Ministry of Higher Education
Higher Institute for Engineering and
Technology in New Damietta
ENGINEERING MECHANICS 2
DYNAMICS
Prepared by
Prof. Dr. Mohamed Saad El Kady
Dean of the Institute
CONTENTS
1.
Chapter 1: Introduction
1.1.
1.2.
1.3.
1.4.
1.5.
1.6.
1.7.
2.
Chapter 2: Motion in One dimension
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
2.8.
3.
17
Displacement
Velocity
Acceleration
Motion with constant velocity
Motion with constant acceleration
Free-fall under gravity
Solved Examples
Problems
Chapter 3: Motion in Three dimensions
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
3.8.
3.9.
3.10.
3.11.
3.12.
3.13.
3.14.
3.15.
3.16.
3.17.
3.18.
5
Major sources:
What is classical mechanics?
mks units
Standard prefixes
Other units
Precision and significant figures
Dimensional analysis
Introduction
Cartesian coordinates
Vector displacement
Vector addition
Vector magnitude
Scalar multiplication
Diagonals of a parallelogram
Vector velocity and vector acceleration
Motion with constant velocity
Motion with constant acceleration
Projectile motion
Relative velocity
Solved Examples
Normal and Tangential Coordinates
Circular Motion
Polar Coordinates
Solved Examples
Problems
39
4.
Chapter 4: Newton's laws of motion
4.1.
4.2.
4.3.
4.4.
4.5.
4.6.
4.7.
4.8.
4.9.
4.10.
4.11.
5.
Chapter 5: Conservation of energy
5.1.
5.2.
5.3.
5.4.
5.5.
5.6.
5.7.
5.8.
5.9.
5.10.
6.
141
Introduction
Energy conservation during free-fall
Work
Conservative and non-conservative force-fields
Potential energy
Hooke's law
Motion in a general 1-dimensional potential
Power
Solved Examples
Problems
Chapter 6: Conservation of momentum
6.1.
6.2.
6.3.
6.4.
6.5.
6.6.
6.7.
6.8.
6.9.
95
Introduction
Newton's first law of motion
Newton's second law of motion
Hooke's law
Newton's third law of motion
Mass and weight
Strings, pulleys, and inclines
Friction
Frames of reference
Solved Examples
Rroblems
Introduction
Two-component systems
Multi-component systems
Rocket science
Impulses
Collisions in 1-dimension
Collisions in 2-dimensions
Solved Examples
Problems
Chapter 1
187
Introduction
Summary
1.1 What is classical mechanics?
Classical mechanics is the study of the motion of bodies
1.2 mks units
The first principle of any exact science is measurement. In
mechanics there are three fundamental quantities which are
subject to measurement:
1. Intervals in space: i.e., lengths.
2. Quantities of inertia, or mass, possessed by various
bodies.
3. Intervals in time.
Any other type of measurement in mechanics can be reduced
to some combination of measurements of these three
quantities.
Each of the three fundamental quantities --length, mass, and
timeThe mks unit of length is the meter (symbol m),
The mks unit of mass is the kilogram (symbol kg),
The mks unit of time is the second (symbol s),
In addition to the three fundamental quantities, classical
mechanics also deals with derived quantities, such as velocity,
acceleration, momentum, angular momentum, etc. Each of
these derived quantities can be reduced to some particular
combination of length, mass, and time.
For instance, a velocity can be reduced to a length divided by
a time. Hence, the mks units of velocity are meters per second:
(1)
Here, stands for a velocity, for a length, and for a time,
whereas the operator […] represents the units, or dimensions,
of the quantity contained within the brackets. Momentum can
be reduced to a mass times a velocity. Hence, the mks units of
momentum are kilogram-meters per second:
(2)
Here, p stands for a momentum, and
for a mass. In this
manner, the mks units of all derived quantities appearing in
classical dynamics can easily be obtained
Chapter 2
Motion in One dimension
Summary
2.1 Displacement
Consider a body moving in 1 dimension. With some
arbitrarily chosen reference point located on the track on
which it is constrained to move. This point is known as the
origin of our coordinate system.
A positive value implies that the body is located
meters to the right of the origin,
A negative value implies that the body is located │x│
meters to the left of the origin.
Here,
is termed the displacement of the body from
the origin.
2.2 Velocity
Velocity is the rate of change of displacement with time.
This definition implies that
(12)
where
is the body's velocity at time , and
is the change
in displacement of the body between times and t+ ∆t.
For instantaneous velocity,
must be kept sufficiently small
that the body's velocity does not change appreciably between
times and t+ ∆t. If
is made too large then formula (12)
becomes invalid; the velocity of the body is always
approximately constant in the interval to t+ ∆t. Thus,
(13)
Where dx/dt represents the derivative of
with respect to .
2.3 Acceleration
Acceleration is the rate of change of velocity with time.
This definition implies that
(15)
where is the body's acceleration at time , and
is the
change in velocity of the body between times and t+ ∆t.
A general expression for instantaneous acceleration, which is
valid irrespective of how rapidly or slowly the body's
acceleration changes in time, can be obtained by taking the
limit of Eq. (15) as
approaches zero:
(16)
Poblems
2-1 The acceleration of a p article as it moves alone a
straight line is given by a = (2t-1) m/s2, where t is in
seconds. If s = 1 m and v = 2m/s when t=0. Determine the
particle’s velocity and position when t=6s. Also determine
the total distance the particles travel during this time period.
2-2 The velocity of a particle traveling in a straight line is
given by v= (6t- 3t2) m/s, where t is in seconds. If s=0
when t = 0, determine the particle’s deceleration and when
t = 3 s. How far has the particle traveled during the 3-s time
interval, and what is its average speed?
2-3 A particle has an initial speed of 27 m/s. If it experiences
a deceleration of a = (-6t) m/s2, where t is in seconds.
Determine the distance traveled before it stops.
2-4 A particle has an initial speed of 27 m/s. If it experiences
a deceleration of a = (-6t) m/s2, where t is in seconds.
Determine the velocity when it travels 10 m. How much time
does this take?
2-5 A car moves in a straight line such that for a short time its
velocity is defined by v = (3t2 + 2t) ft/s, where t is in seconds.
Determine its position and acceleration when t = 3 s. When t=0,
s=0.
2-6 A particle is moving along a straight line such that its
acceleration is defined as a = -2v m/s2 where v is in meters
per second. If v = 20 m/s when s = 0 and t = 0, determine
the particle’s velocity as a function of position and the
distance the particle moves before it stops.
2-7 The acceleration of a rocket traveling upward is given by
a= (6 + 0.02 s) m/s2, where s is in meters, determine the
rocket’s velocity when s = 2 km and the time needed to reach
this altitude. Initially v = 0, s = 0 when t = 0.
2-8 A particle is moving along a straight line such that its
acceleration is defined as a = (4 s2) m/s2, where s is in meters.
If v = -100 m/s when s = 10 m and t = 0, determine the
particle’s velocity as a function of position
2.4 Motion with constant velocity
The simplest type of motion consists of motion with
constant velocity. i.e the displacement is time relationship is a
straight-line. This line can be represented algebraically as
(18)
Here, xo is the displacement at time
:
2.5 Motion with constant acceleration
Motion with constant acceleration occurs in everyday
life whenever an object is dropped: the object moves
downward with the constant acceleration
, under
the influence of gravity.
Set of three useful formulae which characterize motion with
constant acceleration are as follows:
(21)
(22)
(23)
Here, s = x - xo is the net distance traveled after
seconds.
2.6 Free-fall under gravity
Galileo Galilei was the first scientist to appreciate that,
neglecting the effect of air resistance, all bodies in free-fall
close to the Earth's surface accelerate vertically downwards
with the same acceleration: namely, g = 9.81 ms-2.
Equations (21)-(23) can easily be modified to deal with the
special case of an object free-falling under gravity:
(24)
(25)
(26)
Here, g = 9.81 ms-2 is the downward acceleration due to
gravity,
is the distance the object has moved vertically
between times
and (if s > 0 then the object has risen
meters, else if s < 0 then the object has fallen s meters), and
vo is the object's instantaneous velocity at
. Finally, is
the object's instantaneous velocity at time .
Poblems
2-9 Traveling with an initial speed of 70 km/h, a car
accelerates at 6000 km/h2 along a straight road. How long
will it take to reach a speed of 120 km/h? Also, through
what distance does the car travel during this time?
2-10 Ball A is released from rest at height of 40 ft at the same
time that a second ball B is thrown upward 5 ft from the
ground. If the balls pass one another at a height of 20 ft,
determine the speed at which ball B was thrown upward.
2-11 A car can have an acceleration and deceleration of 5
m/s2. If it starts from rest, and can have a maximum speed of
60 m/s, determine the shortest time it can travel a distance of
1200 m when it stops
Chapter 3
Motion in Three dimensions
Summary
3.1 Introduction
The purpose of this section is to generalize the
previously introduced concepts of displacement, velocity, and
acceleration in order to deal with motion in 3 dimensions.
3.2 Cartesian coordinates
Our first task, when dealing with 3-dimensional motion,
is to set up a suitable coordinate system. The most straightforward type of coordinate system is called a Cartesian
system, after René Descartes. A Cartesian coordinate system
consists of three mutually perpendicular axes, the -,y-, and
-axes (say). 3.3 Vector displacement
Consider the motion of a body moving in 3 dimensions.
The body's instantaneous position is most conveniently
specified by giving its displacement from the origin of our
coordinate system. Note, however, that in 3 dimensions such a
displacement possesses both magnitude and direction. In other
words, we not only have to specify how far the body is
situated from the origin, we also have to specify in which
direction it lies. A quantity which possesses both magnitude
and direction is termed a vector. By contrast, a quantity which
possesses only magnitude is termed a scalar. Mass and time
are scalar quantities. However, in general, displacement is a
vector.
The vector displacement of some point I from the origin O
can be visualized as an arrow running from point to point
. See Fig. 11. Note that in typeset documents vector
quantities are conventionally written in a bold-faced font (e.g.,
) to distinguish them from scalar quantities. In free-hand
notation, vectors are usually under-lined (e.g., r).
Figure 11: A vector
displacement
The vector displacement can also be specified in terms of its
coordinates:
(30)
3.8 Vector velocity and vector acceleration
Consider a body moving in 3 dimensions. Suppose that
we know the Cartesian coordinates, , y, and , of this body
as time, , progresses. Let us consider how we can use this
information to determine the body's instantaneous velocity and
acceleration as functions of time.
The vector displacement of the body is given by
(43)
By analogy with the 1-dimensional equation (13), the body's
vector velocity v(vx, vy, vz) is simply the derivative of with
respect to . In other words,
(44)
When written in component form, the above definition yields
(45)
(46)
(47)
Thus, the
-component of velocity is simply the time
derivative of the -coordinate, and so on.
By analogy with the 1-dimensional equation (16), the body's
vector acceleration a = (ax, ay, az) is simply the derivative of
with respect to . In other words,
(48)
When written in component form, the above definition yields
(49)
(50)
(51)
3.9 Motion with constant velocity
An object moving in 3 dimensions with constant
velocity possesses a vector displacement of the form
(61)
where the constant vector ro is the displacement at time
. Note that dr / dt = v and d2 r / dt2 = 0, as expected. As
illustrated in Fig. 14, the object's trajectory is a straight-line
which passes through point ro at time
and runs parallel
to vector .
Problems
3-1 A particle originally at rest and located at point (3 ft, 2
ft, 5 ft), is subjected to an acceleration of a = {6t i + 12t2 k}
ft/s2. Determine the particle’s position (x, y, z) at t = 1 s.
3-2 The velocity of a particle is given by v = {16t 2 i+ 4 t3 j
+ (5t + 2) k} m/s, where t is in seconds. If the particle is at
the origin when t=0, determine the magnitude of the
particle’s acceleration when t = 2 s, Also, what is the x, y, z
coordinate position of the particle at this instant
3-3 A particle moves along the curve y=e2x such that its
velocity has a constant magnitude of v=4 ft/s. Determine
the x and y components of velocity when the particle is at y
= 5 ft.