SUBJECT: CE 223 and Dynamics of Rigid Bodies
MODULE 2: Kinetics of a Particle: Force and Acceleration
INTRODUCTION
The Mechanics of Statics of Rigid Bodies is defined as that branch of physical science which is
concerned with the resultant effect of forces on bodies, both in a state of rest or in motion. Mechanics is
subdivided into three branches viz. Mechanics of Rigid Bodies, Mechanics of Deformable Bodies and
Mechanics of Fluids. This module will focus on the Dynamics of Rigid Bodies
OBJECTIVES
Upon successful completion of this module, students should be able to:
⚫
To state Newton's Second Law of Motion and to define mass and weight.
⚫
To analyze the accelerated motion of a particle using the equation of motion with different coordinate
systems
⚫
To investigate central-force motion and apply it to problems in space mechanics.
DISCUSSION PROPER
Topic 1 Newton’s Second Law of Motion
Kinetics is a branch of dynamics that deals with the relationship between the change in motion of a body and
the forces that cause this change. The basis for kinetics is Newton's second law, which states that when an
unbalanced force acts on a particle, the particle will accelerate in the direction of the force with a magnitude
that is proportional to the force.
F = ma
;
equation of motion
Newton's Law of Gravitational Attraction
Shortly after formulating his three laws of motion, Newton postulated a law governing the mutual attraction
between any two particles. In mathematical form this law can be expressed as
Where:
F = force of attraction between the two particles
G = universal constant of gravitation; according to experimental evidence
G = 66.73(10-12) m3 / (kg · s2)
m1 , m2 = mass of each of the two particles
r = distance between the centers of the two particles
In the case of a particle located at or near the surface of the earth, the only gravitational force having any
sizable magnitude is that between the earth and the particle. This force is termed the "weight" and, for our
purpose, it will be the only gravitational force considered.
W = mg
In the SI system the mass of the body is specified in kilograms
In the FPS system the weight of the body is specified in pounds. The mass is measured in slugs, a term
derived from "sluggish" which refers to the body's inertia.
The Equation of Motion
Inertial Reference Frame
When applying the equation of motion, it is important that the acceleration of the particle be measured
with respect to a reference frame that is either fixed or translates with a constant velocity. In this way, the
observer will not accelerate and measurements of the particle's acceleration will be the same from any
reference of this type.
In this way, the observer will not accelerate and
measurements of the particle's acceleration will be
the same from any reference of this type. Such a
frame of reference is commonly known as a
Newtonian or inertial reference frame.
Equations of Motion: Rectangular Coordinates
When a particle moves relative to an inertial x, y, z frame of reference, the forces acting on the particle, as
well as its acceleration, can be expressed in terms of their i, j, k components. Applying the equation of
motion, we have
 F = ma ;  F i +  F j +  F k = m(a i + a j + a k)
x
y
z
x
y
z
For this equation to be satisfied, the respective i,j, k components on the left side must equal the
corresponding components on the right side.
Equations of Motion
• If the forces can be resolved directly from the free-body diagram, apply the equations of motion in their
scalar component form.
• If the geometry of the problem appears complicated, which often occurs in three dimensions, Cartesian
vector analysis can be used for the solution.
• Friction. If a moving particle contacts a rough surface, it may be necessary to use the frictional equation,
which relates the frictional and normal forces Ff and N acting at the surface of contact by using the
coefficient of kinetic friction, i.e., Ff =  k N. Remember that Ff always acts o n the free body diagram such
that it opposes the motion of the particle relative to the surface it contacts. If the particle is on the verge of
relative motion, then the coefficient of static friction should be used.
• Spring. If the particle is connected to an elastic spring having negligible mass, the spring force Fs can be
related to the deformation of the spring by the equation Fs = ks. Here k is the spring's stiffness measured as a
force per unit length, and s is the stretch or compression defined as the difference between the deformed
length I and the undeformed length l0, i.e.,
s = l - lo
Kinematics
• If the velocity or position of the particle is to be found, it will be necessary to apply the necessary kinematic
F = ma
equations once the particle's acceleration is determined from :

• If acceleration is a function of time, use a = dv/dt and v = ds/dt which, when integrated, yield the particle's
velocity and position, respectively.
• If acceleration is a function of displacement, integrate a ds = v dv to obtain the velocity as a function of
position.
• If acceleration is constant, use v = v0 + act, s = so + v0t +
velocity or position of the particle.
1 2 2
act , v = v02 + 2ac( s - so) to determine the
2
SUBJECT: CE 223 and Dynamics of Rigid Bodies
MODULE 3: Kinetics of a Particle: Work and Energy
INTRODUCTION
In this chapter, we will analyze motion of a particle using the concepts of work and energy. The
resulting equation will be useful for solving problems that involve force, velocity, and displacement.
OBJECTIVES
Upon successful completion of this module, students should be able to:
•To develop the principle of work and energy and apply it to solve problems that involve force, velocity, and
displacement.
• To study problems that involve power and efficiency.
• To introduce the concept of a conservative force and apply the theorem of conservation of energy to solve
kinetic problems.
DISCUSSION PROPER
Kinetics of Particles
Kinetics is the study of the relations between unbalanced forces and the resulting changes in motion
WORK (U) - the amount of energy transferred by a force acting through a distance. Work is only done by
forces that are in the direction of or opposing the direction of Motion
The KINETIC ENERGY (T) of an object is the energy that it possesses due to its motion. It is the work needed
to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its
acceleration, the body maintains this kinetic energy unless its speed changes.
The POTENTIAL ENERGY is the energy held by an object because of its position relative to other objects,
stresses within itself, its electric charge or other factors. It is associated with forces that act on a body in a
way that the total work done by these forces on the body depends only on the initial and final positions of
the body in space.
The cars of a roller coaster reach their maximum kinetic energy
when at the bottom of the path from the top. When they start
rising, the kinetic energy begins to be converted to gravitational
potential energy
When the archer does work on the bow, drawing the
string back, some of the chemical energy of the archer’s
body is transformed into elastic potential energy in the
bent limb of the bow. When the string is released, the
force between the string and the arrow does work on the
arrow. The potential energy in the bow limbs is
transformed into the kinetic energy of the arrow as it take
flights.
PRINCIPLE OF WORK & KINETIC ENERGY
1.) WORK – ENERGY EQUATION
- The total work (U) done by all forces acting on a particle as it moves from point 1 to point 2 equals the
corresponding change in Kinetic Energy of the particle
Kinetic Energy Expression:
T – Kinetic Energy (N.m or Joules/ ft.lb)v - velocity
Frequently occurring Forces
A. WORK ASSOCIATED WITH CONSTANT EXTERNAL FORCE
Work is a scalar quantity. It is defined as the product of the force and the displacement in the direction of
the force. It is denoted by letter U. Units of work are N.m or Joule (J).
A special case, when ϴ=0 i.e. force acts along the placement, then work by force U=PL
Also when ϴ=90 i.e. force is ꓕ to the displacement, then work by force = 0
U1-2 = PLcosα
B. WORK ASSSOCIATED WITH A SPRING FORCE
- The basic force on a spring force is proportional to the change of the spring length.
F=kx
k= spring modulus, it represents the force required to deform a given spring through
a unit distance.
x= displacement from the unstretched position
U1-2 =½k(x1²-x2²) –by Meriam and Kraige
or
U1-2 =-(½kx2²-½x1²) – by Hibbeler
C. WORK ASSOCIATED WITH WEIGHT
Case A: g = Constant
If the altitude variation is sufficiently small so that the acceleration of gravity g may be considered constant.
z
U1-2 = -mg (y2-y1)
Take note that if the body rises then (y2-y1) > 0 and this work is negative. If the body falls, (y2-y1) < 0 then
the work is positive
Note that the work is from point 1 (lower point) to point 2 (upper point)
Case B: g ≠ Constant
If large changes in altitude occur, then the weight (gravitational force) is no longer constant.
x
where:
g = acceleration of gravity at the earth’s surface
R= radius of the earth
r= radial distance from the center of the earth
D. WORK AND CURVILINEAR MOTION
Work done on a particle of mass m, moving along a curved path under the action of the force F, which stands
for the resultant ∑F of all forces acting on the particle.
U1-2 =½ m (v2²-v1²)
E. WORK OF A FRICTION FORCE
Consider a block of mass m slide down distance s on an inclined rough plane. If us and uk are the coefficient
of static and kinetic friction, the block's motion would be resisted by the frictional force = ukN.
U1-2 =-ukN x s
F.) POWER And EFFICIENCY
Consider a case of two persons in a race, set to climb the stairs and reach the top of a 10 storied building.
Here both the persons would be doing an equal amount of work in reaching the top, but if one person
reaches earlier than the other, he would be said to have exerted a greater power than the other one as he
has completed the work in lesser time. Thus the rate at which the work done is equally important
- The capacity of a machine is measured by the time rate at which it can do work or deliver energy.
- The capacity of a machine is rated by its power, which is defined as the time rate of doing work
The mechanical efficiency of a machine is the ratio of the useful power produced (output power) to the
power supplied to the machine ( input power)
If the time interval for output and input is the same
e < 1 (always less than 1)
Advantages of the Work- Energy Method
It avoids the necessity of computing the acceleration and leads directly to the velocity changes as
functions of the forces which do work
This principle is useful whenever the problem involves known or unknown parameters like forces,
mass, velocity and displacement.
Examples:
Bn
lll
SUMMARY
SUBJECT: CE 223 and Dynamics of Rigid Bodies
MODULE 4: Planar Kinematics of a Rigid Body
INTRODUCTION
In this chapter, the planar kinematics of a rigid body will be discussed. This study is
important for the design of gears, cams, and mechanisms used for many mechanical operations.
Once the kinematics is thoroughly understood, then we can apply the equations of motion, which
relate the forces on the body to the body's motion.
OBJECTIVES
Upon successful completion of this module, students should be able to:
⚫
To classify the various types of rigid-body planar motion.
⚫
To investigate rigid-body translation and angular motion about a fixed axis.
⚫
To study planar motion using an absolute motion analysis.
⚫
To provide a relative motion analysis of velocity and acceleration using a translating frame of
reference.
⚫
To show how to find the instantaneous center of zero velocity and determine the velocity of a
point on a body using this method.
⚫
To provide a relative-motion analysis of velocity and acceleration using a rotating frame of
reference.
DISCUSSION PROPER
Topic 1 Planar Rigid-Body Motion
The planar motion of a body occurs when all the particles of a rigid body move along paths which
are equidistant from a fixed plane. There are three types of rigid body planar motion, in order of
increasing complexity, they are
•
Translation
This type of motion occurs when a line in the body remains parallel to its original
orientation throughout the motion. When the paths of motion for any two points on the
body are parallel lines, the motion is called rectilinear translation, Fig. a. If the paths of
motion are along curved lines which are equidistant, the motion is called curvilinear
translation, Fig. b.
•
Rotation about a fixed axis
When a rigid body rotates about a fixed axis, all the particles of the body, except those
which lie on the axis of rotation, move along circular paths, Fig. c.
•
General plane motion
When a body is subjected to general plane motion, it undergoes a combination of
translation and rotation, Fig. d. The translation occurs within a reference plane, and the
rotation occurs about an axis perpendicular to the reference plane.
A. Translation
Position. The locations of points A and B on the body are defined with respect to fixed x, y
reference frame using position vectors rA and rB. The translating x', y' coordinate system is fixed in
the body and has its origin at A, hereafter referred to as the base point. The position of B with
respect to A is denoted by the relative-position vector rB/A ("r of B with respect to A"). By vector
addition,
rB = rA + rB/A
Velocity. A relation between the instantaneous velocities of A and B is obtained by taking the time
derivative of this equation, which yields vB = vA + drB/A/dt. Here vA and vB denote absolute velocities
since these vectors are measured with respect to the x, y axes. The term drB/A/dt = 0, since the
magnitude of rB/A is constant by definition of a rigid body, and because the body is translating the
direction of rB/A is also constant. Therefore,
vB = vA
Acceleration. Taking the time derivative of the velocity equation yields a similar relationship
between the instantaneous accelerations of A and B:
aB =aA
The above two equations indicate that all points in a rigid body subjected to either rectilinear or
curvilinear translation move with the same velocity and acceleration.
B. Rotation about a Fixed Axis
When a body rotates about a fixed axis, any point P located in
the body travels along a circular path. To study this motion, it
is first necessary to discuss the angular motion of the body
about the axis.
Angular Motion. Since a point is without dimension, it cannot have angular motion. Only lines or
bodies undergo angular motion. For example, consider the body shown above and the angular
motion of a radial line r located within the shaded plane.
Angular Position. At the instant shown, the angular position of r is defined by the angle 𝜃,
measured from a fixed reference line to r.
Angular Displacement. The change in the angular position, which can be measured as a differential
d𝜽, is called the angular displacement.
This vector has a magnitude of d𝜃, measured in degrees, radians, or revolutions, where 1 rev = 2𝜋
rad. Since motion is about a fixed axis, the direction of d𝜽 is always along this axis.
the direction is determined by the right-hand rule; that is, the
fingers of the right hand are curled with the sense of rotation, so
that in this case the thumb, or d𝜽, points upward, Fig. a. In two
dimensions, as shown by the top view of the shaded plane, in Fig. b,
both 𝜃 and d 𝜃 are counterclockwise, and so the thumb points
outward from the page.
Angular Velocity. The time rate of change in the angular position is called the angular velocity 𝝎
(omega). Since d𝜽 occurs during an instant of time dt, then,
This vector has a magnitude which is often measured in rad/s,
Angular Acceleration. The angular acceleration α (alpha) measures the time rate of change of the
angular velocity. The magnitude of this vector is
it is also possible to express a as
The line of action of α is the same as that for 𝝎, Fig. a; however, its sense of direction depends on
whether 𝝎 is increasing or decreasing. If 𝝎 is decreasing, then α is called an angular deceleration and
therefore has a sense of direction which is opposite to 𝝎
Constant Angular Acceleration. If the If the body's angular acceleration is constant, α = αC then the
following equations can be used:
Motion of Point P.
As the rigid body in Fig. c rotates, point P travels along a
circular path of radius r with center at point O. This path is
contained within the shaded plane shown in top view, Fig.
d.
Position and Displacement. The position of P is defined by the position vector r, which extends from 0 to P.
If the body rotates d𝜃 then P will displace ds = rd𝜃 .
Velocity. The velocity of P has a magnitude which can be found by dividing ds = rd𝜃 by dt so that
As shown in Figs. c and d, the direction of v is tangent to the circular path.
Both the magnitude and direction of v can also be accounted for by using the cross product of 𝝎 and rp.
Here, rp is directed from any point on the axis of rotation to point P, Fig. c.
Acceleration. The acceleration of P can be expressed in terms of its normal and tangential components.
Since at = dv/ dt and an = v2/ 𝜌, where 𝜌 = r, v = 𝜔r, and α = d𝜔/dt, we have
The tangential component of acceleration, Figs. e and f,
represents the time rate of change in the velocity's
magnitude. If the speed of P is increasing, then at acts in
the same direction as v; if the speed is decreasing, at acts in
the opposite direction of v; and finally, if the speed is
constant, at is zero.
The normal component of acceleration represents the time
rate of change in the velocity's direction. The direction of an
is always toward 0, the center of the circular path, e and f.
Since at and an are perpendicular to one another, if needed
the magnitude of acceleration can be determined from the
Pythagorean theorem; namely
C. Absolute Motion Analysis
A body subjected to general plane motion undergoes a simultaneous translation and rotation. If the body is
represented by a thin slab, the slab translates in the plane of the slab and rotates about an axis
perpendicular to this plane. The motion can be completely specified by knowing both the angular rotation of
a line fixed in the body and the motion of a point on the body. One way to relate these motions is to use a
rectilinear position coordinate s to locate the point along its path and an angular position coordinate 𝜃 to
specify the orientation of the line. The two coordinates are then related using the geometry of the problem.
By direct application of the time-differential equations v = ds/dt, a = dv/dt, 𝜔 = d 𝜃 /dt, and a = d𝜔/ dt, the
motion of the point and the angular motion of the line can then be related.
Topic 2 Relative-Motion Analysis: Velocity
The general plane motion of a rigid body can be described as
a combination of translation and rotation. To view these
"component" motions separately we will use a relativemotion analysis involving two sets of coordinate axes. The x, y
coordinate system is fixed and measures the absolute position
of two points A and B on the body, here represented as a bar,
Fig. a.
Position The position vector rA in Fig. a specifies the location of the "base point" A, and the relativeposition vector rB/A locates point B with respect to point A. By vector addition, the position of B is
then
rB = rA + rB/A
Displacement During an instant of time dt, points A and B undergo displacements drA and drB as
shown in Fig.b. If we consider the general plane motion by its component parts then the entire bar
first translates by an amount drA so that A, the base point, moves to its final position and point B moves to
B’, Fig. c. The bar is then rotated about A by an amount d𝜃 so that B' undergoes a relative displacement drB/A
and thus moves to its final position B. Due to the rotation about A, drB/A = r B/A d𝜃 and the displacement of B
is
Velocity To determine the relation between the velocities of points A and B, it is necessary to take
the time derivative of the position equation, or simply divide the displacement equation by dt.
the equation states is that the velocity of B, Fig. d, is determined by considering the entire bar to translate
with a velocity of vA, Fig. e, and rotate about A with an angular velocity 𝝎, Fig. f. Vector addition of these two
effects, applied to B, yields vB, as shown in Fig. g. Since the relative velocity vB/A represents the effect of
circular motion, about A, this term can be expressed by the cross product vB/A = 𝝎 x rB/A ,
Topic 3 Instantaneous Center of Zero Velocity
The velocity of any point B located on a rigid body can be obtained in a very direct way by choosing the base
point A to be a point that has zero velocity at the instant considered. In this case, vA = 0, and therefore the
velocity equation, vB = vA + 𝝎 x rB/A, becomes vB = 𝝎 x rB/A . For a body having general plane motion, point A
so chosen is called the instantaneous center of zero velocity (IC), and it lies on the instantaneous axis of zero
velocity.
Location of the IC. To locate the IC we can use the fact that the velocity of a point on the body is always
perpendicular to the relative position vector directed from the IC to the point. Several possibilities exist:
• The velocity vA of a point A on the body and the angular velocity 𝝎 of the body are known, Fig. a. In this
case, the IC is located along the line drawn perpendicular to vA at A, such that the distance from A to the IC is
rA/IC = vA/ 𝜔 . Note that the IC lies up and to the right of A since vA must cause a clockwise angular velocity 𝝎
about the IC.
• The lines of action of two nonparallel velocities vA and vB are known, Fig b. Construct at points A and B line
segments that are perpendicular to vA and vB . Extending these perpendiculars to their point of intersection
as shown locates the IC at the instant considered.
• The magnitude and direction of two parallel velocities vA and vB are known. Here the location of the IC is
determined by proportional triangles. Examples are shown in Fig. c and d. In both cases rA/IC = vA/ 𝜔 and rB/IC =
vB/ 𝜔 If d is a known distance between points A and B, then in c, rA/IC + rB/IC = d and in Fig. d, rB/IC - rA/IC = d.
Topic 4 Relative – Motion Analysis: Acceleration
An equation that relates the accelerations of two points on a bar (rigid
body) subjected to general plane motion may be determined by
differentiating vB = vA + vB/A with respect to time. This yields
Since the relative-acceleration components represent the effect of circular motion observed from translating
axes having their origin at the base point A, these terms can be expressed as (aB/A)t = 𝜶 x rB/A and (aB/A)n = 𝜔2rB/A.
Topic 5 Relative – Motion Analysis using Rotating axes
Rigid bodies (mechanisms) are constructed such that sliding will occur at their connections. The kinematic
analysis for such cases is best performed if the motion is analyzed using a coordinate system which both
translates and rotates.
Position
Consider the two points A and B shown in the Fig a. Their location is specified by the position vectors rA and
rB which are measured with respect to the fixed X, Y, Z coordinate system. As shown in the figure, the "base
point" A represents the origin of the x, y, z coordinate system, which is assumed to be both translating and
rotating with respect to the X, Y, Z system. The position of B with respect to A is specified by the relativeposition vector rB/A. The components of this vector may be expressed either in terms of unit vectors along
the X, Y axes, i.e., I and J, or by unit vectors along the x, y axes, i.e., i and j. For the development which
follows, rB/A will be measured with respect to the moving x, y frame of reference. Thus, if B has coordinates
(xB, yB), Fig. 16-32a, then
Velocity
Acceleration
SUMMARY