cylindrical shell of mass M and radius R with respect to its axis.
4.2.5 Circular Homogeneous Disc or Cylinder of Radius R and Mass M
Let us compute the moment of inertia of a circular disc with respect to an axis Cz
perpendicular to it trough its centre (Fig. 17b).
Consider
an
elemental
ring
of
radius r and width dr.
Its area
M
2 rdr , and its mass dm 2 2 rdr , where 2
is the mass of a unit area of
R2
the disc. From Eq. (7) we have for elemental ring
dJ c r 2 dm 2 2 r 3dr ,
and for the whole disc
R
1
J c 2 2 r 3dr 2 R 4
2
0
Substituting the expression for 2 we obtain finally
1
J C MR 2
2
(8)
It is evident that the same formula is obtained for the moment of inertia J of a
homogeneous circular cylinder of mass M and radius R with respect to its axis Cz
(Fig. 17c)
4.2.6 Rectangular Lamina, Cone, and Sphere.
Omitting the computations, here are the equations of the moments of inertia of
several bodies (the students is invited to deduce these formulas independently).
(a) uniform rectangular lamina of mass M with sides of length a and b (axis x is
coincident with side a, axis y with side b):
1
1
J x Mb 2 , J y Ma 2 ;
3
3
(b) uniform right circular cone of mass M and base radius R (axis z is coincident
with the axis of the cone):
J z 0.3MR 2
(c) uniform sphere of mass M and radius R (axis z is coincident with a diameter):
J z 0.4MR 2
4.2.7 Moments of Inertia of a Body about Parallel Axes
The Parallel Axis (Huygens') Theorem
In the most general case, the moments of inertia of the same body with respect
to different axes are different. Let us see how to determine the moment of inertia
of a body with respect to any axis if its moment of inertia with respect to a parallel
axis through the body is known.
Fig
18
Draw through the centre of mass of a body C arbitrary axes Cx ' y ' z ' , and through
an arbitrary point 0 on axis Cx axes 0xyz, so that Oy // Cy ' and Oz // Cz ' (Fig. 18).
Denoting the distance between axes Cz' and 0z by d, from Eqs. (3)
J OC mk ( xk2 yk2
J OZ ' mk ( x 'k2 y 'k2 )
But it is apparent from the drawing that for any point of the body
xk x 'k d , or xk2 x 'k2 d 2 2 x 'k d and yk y 'k . Substituting these expressions for xk
and
yk
into the expression for JOZ
and taking the common multipliers d and 2d
outside the parentheses, we obtain:
JOZ mk ( x 'k2 y 'k2 ) ( mk )d 2 2d ( mk x 'k2 ).
The first summation in the right member of the equation is equal to J Cz’ , and the second to
the mass M of the body. Let us find the value of the third summation. From Eq. (1) we
know that, for the coordinates of the centre of mass, mk x 'k Mx 'C . But since in our case
point C is the origin, x 'C 0,
, and consequently
m x '
k
k
0
We finally obtain
J oz J Cz ' Md 2
(9)
Equation (9) expresses the parallel axis theorem enunciated by Huygens*: The moment of
inertia of a body with respect to any axis is equal to the moment of inertia of the body with
respect to a parallel axis through the centre of mass of the body plus the product of the
mass of the body and the square of the distance between the two axes.
It follows from Eq. (9) that Joz = JCz’ . Consequently, of all the axes of the same direction,
the moment of inertia is least with respect to the one through the centre of mass.
The parallel axis theorem can also be used to determine the moment of inertia of a body
with respect to a given axis Oz1 if its moment of inertia with respect to any parallel axis
Az2 and the distances d1 and d2 of each axis from the body's centre of mass are known.
Hence, knowing J Az2 and d 2 we obtained J Cz ' from Eq. (9) and, applying the same
formula, determine the required moment of inertia J Oz1
DO THIS
Exercise
1
Determine the moment of inertia of a thin rod with respect to an axis C
perpendicular to it through its centre of mass.
2
Determine the moment of inertia of a cylinder with respect to an axis
its generator (see Fig. 16b).
Module Development Template
through
124
4.3
Product of Inertia
4.3.1 Principal Axes of Inertia of a Body
The moment of inertia of a body with respect to an axis also does not completely
characterize the distribution of mass of the system. For example, if the rod DE in Fig.
below
is turned in plane 0yz so as to make other than a right angle with axis 0z and the distance h
of spheres A and B from the axis is kept the same by moving them outward, neither the
location of the centre of mass nor moment of inertia of the spheres with respect to axis 0z
will change. Yet the distribution of mass will have changed (the symmetry with respect to
axis 0z being disturbed), and this will affect the system's rotation about the axis (additional
lateral stresses will appear in the bearing).
Accordingly, the concept of the product of inertia is introduced as characterizing such
asymmetry in the distribution of mass. Drawing coordinate axes 0xyz through point 0, the
products of inertia with respect to those axes are the quantities J xy , J yz , J zx given by the
following equations:
J xy mk xk yk , J yz mk yk zk , J zx mk zk xk ,
where mk
is the mass of the points and
xk , yk , zk
(10)
are their coordinates.
Obviously J xy J yx etc , .
For solid bodies, Eqs. (10) by analogy with (5') take the form:
J xy xydV
(10’)
V
unlike axial moments of inertia, products of inertia can be either positive, negative or, in
selected coordinates, zero.
Consider a homogenous body having an axis of symmetry. Draw the coordinate axes 0xyz
so that axis z is directed along the axis of symmetry. By virtue of symmetry, to each point
of mass mk with coordinates xk , yk , zk , there corresponds another point of equal mass
with coordinates
xk , yk , zk
. Consequently,
m x z
k k k
0 and
m y z
k
k k
0 , and
taking into account Eqs. (10), we obtain:
J xz 0,
J yz 0
Module Development Template
(11)
125
Thus, symmetry in the distribution of mass with respect to axis 0z is characterised by two
centrifugal moments of inertia, J xz and J yz ,
becoming zero. Axis 0z with respect to
which the products of inertia Jxz whose subscripts contain the notation of that axis, are zero
is called the principal axis of inertia of the body with respect to point 0
It follows from what has been said that if a body has an axis of symmetry, that axis is the
principal axis of inertia with respect to any of its points.
The principal axis of inertia is not necessarily the axis of symmetry.
Consider a homogenous body having a plane of symmetry. Drawing axes 0xy in that plane
and axis 0z perpendicular to them, by virtue of symmetry, to every point of mass mk with
coordinates xk , yk , zk , there corresponds a point of same mass with coordinates
xk , yk , zk ,.
m x z
k k k
Consequently,
0 and
m y z
k
k k
as
in
the
previous
case,
we
find
that
0 , or J xz 0, J yz 0 .
Thus, if a body has a plane of symmetry, any axis perpendicular to that plane is the
principal axis of inertia of the body with respect to the point 0 at which the axis intersects
the plane.
Eqs. (11) express the conditions that axis 0z is the principal axis of inertia of a body with
respect to point 0 (the origin of the coordinate system). Similarly, if J xy 0, J xz 0
,
axis 0x will be the principal axis of inertia with respect to point 0, etc. Consequently, if all
the products of inertia are zero, i.e.,
J xy 0, J yz 0,
J zx 0 ,
(12)
each of the coordinate axes 0xyz is a principal axis of inertia with respect to point 0 (the
origin of the coordinate system).
Let us show that principal axes of inertia exist at any point of a body. For this let us first
prove the following theorem:
If the moment of inertia with respect to an axis 0z is greater or smaller than the moment of
inertia with respect to any neighboring axis through 0, that axis (0z) is the principal axis of
inertia of the body with respect to 0.
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4.4
D'ALEMBERT'S PRINCIPLE
Compulsory Reading:
Classical Mechanics Page 260
D'Alembert's Principle
http://www. en.wikipedia.org/wiki/D%27Alembert%27s_principle
4.4.1 Forces Acting on the Axis of a Rotating Body
D'Alembert's Principle
All the methods of solving the problems of dynamics examined up till now were based on
equations derived either directly from Newton's laws or from the general theorems, which
are corollaries of those laws. However, the equations of motion or equilibrium of other
general propositions called the principles of mechanics. We shall see that in many cases
application of those principles offers better methods of problem solutions. In this case we
shall examine one of the general principles of mechanics known as D' Alembert's principle.
Let there be a system of n material particles. Selecting any particle of mass m k , assume it
to be acted upon by external and internal forces Fkext and Fkint (which include both active
forces and the reactions of constraints), which impart it an acceleration
with
k
respect to an inertial reference frame.
Let us introduce the quantity
Fki mk k
(96)
with the dimension of force. The vector quantity equal in magnitude to the product of the
particle's mass and acceleration and directed in the opposite sense of the acceleration and
directed in the opposite sense of the acceleration is called the force of inertia of that particle
(sometimes the D'Alembert inertia force).
The motion of a particle, we then find, satisfies the following D'Alembert's principle for a
material particle: If, at any moment of time, to the effective forces Fkext and Fkint acting on
the particle is added the inertia force Fki , the resultant force system will be in equilibrium,
i.e.,
Fkext Fkint Fki 0
(97)
It will be readily observed that D'Alembert's principle is equivalent to Newton's second law,
and vice versa. For Newton's second law gives for this particle mk k Fkext Fkint .
Transferring mkk to the right-hand side of the equation, and taking into account the
notation (96), we arrive at Eq. (97), we obtain the formula expressing Newton's second law.
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Reasoning similarly for all the particles of the system, we arrive at the following result,
which expresses D'Alembert's principle for a system: If, at any moment of time, to the
effective external and internal forces acting on every particle of a system are added the
respective inertia forces, the resultant force system will be in equilibrium, and the equations
of statistics will apply to it.
Mathematically D'Alembert's principle is expressed by a set of n simultaneous vector
equations of the form (97) which, apparently, are equivalent to the differential equations of
motion of a system. D'Alembert's principle can be used to derive all the general theorems
of dynamics. The value of D'Alembert's principle is that, when directly applied to the
problems of dynamics, the equations of motion of a system can be written in the form of the
well-known equations of equilibrium; this makes for uniformity in the approach to problem
solutions and usually greatly simplifies the computations. Furthermore, when used in
conjunction with the principle of virtual displacement, which will be examined in the next
section, D' Alembert's principle yields a new general method of solution of problems of
dynamics
In applying D'Alembert's principle it should be remembered that, like the fundamental law of
dynamics, it refers to motion considered with respect to an inertial frame of reference. That
means that acting on the particles of the mechanical system whose motion is being
investigated are only the external and internal forces that appear as a consequence of the
interactions of the particles of the system among themselves and with bodies not belonging
to a system; it is under the action of those forces that the particles of the system are
moving with their respective accelerations k . The inertia forces mentioned in
D'Alembert's principle do not act on the moving particles [otherwise, by Eqs. (97), the
points would be at rest or in uniform motion in which case, as is apparent from Eq. (96),
there would be no inertia forces]. The introduction of inertia forces is but a device making it
possible to examine the equations of dynamics by the simpler methods of statics.
We know from Statics that the geometrical sum of balanced forces and the sum of their
moments with respect to any centre 0 are zero; we know, further, from the principle of
solidification that this holds not only for forces acting on a rigid body but for any deformable
system. Thus, according to D'Alembert 's principle, we must
(F
ext
int
i
mo ( Fk ) mo ( Fk ) mo (Fk ) 0
let us introduce the following notation:
ext
k
Fkint Fki 0
Ri Fki , M oi mo ( Fki ).
(98)
(99)
The quantities R i and M oi are respectively the principal vector of the inertia forces and
their principal moment with respect to a centre 0. Taking into account that the sum of the
internal forces and the sum of their moments are each zero we obtain:
(100)
Fkext Ri 0, mo (Fkext ) M oi
The use of Eqs. (100), which follow from D'Alembert's principle, simplifies the process of
problem solution because the equations do not contain the internal forces. Actually, Eqs.
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(100) are equivalent to the equations expressing the theorems of the change in the
momentum of a system, differing from them only in form.
Eqs. (100) are especially convenient in investigating the motion of a rigid body or a system
of rigid bodies. For the complete investigation of any deformable system these equations,
however, are insufficient*).
For the projections on a set of coordinate axes, Eqs. (100) give equations analogous to the
corresponding equations of statics. To use these equations for solving problems we must
know the principal moment of the inertia forces.
4.4.2 The Principal Vector and the Principal Moment of the Inertia Forces of a Rigid
Body
It follows from Eqs. (99) that a system of inertia forces applied to a rigid body can be
replaced by a single force equal to
Ri and applied at the centre 0, and a couple of
moment M oi . The principal vector of a system, it will be recalled, does not depend on the
centre of reduction and can be computed at once. As Fki mk k
the equation of motion of centre of mass
Ri mkk Mc
m
k
k
taking into account
M c we will have
(101)
Thus, the principal vector of inertia forces of a moving body is equal to the product of the
mass of the body and the acceleration of its centre of mass, and is opposite in direction to
the acceleration.
If we resolve the acceleration c into its tangential and normal components, then vector R i
will resolve into components.
Ri M c and Rni M cn
Let us determine the principal moment of the inertia forces for particular types of motion.
Translational Motion. In this case a body has no rotation about its centre of mass C, from
which we conclude that mC ( Fkext ) 0 and Eq. (100) gives M C 0 .
Thus, in translational motion, the inertia forces of a rigid body can be reduced to a single
resultant R i through the centre of mass of the body.
Plane Motion. Let a body have a plane of symmetry, and let it be moving parallel to the
plane. By virtue of symmetry, the principal vector and the resultant couple of inertia forces
lie, together with the centre of mass C, in that plane.
Therefore, placing the centre of reduction in point C, we obtain form Eq. (100)
M Ci mc ( Fkext ) (insert Eq.) On the other hand (see 156), from the last of Eqs. (68)
m (F
C
ext
k
) JC . We conclude from this that
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M Ci J C
(102)
Thus, in such motion of a system of inertia forces can be reduced to a resultant force
R i [Eq. (101)] applied at the centre of mass C (Fig. 22) and a couple in the plane of
symmetry of the body whose moment is given by Eq. (102). The minus sign shows that
the moment M Ci is in the opposite direction of the angular acceleration of the body.
Fig 22
4.4.3 Rotation About an Axis Through the Centre of Mass.
Let a body have a plane of symmetry, and let the axis of rotation Cz be normal to the
plane through the centre of mass. This case will thus be a particular case of the previous
motion. But here C 0 and consequently R i 0 , Thus, in this case a system of inertia
forces can be reduced to a resultant force couple in the plane of symmetry of the body of
moment
M zi J z
(102’)
in applying Eqs. (101) and (102) to problem solutions, the magnitudes of the respective
quantities are computed and the directions are shown in fig. 22
DO THIS
Exercise
Use D’Alembert’s principle to solve the following:
a) A load of weight P attached to a string of length I is displaced through an angle from
the vertical to a position M 0 and released from rest ( see Fig below). Determine the
tension in the thread when the load is in the lowest position M 1
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b)
c)
d)
Fig. 19
(a)
Two weights P1 and P2 are connected by a thread and move along a horizontal plane
under the action of forces Q applied to the first weight. The coefficient of friction of the
weights on the plane is f. Determine the accelerations of the weights and the tensions
in the thread
Fig 20
Wound on a drum of weight P and radius r is a string carrying a load A of weight Q.
Neglecting the mass of the string and friction, determine the angular acceleration of
the drum when the load falls, the radius of gyration of the drum with respect to its axis
is. Also determine the tension in the thread.
A homogeneous rod bent at a right angle as shown in Fig 21 rotates in a horizontal
plane about its end A with an angular velocity and angular acceleration. The
respective distances AB = a and BC = b; the mass of unit length of the rod is 1.
Determine the stresses at a cross section of the rod at point D at a distance h from
end B.
Fig. 21
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References
Michon,G. P, Rigid Bodies
http://home.att.net/~numericana/answer/rigid.htm
Ruina, A. and Pratap, R. Introduction to Statics and Dynamics
Targ, S. Theoretical Mechanics : A Short Course Mir Publishers Moscow (1988)
Mariam J.L. and Kraige L.G. Engineering Mechanics Volume 2: Dynamics.
D'Alembert's Principle
http://en.wikipedia.org/wiki/D%27Alembert%27s_principle
Moment of inertia
http://en.wikipedia.org/wiki/Moment_of_inertia
Dynamics of Rigid Bodies
http://www.engin.brown.edu/courses/en4/notes/RigidKinematics/rigkin.htm
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11. COMPILED LIST OF ALL KEY CONCEPTS (GLOSSARY)
Note: this glossary contains only those basic concepts that span the module. Most of the
important concepts have been singularly defined and explained in the relaevant units.
Should the student have any problem with the definitions, they are advised to visit the site
Wikipedia [www.en.wikipedia.org]. If this is being viewed as a computer file, key words can
be clicked to link with the relavent Wikipedia entry.
Central field
In atomic physics, the central field approximation for many-electron atoms takes the
combined electric fields of the nucleus and all the electrons acting on any of the electrons
to be radial and to be the same for all the electrons in the atom. That is, every electron
sees an identical potential U(r) that is only a function of its distance from the nucleus. This
facilitates an approximate analytical solution to the eigenvalue problem for the Hamiltonian
operator.
Circular Motion
In physics, circular motion is rotation along a circle: a circular path or a circular orbit. The
rotation around a fixed axis of a three-dimensional body involves circular motion of its parts.
We can talk about circular motion of an object if we ignore its size, so that we have the
motion of a point mass in a plane.
Examples of circular motion are: an artificial satellite orbiting the Earth in geosynchronous
orbit, a stone which is tied to a rope and is being swung in circles (cf. hammer throw), a
racecar turning through a curve in a racetrack, an electron moving perpendicular to a
uniform magnetic field, a gear turning inside a mechanism.
Circular motion involves acceleration of the moving object by a centripetal force which pulls
the moving object towards the center of the circular orbit. Without this acceleration, the
object would move inertially in a straight line, according to Newton's first law of motion.
Circular motion is accelerated even if the speed is constant, because the object's velocity
vector is constantly changing direction.
Coriolis Force and Effect
The Coriolis effect is an apparent deflection of moving objects from a straight path when
they are viewed from a rotating frame of reference. The effect is named after GaspardGustave Coriolis, a French scientist who described it in 1835, though the mathematics
appeared in the tidal equations of Pierre-Simon Laplace in 1778. The Coriolis effect is
caused by the Coriolis force, which appears in the equation of motion of an object in a
rotating frame of reference. Sometimes this force is called a fictitious force (or pseudo
force), because it does not appear when the motion is expressed in an inertial frame of
reference, in which the motion of an object is explained by the real impressed forces,
together with inertia. In a rotating frame, the Coriolis force, which depends on the velocity of
the moving object, and centrifugal force, which does not, are needed in the equation to
correctly describe the motion.
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Force: this is the vector action of one body upon another.
Harmonic Oscillator and Simple Harmonic Motion
In order for mechanical oscillation to occur, a system must posses two quantities: elasticity
and inertia. When the system is displaced from its equilibrium position, the elasticity
provides a restoring force such that the system tries to return to equilibrium. The inertia
property causes the system to overshoot equilibrium. This constant play between the
elastic and inertia properties is what allows oscillatory motion to occur. The natural
frequency of the oscillation is related to the elastic and inertia properties by:
The simplest example of an oscillating system is a mass connected to a rigid foundation by
way of a spring. The spring constant k provides the elastic restoring force, and the inertia of
the mass m provides the overshoot. By applying Newton's second law F=ma to the mass,
one can obtain the equation of motion for the system:
where
takes the form
is the natural oscillating frequency. The solutions to this equation of motion
where xm is the amplitude of the oscillation, and φ is the phase constant of the oscillation.
Both xm and φ are constants determined by the initial condition (intial displacement and
velocity) at time t=0 when one begins observing the oscillatory motion.
Kepler’s Three Laws
1. The orbit of every planet is an ellipse with the sun at one of the foci. An ellipse is
characterized by its two focal points; see illustration. Thus, Kepler rejected the
ancient Aristotelean, Ptolemaic,and Copernican belief in circular motion.
2. A line joining a planet and the sun sweeps out equal areas during equal intervals of
time as the planet travels along its orbit. This means that the planet travels faster
while close to the sun and slows down when it is farther from the sun. With his law,
Kepler destroyed the Aristotelean astronomical theory that planets have uniform
velocity.
3. The squares of the orbital periods of planets are directly proportional to the cubes of
the semi-major axes (the "half-length" of the ellipse) of their orbits. This means not
only that larger orbits have longer periods, but also that the speed of a planet in a
larger orbit is lower than in a smaller orbit.
Mass: this is defined as the quantitative measure of inertia or resistance to change in the
motion of a body and also gives rise to Gravitational attraction.
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Mechanical system A mechanical system is defined as such a collection of material points
(particles) or bodies in which the position or motion of each particle or body of the system
depends on the position and motion of all the other particles or bodies. A classical example
of a mechanical system is the solar system, all the component bodies of which are
connected by the forces of their mutual attraction. Other examples of mechanical systems
are machines or any mechanism whose members are connected with pins, rods, cables,
belts, belts, etc.
Natural Circular Frequency
The fundamental tone, often referred to simply as the fundamental and abbreviated fo, is
the lowest frequency in a harmonic series.
The fundamental frequency (also called a natural frequency) of a periodic signal is the
inverse of the pitch period length. The pitch period is, in turn, the smallest repeating unit of
a signal. One pitch period thus describes the periodic signal completely. The significance of
defining the pitch period as the smallest repeating unit can be appreciated by noting that
two or more concatenated pitch periods form a repeating pattern in the signal. However,
the concatenated signal unit obviously contains redundant information.
In terms of a superposition of sinusoids (for example, fourier series), the fundamental
frequency is the lowest frequency sinusoidal in the sum.
Newton’s Laws of Motion
Newton's laws of motion are three physical laws which provide relationships between the
forces acting on a body and the motion of the body. They were first compiled by Sir Isaac
Newton in his work Philosophiae Naturalis Principia Mathematica (1687). The laws form the
basis for classical mechanics and Newton himself used them to explain many results
concerning the motion of physical objects. In the third volume of the text, he showed that
these laws of motion, combined with his law of universal gravitation, explained Kepler's
laws of planetary motion.
Briefly stated, the three laws are:
1. A physical body will remain at rest, or continue to move at a constant velocity, unless
an unbalanced net force acts upon it.
2. The net force on a body is equal to its mass multiplied by its acceleration.
3. For every action there is an equal and opposite reaction.
Particle: is a body of negligible dimensions. Also when the dimensions of a body are
irrelevant to the description of its motion or the action of forces upon it, the body may be
treated as a particle.
Position: Position in space is determined relative to some geometric referrence system by
means of linear and/or angular measurements. The absolute frame of referrence is an
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imaginary set of rectangular axes assumed to have no translation or rotation effect in space
for other motions, such as rocket motion, relative motion of the earth has to be taken into
account and relative corrections made to equations describing the laws of Newtonian
mechanics
Rectilinear translation Motion in which every line in the body remains parallel to its
original position. The motion of the body is completely specified by the motion of any point
in the body. All points of the body have the same velocity and same acceleration.
Rigid Body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation
is neglected. In other words, the distance between any two given points of a rigid body
remains constant in time regardless of external forces exerted on it. In classical mechanics
a rigid body is usually considered as a continuous mass distribution, while in quantum
mechanics a rigid body is usually thought of as a collection of point masses. For instance,
in quantum mechanics molecules (consisting of the point masses: electrons and nuclei) are
often seen as rigid bodies (see classification of molecules as rigid rotors). The position of a
rigid body is determined by the position of its center of mass and by its orientation (at least
six parameters in total).
Simple Pendulum
A pendulum is an object that is attached to a pivot which the pendulum can swing freely.
This object is subject to a restoring motion that will accelerate it toward an equilibrium
position. When the pendulum is displaced from its place of rest, the restoring force will
cause the pendulum to oscillate about the equilibrium position.
A basic example is the simple gravity pendulum or bob pendulum. This is a weight (or bob)
on the end of a massless string, which, when initially displaced, will swing back and forth
under the influence of gravity over its central (lowest) point.
The regular motion of pendulum can be used for time keeping, and pendulum are used to
regulatee pendulum clocks.
Space: This is the geometric region occupied by bodies. Position in space is determined
relative to some geometric reference system by means of linear and angular
measurements.
Time: time is the measure of the succesion of events which is considered to be an absolute
quantity in Newtonian mechanics.
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Vectors and Scalars
Scalars have magnitude only. Temperature, speed, mass, and volume are examples of
scalars.
Vectors have magnitude and direction. The magnitude of is written | | v . Position,
displacement, velocity, acceleration and force are examples of vector quantities. Vectors
have the following properties:
1.
2.
3.
4.
Vectors are equal if they have the same magnitude and direction.
Vectors must have the same units in order for them to be added or subtracted.
The negative of a vector has the same magnitude but opposite direction.
Subtraction of a vector is defined by adding a negative vector:
5.
+ (-
)
Multiplication or division of a vector by a scalar results in a vector for which
(a)
(b)
6.
=
only the magnitude changes if the scalar is positive
the magnitude changes and the direction is reversed if the scalar is negative.
The projections of a vector along the axes of a rectangular co-ordinate system are
called the components of the vector. The components of a vector completely define
the vector.
Figure 3.1: Projections of a vector in 2-D.
cos
=
Ax = Acos
sin
=
Ay = Asin
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We can invert these equations to find A and as functions of Ax and Ay . By
Pythagoras we have,
A=
and from the diagram,
tan
=
=
.
7.
To add vectors by components: = + + +...
(a)
Find the components of all vectors to be added.
(b)
Add all x components to get Rx = Ax + Bx + Cx + ...
(c)
Add all y components to get Ry = Ay + By + Cy +...
Then:
| |=
=
.
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12. COMPILED LIST OF COMPULSORY READINGS (Complete reference +
abstract/rationale)
Fitzpatrick, R. (2001). Classical Mechanics: An Introductory Course. Austin, Texas.UTP
Rationale
Fitzpartrick has the traditional approach to mechanics and dwells only briefly on vector
algebra and calculus which forms the modern basis for teaching mechanics. He has
adequately covered some aspects of the content in our curriculum and will make very good
basic reading for the student. However, there are no exercises for the student and only a
few examples are given at the end of each chapter.
The student should also note the interchangeable use of Imperical Units and S.I. Units.
We have, however, supplemented this compulsory reading material by including those
topics which are not found in the compulsory reading book.
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13. COMPILED LIST OF (OPTIONAL) MULTIMEDIA RESOURCES (Complete reference
+ abstract/rationale)
Reading # 1: Wolfram MathWorld (visited 03.11.06)
Complete reference : http://mathworld.wolfram.com
Abstract : Wolfram MathWorld is a specialised on-line mathematical encyclopedia.
Rationale: It provides the most detailed references to any mathematical topic. Students
should start by using the search facility for the module title. This will find a major article. At
any point students should search for key words that they need to understand. The entry
should be studied carefully and thoroughly.
Reading # 2: Wikipedia (visited 03.11.06)
Complete reference : http://en.wikipedia.org/wiki
Abstract : Wikipedia is an on-line encyclopedia. It is written by its own readers. It is
extremely up-to-date as entries are contunally revised. Also, it has proved to be extremely
accurate. The mathematics entries are very detailed.
Rationale: Students should use wikipedia in the same way as MathWorld. However, the
entries may be shorter and a little easier to use in the first instance. Thy will, however, not
be so detailed.
Reading # 3: MacTutor History of Mathematics (visited 03.11.06)
Complete reference :
http://www-history.mcs.standrews.ac.uk/Indexes
Abstract : The MacTutor Archive is the most comprehensive history of mathematics on the
internet. The resources are organsied by historical characters and by historical themes.
Rationale: Students should search the MacTutor archive for key words in the topics they
are studying (or by the module title itself). It is important to get an overview of where the
mathematics being studied fits in to the hostory of mathematics. When the student
completes the course and is teaching high school mathematics, the characters in the
history of mathematics will bring the subject to life for their students. Particularly, the role of
women in the history of mathematics should be studied to help students understand the
difficulties women have faced while still making an important contribution.. Equally, the role
of the African continent should be studied to share with students in schools: notably the
earliest number counting devices (e.g. the Ishango bone) and the role of Egyptian
mathematics should be studied.
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14. COMPILED LIST OF USEFUL LINKS (Name of page/site; URL;
Description/Rationale)
A.
Circular motion
http://en.wikipedia.org/wiki/Circular_motion
Rationale
This link is useful because it gives the definition of circular motion and discusses on
other related issues.
B.
Circular motion
http://hyperphysics.phy-astr.gsu.edu/HBASE/circ.html
C.
Circular motion
www.phy.ntnu.edu.tw/java/shm/shm.html
D.
Planetary motion
http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/kepler6.html
Rationale
This link is useful because it give animations on planetary motion which help the
learner to picture how the motion of planets occurs.
E.
Planetary motion
http://www-istp.gsfc.nasa.gov/stargaze/kep3laws.htm
F.
Planetary motion
http://en.wikipedia.org/wiki/Kepler’s_laws_of_planetary_motion
Rationale
This link is useful because it gives the definition of planetary motion and helps the
learner to understand some issues related to planetary motion.
G.
Planetary motion
www.windows.ucar.edu/tour/link=/the_universe/uts/planets.html
H.
Simple harmonic motion
http://en.wikipedia.org/wiki/Simple_harmonic_motion
Rationale
This link is useful because it gives the definition of simple harmonic motion and goes
on to explain some issues related to simple harmonic motion.
I.
Simple harmonic motion
www.phy.ntnu.edu.tw/java/shm/shm.html
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J.
Simple harmonic motion
http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html
K.
Simple harmonic motion
www.kettering.edu/~drussell/Demos/SHO/mass.html
L.
Simple harmonic motion
http://theory.unwinnipeg.ca/physics/shm/nodes2.html
M.
Classical Mechanics Lecture notes
http://farside.ph.utexas.edu/teaching/301/lectures/lectures.html
N.
Newtonian Physics
http://www.lightandmatter.com/arealbook1.html
O.
Classical Mechanics: An Introductory Course
http://www.lulu.com/content/159798
P.
Centripetal Force
http://en.wikipedia.org/wiki/Centripetal_force
Q.
Angular Momentum
http://en.wikipedia.org/wiki/Angular_momentum
R.
D’Alembert’s Principle
http://en.wikipedia.org/wiki/D%27Alembert%27s_principle
T.
Vectors and Scalars
http://en.wikipedia.org/wiki/Vector_(spatial)
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15. SYNTHESIS OF THE MODULE
In this module you familiarized yourself, initially, with an introduction to vector calculus
which led to the vector definitions of the basic concepts of plane and curvilinear motion,
position, vector and acceleration. In most of this section you considered kinematics, the
motion particles /bodies without referring to the force producing the motion.
You should, by now be convercant with Newton’s laws of motion which has the concept of
inertia as is concept and their applications to all the other sections of mechanics such as
oscillations, rigid bodies and dynamics.
The crutial concept that runs through this module is that of force and the resultant sections
of different forces.
By this time you should also be able to model real-life situations in mathematical terms and
solve the equations thus produced. Whenever and wherever possible, we also tried to
relate the content to high school mathematics and started with the section headed
‘’Bridging the gap’’ of common place situations that we should be able to explain
mathematically. You should now be comfortable with the rubic of mechanics and be able to
teach the relevant content to High School students.
Good Luck!
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16. SUMMATIVE EVALUATION
DYNAMICS
Question 1
A car travelling at 10ms-1 collides with a tree. (a) A passenger without a seat belt strikes the
wind shield headfirst and comes to rest in 0.002 s. The contact area between the head and
the windshield is 6 104 m2 , and the mass of the head is 5 kg. Find the average force and
the force per unit area exerted on the head. (b) A passenger of mass 70 kg wearing a
shoulder harness comes to rest in 0.2 s. The area of the harness in contact with the
passenger is 0.1 m2. Find the average force and the average force per unit area.
Answer
(a) Again we use F t p p . The final momentum is zero, since the windshield is
stationary, and the initial momentum is the head mass times the velocity. Thus the
magnitude of the average force is
p (5kg )(10ms 1 )
F
25000 N
t
(0.002s)
The average force per unit area is
F
25000 N
4.16 107 Nm2
4
2
A 6 10 m
This is a very large force per unit area and is certain to cause serious injury.
(b) The average force is found from the change in momentum of the entire body as the
speed of the car changes from 10 ms-1 to zero. Thus the magnitude of F is
p (70kg )(10ms 1 )
3500 N
t
(0.2s)
This is much smaller than the force exerted on the head of the unrestrained passenger in
part (a). The average force per unit area is
F
F 3500 N
3.5 104 Nm2
A 0.1 m2
Since this is smaller than the force per unit area on the unrestrained passenger by a factor
1200, the chance of serious injury is much smaller for this passenger.
Question 2
A car of mass m = 1000 kg moving at 30 ms-1 collides with a car of mass M = 2000 kg
travelling at 20 ms-1 in the opposite direction. Immediately after the collision, the 1000-kg
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car moves at right angles to its original direction at 15 ms -1. Find the velocity of the 2000-kg
car right after the collision.
Solution
Let us take the x and y axes where the x component of the total momentum of the two
vehicles is conserved, so
mvx MVx mvx' MVx'
Since vx' 0 , we can solve for Vx' and substitute:
Vx'
mvx MVx mvx
Vx
M
M
1000 kg
(30 ms 1 ) (20 ms 1 ) 5 ms 1
2000 kg
The initial momentum components along the y axes were zero, so mv'y MVy' 0 , and
Vy'
mv 'y
M
1000 kg
(15 ms 1 ) 7.5 ms 1
2000 kg
OSCILLATIONS
Question 3
An object is connected to one end of a horizontal spring whose other end is fixed.
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The object is pulled to the right (in the positive x direction) by an externally applied force of
magnitude 20 N causing the spring to stretch 1.000 cm.
(a) Determine the value of the force constant.
(b) If the mass of the object is 4 kg determine with which it oscillates if the applied force
is suddenly removed.
(c) Determine the frequency of the oscillation.
(d) Determine the angular frequency of the oscillation.
(e) Determine the position of the object 0.75 s after it begins oscillating.
(f) Determine the velocity of the object at t = 0.75 s.
(g) Determine the acceleration at t = 0.75 s.
(h) Determine the force exerted on the body by the spring at t = 0.75 s
Solution
(a) the force constant is the k in Hooke’s law
F kx
The force F produced by the spring is F 20.00 N , where the minus sign means
that the force acts to the left (in the negative x direction). Since x 1.000 102
you have
m,
F
20.00 N
2.000 103 N / m
2
x
1.000 10
m
(b) The ratio of the force constant to the mass determines the period T, which
k
k 2.000 103 N / m
5.000 102 N /(m kg )
m
4.000 kg
Thus the period is
T
2
2
0.2810 s
5.000 102 N /( m kg )
(c) Since
1
, you have
T
1
3.559 cycles / s 3.559 Hz
0.2810 s
(Since a cycle is simply a count, it is a dimensionless number and can be inserted at will in
the units of the answer.)
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(d) you have
2 2 rad / cycle 3.559 cycles / s 22.36 rad / s
(e) the oscillation is described by
x A cos(t )
If A and are adjusted to fit the initial conditions and if the proper value of is used. The
initial conditions (dx / dt )0 0 and x0 1.000 102 m require that
0
and
A 1.000 102 m
So x 1.000 102 m cos(22.36 rad / s t )
where the value of the is that found in part d. the position at t 0.7500 s is
x 1.000 102 m cos(22.36 rad / s 0.7500 s )
1.000 102 m cos(16.77 rad )
1.000 102 m (0.4871) 4.871 103 m 0.4871 cm
The argument of the cosine is larger than 2 rad because the oscillation has passed
beyond its first cycle. The minus sign means that the body is to left of its equilibrium
position, and the spring is compressed. Note that for the purpose of evaluation x it is most
convenient to express the cosine in terms of the angular frequency rather than the
frequency .
(f) evaluating the terms you have
dx
A sin(t )
dt
1.000 102 m 22.36 rad / s sin(16.77 rad )
1.000 102 m 22.36 rad / s ( 0.8733)
0.1953 m / s 19.53 cm / s
(A radian is a dimensionless number, being defined as the ratio of two lengths, so it can be
deleted at will from the units.) The positive velocity means the body is moving to the right.
Thus although the spring is compressed, the compression is decreasing.
(g) the acceleration is
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d 2x
A 2 cos(t )
2
dt
1.000 102 m (22.36 rad / s) 2 cos(16.77 rad )
1.000 102 m (22.36 rad / s) 2 (0.4871)
2.435 m / s 2 243 cm / s 2
The body is accelerating to the right since the acceleration is positive.
(h) one way to do this is to apply Newton’s Second law:
d 2x
dt 2
4.000 kg 2.435 m / s 2
9.740 N
F m
d 2x
is that found in part g. the force has a positive value, and so acts to
dt 2
the right since the acceleration is to the right.
where the value of
Another way of finding F is to apply Hooke’s Law:
F kx
2.000 103 N / m (4.871103 m)
9.740 N
where the values of k and x are those found in part a and e .
FORCE ENERGY AND MOTION
Question 4
Give the approximate value of the speed, in m/s, for each object listed below. Also express
each speed as a fraction of the speed of light, which is 3.0 108 ms 1 . In cases where full
information is not provided, make reasoned estimates in order to obtain your results.
(a) An ant crawling
(b) A person walking to work at a comfortable pace
(c) An athlete running the mile
(d) A car on a super highway
(e) A cruising jet (650 miles/ hr, or about 90 % the speed of sound).
(f) A satellite orbital radius of 7000 km, (orbital period of about 90 minutes).
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(g) The moon in its orbit around the earth (orbital radius of about 380 000 miles, orbital
period of 27.3 days).
(h) The earth in its orbit around the sun (orbital radius of approximately 93 000 000
miles, orbital period of 365.3 days).
N.B. [1 inch = 2.54 cm, 1 mile = 1.61 km and 1 foot = 0.3048 m]
Solution
Estimates
(a) 0.1 m/s = 3.3 × 10-10 c
(b)
Question 5
Two dormitory roommates, Hugh and Lou, decide to play an unusual game of catch.
Hugh stands on a balcony, and Lou stands on the ground directly below. The balcony is
10 m above the ground. Hugh and Lou each through the ball directly towards each other
with an initial speed of 15 m/s.
(a) How long does the ball take to travel from Lou up to Hugh? How fast is the ball
travelling when Hugh catches it ?
(b) How long does it take for the ball to descend from Hugh to Lou? How fast is the ball
travelling when Lou catches it ?
(c) Compare the round-trip travel time with the time that would be required if the ball
traveled with a constant speed of 15 m/s in both directions.
(d) Alice, another resident at the dormitory, lives 15 m above the two roommates. She
decides to douse Hugh and Lou with two water filled balloons. She watches them
toss the ball back and forth several times and learns to anticipate when Hugh will
release the ball. Alice wants to drop the balloons so that Hugh and Lou will be hit
simultaneously, just as the ball is reaching Lou. When should Alice drop each
balloon? Must she drop both the balloons before Hugh throws the ball downward?
How long after Hugh throws the ball will the first balloon fall past him? How long after
that will the second balloon strike him?
(e) Alice’s diabolical spot works perfectly. However, Hugh and Lou are accustomed to
Alice’s pranks, and each of them has rotten tomato handy. They hold the tomatoes ,
count to three, and simultaneously hurl the tomatoes at Alice. Each tomato has an
initial speed of 30 m/s. How soon after the tomatoes are thrown does Alice need to
be out of the way?
(f) Both tomatoes miss on the way up, but then Alice makes a mistake. She leans out to
gloat at her wet victims. Both tomatoes strike her on the back of the head. Whose
tomato hits Alice first? When does it hit her, and how fast is it travelling? When does
the other tomato hit, and how fast is it travelling?
Solution
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(a) 0.981 s, 5.39 m/s
(b) 0.563 s, 20.5 m/s
(c) Actual round-trip time is 1.54 s, while constant-speed time would be 1.33 s
(d) Balloon intended for Lou must be dropped 1.70 s before Hugh throws the ball;
balloon intended for Hugh must be dropped 1.19 s before Hugh throws the ball, Yes.
0.05 s, 0.51 s
(e) 0.55 s
(f) Lou’s tomato strikes first, 5.13 s after it is thrown (and 4.13 s after missing on the
way up); it hits her at 20.3 m/s. Hugh’s tomato hits 0.44 s later (5.57 s after it is
thrown); it hits at 24.6 m/s
Question 6
A 3000-pound vehicle is negotiating a circular interchange of radius 300 feet at 30 miles
per hour. Assuming the roadway to be level, find the force between the tires and the road
so that the car stays on the circular path and does not skid. (Use F = ma, where m =
3000/32). Find the angle at which the roadway should be banked so that no lateral frictional
force is exerted on the tires of the vehicle.
Question 7
The path of a shot thrown at an angle θ is
1
r (t ) (v0 cos )ti h (v0 sin )t gt 2 j
2
where v0 is the initial speed, h is the initial height, t is the time in seconds, and g is the
acceleration due to gravity. Verify that the shot will remain in the air for a total of
t
v0 sin v0 2 sin 2 2 gh
g
seconds
and will travel a horizontal distance of
v0 2 cos
g
2 gh
2
sin sin 2 feet .
v0
N.B. [1 inch = 2.54 cm, 1 mile = 1.61 km and 1 foot = 0.3048 m]
Question 8
Determine the moments of inertia of the homogeneous rectangular parallelepiped of mass
m about the centroidal x0 and z axes and about the x axis through one end.
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Solution
A transverse slice of thickness dz is selected as the element of volume. The moment of
inertia of this slice of infinitesimal thickness equals the moment of inertia of the area of the
section times the mass per unit area dz . Thus the moment of inertia of the transverse slice
about the y ' axis is
1
dI y ' y ' dz ab3
12
and that about the x ' axis is
1
dI x ' x ' dz a 3b
12
As long as the element is a plate of differential thickness, we have
ab 2
a b2
12
where m is the mass of the block.
dI zz dI x ' x ' dI y ' y ' dz
By interchanging symbols the moment of inertia about the x0 axis is
I 0 x0
1
m( a 2 l 2 )
12
The moment of inertia about the x axis may be found by the Parallel-axis theorem.
Thus
2
I xx I x0 x0
1
l
m m a 2 4l 2
2 12
This last result may be obtained by expressing the moment of inertia of the elemental slice
about the x axis and integrating the expression over the length of the bar. Again by the
parallel-axis theorem
a2
1 3 2
dI xx dI x ' x ' z dm dz a b z abdz ab z 2 dz
12
12
2
Integrating gives the result obtained previously
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a2
abl 2 a 2 1
2
2
2
I xx ab z dz
l m a 4l
12
3
4 12
0
1
The expression for I xx may be simplified for a long prismatical bar or slender rod whose
transverse dimensions are small compared with the length. In this case a 2 may be
neglected compared with 4l 2 , and the moment of inertia of such a slender bar about an axis
1
through one end normal to the bar becomes I ml 2 . By the same approximation the
3
1
moment of inertia about a centroidal axis normal to the bar is I ml 2 .
12
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17. REFERENCES
Duncan, T (1981) Advanced Physics, London. John Murray.
Edwards and Penney (1982) Calculus and Analytic Geometry, New Jersey. Prentice-Hall.
Eisberg, R.M and Lerner, L.S. Physics: Foundations and Applications, New York. McGraw
Hill.
Fadell, E.R. Fadell, A.G. (1970) Calculus. New York .Van Nostrand Reinhold.
Kane, J.W. and Sternheim (1978) Physics (2nd Ed). New York ,John Wiley and Sons.
Learson, Hostetter and Edwards (1998) Calculus (6th Ed) ,Boston. Houghton Miflin.
Meriam, J.L. and Kraige L.G. (1987) Engineering Mechanics Vol 2 Dynamics, New York.
John Wiley and sons.
Sears, F.W., Zemansky M.N. and Young, H.D. (1982) College Physics (5th Ed), Reading.
Addison-Wesley.
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18. MAIN AUTHOR OF THE MODULE
Team
Mr. Tendayi Chihaka (main author)
Mr. Blessing Mufoya (co-author)
Mr. Admire Kurira (co-author)
This module is a product of collaborative work by a team of three members of the
Department of Science and Mathematics Education, at the University of Zimbabwe. The
main Author, Tendayi Chihaka, holds a Masters in Mathematics Education and has vast
experience in pre-service teacher education, having trained pre-service secondary
mathematics teachers from 1978 to 1997 – a period of twenty years. He joined the
Department of Science and Mathematics Education in 2003 where he was elected
chairman of Department in 2004, the position he currently holds. He is married and has
three children.
Mr Kurira holds a Masters in Mathematics and Physics and a Post Graduate Certificate in
Education. He has taught Engineering at Harare Polytechnic 1996-1999. He joined the
University of Zimbabwe, Department of Science and Mathematics in August 1999 as an
ICT and Mathematics lecturer. He is married with 3 children.
Mr Mufoya holds a Masters in Mathematics and a Post Graduate Diploma in Education. He
joined the University of Zimbabwe, Department of Science and Mathematics in November
2005. He in currently single.
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