Chapter 13 Kinetics of Particles: Energy and Momentum Methods
3.1 Introduction
• Previously, problems dealing with the motion of particles were solved through the fundamental equation of motion,Current chapter introduces two additional methods of analysis.
.amF
• Method of work and energy: directly relates force, mass, velocity and displacement.
• Method of impulse and momentum: directly relates force, mass, velocity, and time.
Work of a Force
• Differential vector is the particle displacement.rd
• Work of the force is
dzFdyFdxFdsF
rdFdU
zyx
cos
• Work is a scalar quantity, i.e., it has magnitude and sign but not direction.
force.length • Dimensions of work are Units are J 1.356lb1ftm 1N 1 J 1 joule
Work of a Force
• Work of a force during a finite displacement,
2
1
2
1
2
1
2
1
cos
21
A
Azyx
s
st
s
s
A
A
dzFdyFdxF
dsFdsF
rdFU
• Work is represented by the area under the curve of Ft plotted against s.
• Work of a constant force in rectilinear motion, xFU cos21
• Work of the force of gravity,
yWyyW
dyWU
dyW
dzFdyFdxFdU
y
y
zyx
12
212
1
• Work of the weight is equal to product of weight W and vertical displacement y.
• Work of the weight is positive when y < 0, i.e., when the weight moves down.
Work of a Force
• Magnitude of the force exerted by a spring is proportional to deflection,
lb/in.or N/mconstant spring
kkxF
• Work of the force exerted by spring,
222
1212
121
2
1
kxkxdxkxU
dxkxdxFdUx
x
• Work of the force exerted by spring is positive when x2 < x1, i.e., when the spring is returning to its undeformed position.
• Work of the force exerted by the spring is equal to negative of area under curve of F plotted against x,
xFFU 2121
21
Work of a Force
Work of a gravitational force (assume particle Moccupies fixed position O while particle m follows path shown),
12221
2
2
1r
MmGr
MmGdrr
MmGU
drr
MmGFdrdU
r
r
Work of a Force
Forces which do not do work (ds = 0 or cos :
• weight of a body when its center of gravity moves horizontally.
• reaction at a roller moving along its track, and
• reaction at frictionless surface when body in contact moves along surface,
• reaction at frictionless pin supporting rotating body,
Work of a Force
Particle Kinetic Energy: Principle of Work & Energy
dvmvdsFdsdvmv
dtds
dsdvm
dtdvmmaF
t
tt
• Consider a particle of mass m acted upon by force F
• Integrating from A1 to A2 ,
energykineticmvTTTU
mvmvdvvmdsFv
v
s
st
221
1221
212
1222
12
1
2
1
• The work of the force is equal to the change in kinetic energy of the particle.
F
• Units of work and kinetic energy are the same:
JmNmsmkg
smkg 2
22
21
mvT
Applications of the Principle of Work and Energy
• Wish to determine velocity of pendulum bob at A2. Consider work & kinetic energy.
• Force acts normal to path and does no work.
P
glv
mvml
TUT
2210
2
22
2211
• Velocity found without determining expression for acceleration and integrating.
• All quantities are scalars and can be added directly.
• Forces which do no work are eliminated from the problem.
• Principle of work and energy cannot be applied to directly determine the acceleration of the pendulum bob.
• Calculating the tension in the cord requires supplementing the method of work and energy with an application of Newton’s second law.
• As the bob passes through A2 ,
mglglmmgP
lvmmamgP
amF
n
nn
32
22
glv 22
Applications of the Principle of Work and Energy
Power and Efficiency
• rate at which work is done.
vFdt
rdFdt
dUPower
• Dimensions of power are work/time or force*velocity. Units for power are
smN 1
sJ1 (watt) W 1
•
inputpower outputpower
input workkoutput wor
efficiency
Sample Problem 13.2
Sample Problem 13.3
Sample Problem 13.4