CHAPTER 14

Kinetics of a particle:Work and Energy14.1 The Work of a Force1. Definition of Work A force does work in a particle only when the particle

undergoes a displacement in the direction of the force.

Here =force acts on the particle

= - =displacement of the particle

= =angle between and

F

r rdr

ds

F

r rdsdr

rd

rd

F

Unit of Work (SI)

1 N*m=1 joule (J)

1 ft*lb=

2. Work of a Variable Force

0

negtive

positive

dU

0.90

18090

900

0

00

00

rd

rdF

dsFdu

*

cos

SFF

F

cosF

1S

2S

1r2r

dssFs

srdF

r

r

dUU

dsF

rdFdU

cos)(

1

2

cos

2

1

2

1

21

cosSF

cosSF

S1S 2S

ds

3. Work of a Constant Force moving along a straight live

4. Work of a Weight

1S 2ScoscF

cF

S 12

1

2

1

221

cos

cos

cos

SSF

dss

sF

dsFs

sU

1S

2S

1r2r

W

z

y

S

1y2y

kww

kdzjdyidxrd

kzjyixr

0

0

121

2

2

1

1

2

1

221

Znegtive

ZpositiveZw

zzwdzz

zwwdz

z

z

kdzjdyidxkwr

r

rdwr

rrdFU

(position)

(displacement)

5. Work of a spring forces

Consider a linear spring force(1) Work done on the spring

(2)Work of a spring done on the particle A particle (or only) attached to a spring

F.B.D of spring and particle

spring force on the particle

work of a spring on the particle is

ksFs

1S

2S

Sds

)( 22212

1212

221

1

2

1

221

sskksks

ksdss

sFsds

s

sU

1S 2SsF

ksFs sF

S

ksFs

0)(

1

2

21

222

1

1

2221

21

ssks

sks

ksdsrdsFU

14.2 Principle of Work and Energy (PWE)1. P.W.E

The particle’s initial kinetic energy plus the work done by all the forces acting on the particle as it moves from its initial to its final position is equal to the particle’s final kinetic energy.

or

Here kinetic energy

Work done by forces.

2. Derivation external force

Equation of motion of particle

2211 TUT 2

221

212

121 mvUmv

221 mvT

21U

y

x

z

tFP R )(

FFR

t

nr

neF )(

Initial frame

FFR

nnRttR

nnttR

uFuF

umaumaamF

)()(

Work done on particle P by external force is

(1)Applying the kinetic equation to the above equation yields

(2)Integrate both sides to yield

or

RF

dsmadsFrdF

dsarda

udsrd

uauaa

dsuFFdsrdF

rdamrdFrdF

tt

t

t

nntt

R

)(

)cos(cos

Vdvdsat mvdvdsFrdF tv

mvdvr

rrdF

r

r

1

2

2

1 12

212

1222

1

1

2221

21 TTmvmvv

vmvU

2211 TUT

dtsd

tt mmaF2

)(

3. Remark

(1) PWE represents an integrated form of equation of motion

(2) PWE provides a convenient substitution for

When solving kinetic problems involving force , velocity and displacement.

Ex:

From P.W.E we have

ttmaF

tt maF

dsvdv

ta ds

dvt mvF

mvdvdsFt mvdv

v

vdsF

s

st

1

2

1

2

122

1212

221

21 TTmvmvU

ttmaF

2211 TUT

sin2

sin

2

222

1212

1

grV

mvmgrmv

r

tmg

n

?T

14.3 Principle of Wok and Energy for a System of Particles

principle of Work and Energy for

Particle

=resultant external force on ith particle

=resultant internal force on ith particle

Since work and energy are scalars both work and kinetic energy applied to each particle of the system may be added together algebraically.

So that

or

iF

if

iS

ti

ni

222

1

1

2

1

2212

1iiix

i

iix

i

iii vmrdf

r

rrdF

r

rvm

iF

n

ijj

iji ff1

222

1

1

2

1

2222

1iiii

i

iii

i

iii vmrdf

r

rrdF

r

rvm

2211 TUT

Here =System’s initial kinetic energy

=System’s final kinetic energy

=Work done by all external and internal forces

Note:

(1) ,since the paths over which corresponding particles travel will be different.

(2) ,if the particles are contained within the boundary of a translating rigid body , or particles connected by inextensible cables.

1T

2T

21U

01

2 iii

i rdfr

r

(Non rigid Body)

Elastic..plastic…i

j

iS

jS

ijf

jif

(1)

ii

jj

Rigid body

(2)

01

2 iii

i rdfr

r

14.4 Power and Efficiency1. Power The amount of work per unit of time.

or

unit of power

2. Efficiency 效率

vFdt

rdF

dt

rdFP

workrdFdv

dt

dvP

t

UPav

WsJ

smN 111

powerinput

tpoweroutpu

tenergyinpu

utenergyoutp

14.5 Conservative Forces and Potential Energy

1. Conservative force

The force moves the particle form one point to another point to produce work which is independent of the path followed by the particle.

(1) Work done by weight

(2) Work done by the spring force on a particle

F 2

1

3

F

rdFrdFrdF

321

Conservative force

yWU w

yw0y

)( 212

1222

1 ksksV 1S 2S

S

Spring force

2. Potential Energy 位能 A measure of the amount of work a conservation force will

do when it moves from a given position to the datum or a

reference plane.

(1) Gravitational potential energy

(2) Elastic Potential Energy

always positive.The spring force has the capacity for always positive work on the particle.

)(保守力F

P

P

F

datum

WyVg (y=positive upward)

datum

w 推回基準作的功

作負功

y

y

WyVg

WyVg

221 ksVe

Unstretched

datum

pull

push

S

S ?eV

3. Potential Function V

The work done by conservation forces(W and Fs) in

moving the particle from point to point

Is

Ex:

VeVgV

),,( 111 zyx ),,( 222 zyx

2121 VVV

0l

datum(Unstretched position)

1S

2S

w

1v

2v212121

222

1212

11221

222

12222

212

11111

212

1222

121

1221

)()(

)(

)()(

)()(

UVU

ksksSSWVV

ksWSVeVgV

ksWSVeVgV

ksksV

SSWyWU

Fsw

Fs

w

14.6 Conservation of Energy The principle of work and energy is rewritten as

Work done by conservative forces

Work done by nonconservative forces

If , then

Conservation of energy

The sum of the particle’s kinetic and potential energy remains

constant during the motion .

Conservation of energy for a system of particles is

end

noncons

cons

nonconscons

U

VVU

TUUT

)(

)(

)()(

21

2121

221211

222111 )( VTUVT noncons

0)( 21 nonconsU

2211 VTVT

2211 VTVT