CHAPTER 13CHAPTER 13

Kinetics of Particles:Kinetics of Particles:

Energy and Momentum Energy and Momentum MethodsMethods

Method of work and energy – Method of work and energy – Involves relations between displacement, velocity, mass, and Involves relations between displacement, velocity, mass, and

force.force.

Method of impulse and momentum Method of impulse and momentum – –

Involves relations between mass, velocity, force, and time.Involves relations between mass, velocity, force, and time.

Some problems become easier to do.

13.1 INTRODUCTION13.1 INTRODUCTION

When a force is applied to a mass and the When a force is applied to a mass and the mass moves through an incremental distance, mass moves through an incremental distance, the work done by the force is the work done by the force is

13.2 WORK OF A FORCE13.2 WORK OF A FORCE

rdFdU

dzFdyFdxFzyx

ds)(cosFrdFdU

090if0dU 090if0dU 090if0dU

)J(JoulemNorlbftUnits r

rdr

rdF

A

A’

2

121 rdFU

To get the total work done along a path To get the total work done along a path

requires requires

2

1

t

2

1

21dsFds)(cosFU

Notice thatNotice that

When using rectangular coordinatesWhen using rectangular coordinates

2

1

zyx21)dzFdyFdxF(UF

rd

ds

tF

0 s1s 2s

The work is the area under the curve.The work is the area under the curve.

Work of a Constant Force in Rectilinear Work of a Constant Force in Rectilinear MotionMotion

x)(cosFU21

)xx)((cosF12

F

x1

2

)kdzjdyidx(jWdU

Work of the Force of GravityWork of the Force of Gravity

WdydU

2

1

21WdyU )yy(W

21

yWU21

y2

y1

y

dy

A1

A

A2W

Work of the Force Exerted by a SpringWork of the Force Exerted by a Spring

kxF

2

1

21kxdxU )kxkx( 2

1212

221

x1

x2

F

x

Spring undeformed

x

F

F1

F2

x1 x2

x

kxF

))(( 121221 xxxxk

Work of a Gravitational ForceWork of a Gravitational Force

2

1221 dr

r

GMmU

12 rGMm

rGMm

Forces which Forces which do notdo not do work do work (ds(ds = 0 or cos = 0 or cos = 0):= 0):

• weight of a body when its center of weight of a body when its center of gravity moves horizontally.gravity moves horizontally.

• reaction at a roller moving along its track, andreaction at a roller moving along its track, and

• reaction at frictionless surface reaction at frictionless surface when body in contact moves along when body in contact moves along surface,surface,

• reaction at frictionless pin supporting rotating body,reaction at frictionless pin supporting rotating body,

13.3 KINETIC ENERGY OF A PARTICLE. 13.3 KINETIC ENERGY OF A PARTICLE. PRINCIPLE OF WORK AND ENERGYPRINCIPLE OF WORK AND ENERGY

tt maF

If the force doing work is the net force thenIf the force doing work is the net force then

dt

dvm

dt

ds

ds

dvm

ds

dvmv

2

1

t21 dsFU 2

1

2

1

mvdvdsds

dvmv

212

1222

121 mvmvU

221 mvT 1221 TTU 2211 TUT

13.4 APPLICATIONS OF THE PRINCIPLE 13.4 APPLICATIONS OF THE PRINCIPLE OF WORK AND ENERGYOF WORK AND ENERGY

Work the pendulum problem in the text.Work the pendulum problem in the text.

• Wish to determine velocity of Wish to determine velocity of pendulum bob at pendulum bob at AA22. Consider . Consider work & kinetic energy.work & kinetic energy.

glv

vg

WWl

TUT

2

2

10

2

22

2211

• Velocity found without determining expression for Velocity found without determining expression for acceleration and integrating.acceleration and integrating.

• All quantities are scalars and can be added directly.All quantities are scalars and can be added directly.

• Forces which do no work are eliminated from the problem.Forces which do no work are eliminated from the problem.

• Principle of work and energy cannot Principle of work and energy cannot be applied to directly determine the be applied to directly determine the acceleration of the pendulum bob.acceleration of the pendulum bob.

• Calculating the tension in the cord Calculating the tension in the cord requires supplementing the method requires supplementing the method of work and energy with an of work and energy with an application of Newton’s second law.application of Newton’s second law.

• As the bob passes As the bob passes through through AA22 , ,

Wl

gl

g

WWP

l

v

g

WWP

amF nn

32

22

glv 22

13.5 POWER AND EFFICIENCY13.5 POWER AND EFFICIENCY

Power is the rate at which work is done.Power is the rate at which work is done.

dt

dUP

dt

rdF

vF

Units – 1 hp = 550 ftUnits – 1 hp = 550 ft lb/s and 1 Watt = 1 J/slb/s and 1 Watt = 1 J/s

EfficiencyEfficiency

inwork

outwork

inpower

outpower

13.6 POTENTIAL ENERGY13.6 POTENTIAL ENERGY

Close to the EarthClose to the Earth

DefineDefine

2

1

21WdyU )yy(W

21

WyVg

ThenThen

2121 )()( gg VVU

g21VU

Not So Close to the EarthNot So Close to the Earth

DefineDefine

ThenThen

2

1221 dr

r

GMmU

12 rGMm

rGMm

2121 )()( gg VVU

g21VU

rGMmVg r

WR2

For a SpringFor a Spring

DefineDefine

ThenThen

2121 )()( ee VVU

e21VU

221 kxVe

2

1

21kxdxU )kxkx( 2

1212

221

Notice that the work done Notice that the work done by each of these three forcesby each of these three forcesis equal to a change in something that is is equal to a change in something that is a function of position only.a function of position only.The idea of a function of positionThe idea of a function of positionis valid as long asis valid as long asF is conservative. is conservative.It would not work for a force like friction.It would not work for a force like friction.This function, This function, V, is call the potential energy., is call the potential energy.Potential energy is an energy of Potential energy is an energy of position orposition orconfiguration.configuration.

13.7 CONSERVATIVE FORCES13.7 CONSERVATIVE FORCES

Work done by conservative forces isWork done by conservative forces isindependent of the pathindependent of the pathover which work is done.over which work is done.

),,(),,( 22211121 zyxVzyxVU

For shortFor short

2121 VVU

0 rdF

For any conservative forceFor any conservative force

),,( zyxdVdU

An elemental work corresponding An elemental work corresponding to an elemental displacementto an elemental displacement

)dzz,dyy,dxx(V)z,y,x(VdU

),,( zyxdVdU

dz

zVdy

yVdx

xVdzFdyFdxF

zyx

x

VFx

y

VFy

z

VFz

k

zVj

yVi

xVkFjFiFF

zyx

The vector in the parentheses is known as the The vector in the parentheses is known as the gradient of the scalar function gradient of the scalar function V..

VF grad

VF

13.8 CONSERVATION OF ENERGY13.8 CONSERVATION OF ENERGY

VU21

T

1221TTVV

2211VTVT

If the only forces doing work on a system of particles areIf the only forces doing work on a system of particles areconservative, then the total mechanical energy is conserved.conservative, then the total mechanical energy is conserved.

Kinetic plus Potential

Pendulum MotionPendulum Motion

2211VTVT

l

0mvWl0 2

21

gl2v

13.9 MOTION UNDER A CONSERVATIVE 13.9 MOTION UNDER A CONSERVATIVE CENTRAL FORCE. APPLICATION CENTRAL FORCE. APPLICATION

TO SPACE MECHANICSTO SPACE MECHANICS

The gravitational attractive force is conservative.The gravitational attractive force is conservative.So, in space mechanics both energy and angular momentumSo, in space mechanics both energy and angular momentumare conserved since this force is are conserved since this force is both conservative and central.both conservative and central.

O

0v

v

r

0r

0

0P

P sinrmvsinmvr

000

VTVT00

rGMmmvr

GMmmv 2

21

0

2

021

13.10 PRINCIPLE OF IMPULSE AND 13.10 PRINCIPLE OF IMPULSE AND MOMENTUMMOMENTUM

)vm(dt

dF

2

1

t

t12 dtFvmvm

21Imp

For a system of particles For a system of particles external impulses are considered onlyexternal impulses are considered only(remember Newton’s third law)(remember Newton’s third law)

2

t

t1 vmdtFvm2

1

12 vmvm

If no external forces act on the particles thenIf no external forces act on the particles then

13.11 IMPULSE MOTION13.11 IMPULSE MOTION

Non-impulsive forces can be neglected for they are small Non-impulsive forces can be neglected for they are small in comparison (usually) to the impulsive forces. in comparison (usually) to the impulsive forces. If it is not known for sure that the forces are small, If it is not known for sure that the forces are small, then include them.then include them.

Impulsive MotionImpulsive Motion• Force acting on a particle during a Force acting on a particle during a

very short time interval that is large very short time interval that is large enough to cause a significant enough to cause a significant change in momentum is called an change in momentum is called an impulsive forceimpulsive force..• When impulsive forces act on a When impulsive forces act on a particle,particle,

• When a baseball is struck by a bat, When a baseball is struck by a bat, contact occurs over a short time contact occurs over a short time interval but force is large enough to interval but force is large enough to change sense of ball motion.change sense of ball motion.

• Nonimpulsive forcesNonimpulsive forces are forces for are forces for whichwhich

is small and therefore, may is small and therefore, may be neglected.be neglected.

21 vmtFvm

tF

• Impact: Impact: Collision between two bodies Collision between two bodies which occurs during a small time which occurs during a small time interval and during which the bodies interval and during which the bodies exert large forces on each other.exert large forces on each other.

• Line of Impact: Line of Impact: Common normal to the Common normal to the surfaces in contact during impact.surfaces in contact during impact.

• Central Impact: Central Impact: Impact for which the Impact for which the mass centers of the two bodies lie on mass centers of the two bodies lie on the line of impact; otherwise, it is an the line of impact; otherwise, it is an eccentric impact.eccentric impact.

Direct Central Direct Central ImpactImpact

• Direct Impact: Direct Impact: Impact for which the Impact for which the velocities of the two bodies are directed velocities of the two bodies are directed along the line of impact.along the line of impact.

Oblique Central Oblique Central ImpactImpact

• Oblique Impact: Oblique Impact: Impact for which one Impact for which one or both of the bodies move along a line or both of the bodies move along a line other than the line of impact.other than the line of impact.

13.12 IMPACT13.12 IMPACT

• Block constrained to move along Block constrained to move along horizontal surface.horizontal surface.

• Impulses from internal forcesImpulses from internal forcesalong the along the nn axis and from external axis and from external forceforceexerted by horizontal surface and exerted by horizontal surface and directed along the vertical to the directed along the vertical to the surface.surface.

FF

and

extF

• Final velocity of ball unknown in Final velocity of ball unknown in direction and magnitude and direction and magnitude and unknown final block velocity unknown final block velocity magnitude. Three equations magnitude. Three equations required.required.

• Bodies moving in the same Bodies moving in the same straight line, straight line, vvA A >> vvB B ..

• Upon impact the bodies undergo aUpon impact the bodies undergo aperiod of deformation,period of deformation, at the end at the end of which, they are in contact and of which, they are in contact and moving at a common velocity.moving at a common velocity.

• A A period of restitution period of restitution follows follows during which the bodies either during which the bodies either regain their original shape or regain their original shape or remain permanently deformed.remain permanently deformed.

• Wish to determine the final Wish to determine the final velocities of the two bodies. The velocities of the two bodies. The total momentum of the two body total momentum of the two body system is preserved,system is preserved,

BBAABBAAvmvmvmvm

• A second relation between the A second relation between the final velocities is required.final velocities is required.

13.13 DIRECT CENTRAL IMPACT13.13 DIRECT CENTRAL IMPACT

Av

Av

• Period of Period of deformation:deformation:

umPdtvmAAA

• Period of Period of restitution:restitution: AAA

vmRdtum • A similar analysis of particle A similar analysis of particle BB

yieldsyields

B

B

vu

uv e

• Combining the relations leads to the Combining the relations leads to the desired second relation between desired second relation between the final velocities.the final velocities.

BAAB

vvevv

• Perfectly plastic impact, e Perfectly plastic impact, e = = 0: 0: vvv

AB

vmmvmvmBABBAA

• Perfectly elastic impact, ePerfectly elastic impact, e = 1: = 1:

Total energy and total momentum Total energy and total momentum conserved.conserved. BAAB

vvvv

nrestitutio of tcoefficien e

Pdt

Rdt e

uv

vu

A

A

1e0

uv

vu

vu

uv e

A

A

B

B

BA

AB

vv

vv

BA

AB

vv

vv e

• Final velocities are Final velocities are unknown in unknown in magnitude and magnitude and direction. Four direction. Four equations are equations are required.required.• No tangential impulse No tangential impulse

component; tangential component; tangential component of momentum component of momentum for each particle is for each particle is conserved.conserved.

tBtBtAtA

vvvv

• Normal component of total Normal component of total momentum of the two momentum of the two particles is conserved.particles is conserved.

nBBnAAnBBnAA

vmvmvmvm

• Normal components of Normal components of relative velocities before relative velocities before and after impact are related and after impact are related by the coefficient of by the coefficient of restitution.restitution.

nBnAnAnB

vvevv

13.14 OBLIQUE CENTRAL IMPACT13.14 OBLIQUE CENTRAL IMPACT

• Tangential momentum of ball Tangential momentum of ball is conserved.is conserved.

tBtB

vv

• Total horizontal momentum of Total horizontal momentum of block and ball is conserved.block and ball is conserved.

xBBAAxBBAA

vmvmvmvm

• Normal component of relative Normal component of relative velocities of block and ball are velocities of block and ball are related by coefficient of related by coefficient of restitution.restitution.

nBnAnAnB

vvevv

• Note: Validity of last expression does not follow from Note: Validity of last expression does not follow from previous relation for the coefficient of restitution. A similar previous relation for the coefficient of restitution. A similar but separate derivation is required. but separate derivation is required.

• Three methods for the analysis of kinetics Three methods for the analysis of kinetics problems:problems:

- Direct application of Newton’s second lawDirect application of Newton’s second law

- Method of work and energyMethod of work and energy

- Method of impulse and momentumMethod of impulse and momentum

• Select the method best suited for the problem or part of Select the method best suited for the problem or part of a problem under consideration.a problem under consideration.

13.15 PROBLEMS INVOLVING ENERGY 13.15 PROBLEMS INVOLVING ENERGY AND MOMENTUMAND MOMENTUM