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