47
C H A P T E R 4
Work and Energy of Particles
4.1 INTRODUCTION
In the foregoing chapters direct application of Newton’s second law of motion and principles
of kinematics enables one to determine the position, velocity, and acceleration of a particle in
motion. In this chapter, methods for relating force, mass, displacement, and velocity are intro-
duced.
Specifically, work of a force and potential energy of a particle are introduced in Section 4.2.
Section 4.3 is concerned with potential energy and strain energy.
Kinetic energy of a particle and the principle of work and energy are presented in Sec-
tion 4.4. e principle of conservation of energy is included in Section 4.5. Power and mechan-
ical efficiency are briefly mentioned in Section 4.6.
4.2 WORK OF A FORCE AND POTENTIAL ENERGY
Consider a particle moving from a point P to a neighboring point P
0
, as shown in Figure 4.1.
F
⃑
dr
⃑
r
⃑ +
dr
⃑
r
⃑
P'
P
O
α
Figure 4.1: Force acting on a particle moving from one point to another.
Let
*
r be the position vector measuring from the origin of the frame of reference to the
particle at P , and d
*
r (strictly speaking, it should be
*
r and in the limit
*
r approaches d
*
r )
be the displacement vector (small vector joining P and P
0
in Figure 4.1) of the particle. If a force
*
F acts on the particle the work of the force associated with the displacement d
*
r may be defined
48 4. WORK AND ENERGY OF PARTICLES
as the dot product of two vectors,
d U D
*
F d
*
r (4.1a)
which is a scalar quantity and this equation can be written as
d U D F ds cos ˛; (4.1b)
where F; ds, and ˛ are, respectively, the magnitudes of the force, displacement, and angle en-
closing by
*
F and d
*
r .
Equations (4.1a) and (4.1b) can be generalized to the case shown in Figure 4.2 in which
the particle at position P
1
moves to position P
2
in a finite displacement. e work in this finite
displacement is denoted by U
1!2
,
U
1!2
D
P
2
Z
P
1
*
F d
*
r (4.2a)
which can be rewritten as
U
1!2
D
s
2
Z
s
1
F cos ˛ ds D
s
2
Z
s
1
F
t
ds (4.2b)
by using Equation (4.1b) since F cos ˛ represents the tangential component F
t
of the force.
Note that in this equation the integration is performed along the path traveled by the particle.
F
⃑
d
r⃑
P
1
P
2
O
α
s
2
s
1
s
P
ds
Figure 4.2: Force acts on a particle that moves in a finite displacement.
e foregoing equations can be applied to the work of a particle of mass m under the
influence of gravity, as shown in Figure 4.3. at is, the work of the weight
*
W D m
*
g of the
particle is obtained by replacing
*
F with
*
W in Equation (4.2a), for example.
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