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Chapter 4 Kinetics of a Particle

0" and 0d

d when occurs, minimum Local

0" and 0d

d when occurs, maximum Local

dd

d sloped aldifferenti

tangentd

dlim slope

fx

f

fx

f

xx

fx f

x

f

x

f

x

f(x)

x

f

max

min

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Integration: the reverse of differentiation

xox+x

xx

f(x)

x)(

)(

)()(lim

)(

)()( lim

: from curve under the area thecalculate To

0

10

xfx

xxf

x

xdxfxdxf

xd

xdxfd

x

xxdxfixixf

xtox

x

x

xx

x

x

x

o

ooo

x

N

io

xi

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constant a is C

Ced e ed

de

Clnd 1

1

d

lnd

1- n ,C1n

dx n d

d

Ccotdcsc csc- dx

cotd

Ctand sec sec d

tand

cosd sin ,sin - dx

cosd

1nn1-n

n

22

22

xxxx

xx

xxxxx

x

xxx

x

x

xx xxx

xxxxx

x

xxxxx

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4

2F

1F RF

zzz

yyy

xxx

yzyyxx

i

FFF

FFFFFF

maFmaFmaF

amFi

21

21

21

where

and ,

Newton’s 2nd law

constant ,0 ,0 ,0 vaamFi

iNewton’s 1st law

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5

Newton’s 3rd law

action = reaction

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6

Work done,cos FdsrdFdU

zFyFxF zyx ddd

rdds

where

rdFU

Total work done

Example 1 What is the work done by a force on a article: (a)in circular motion? (b)horizontal motion? (c)from A to B?

F

rd

v

F

A

B

hg

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7

rd

F

Kinetic energy K.E.

Work done by an external force

K.E.)in (change ..

2

1

2

1

d2

2

) d(

d d

d d

21

22

21

21

21

21

21

2121

2

EK

mvmv

vmvvm

vvmrdtvdm

ramrFW

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8

Power P

Energy dissipated per unit time

vFt

rF

t

U

d

d

d

d

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9

Dissipative force (e.g. friction): work done from one point to another point depends on the path.

f

A

Bpath 1

path 2

e.dissipativ is force themeanssign ""

W Smg -

rdmgW

Smg -

)d toopposite is ( rdmg

dW

)2path (BA

2

)2path (BA

1

)1path (BA

2path

1path

1path

rf

rf

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10

P.E. rF

dF

EKdF

dFdF

dFdF

dFdF

energy, potential of change theas defined isIt only, points end and initial offunction a is dc Therefore,

same. theis rc

0.. force, veconservati aunder path closed a of completionAfter 0 r

rr

rr

rr

pathany

pathany

c

2path reversed c1path c

2path c1path c

2path c1path c

0

0

path close

cF

Non-dissipative force (conservative force): work done from one point to another point is independent on the path.

A

Bpath 1

path 2

P.E. between two points is equal to the work done by an external force against the field of a conservative force for bringing the particle from the starting point to the end point, with the external force = .

,d)c(- pathany rF

c- F

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11

Example 2 (gravitational potential)

r

mMG

r

mMG r

r

mMGrV

V

rrrr

mMGV

rrr

mMGV

rVP.E.

rr

mMGF

rr

2

r

2

r

2

d)(

. 0such that ,at set ispoint reference The

ˆd]ˆ[

d]ˆ[

)(

;ˆ2

R

Mr m

r̂

X=0 X

X

Example 3 Find V of a spring. Ans. kx2/2

Example 4 Potential energy of a mass m, positioned at h from the ground. Ans. mgh

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12

.Δd K.ErF

In general, the two types of forces coexist:

P.E.K.E.rfrF-K.E.rf

K.E.rfF( K.E.rF

ΔΔdd)c(Δd

Δd)c

Δd

If there is no dissipative force, K.E. + P.E. = 0, i.e. conservation of mechanical energy.

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13

ExampleThe rod is released at rest from = 0, find :(a) velocity of m when the rod arrives at the horizontal position.(b) the max velocity of m.(c) the max. value of .

2m

r

m

r

(a) At = 45o, v = 0.865 (gr)1/2

P.E.=-2mgr sin+mg(r-r cos )K.E. = (2m+m)v2/2(P.E.+ K.E.) = 03mv2/2 – mgr(2 sin + cos -1) = 0v = [2gr(2 sin + cos -1)/3]1/2

2/1

max

o2

)(908.0

63.4 θor 2 θ tan 0θsinθcos23

2/

dθ

d (b)

grv

grv

omaxmaxmax

maxmax2

maxmax2

maxmax

9.126θor0θ ,6.0or18.02.0θcos 03θcos2θcos5

θcos1θcos12 01θ cos θsin 2

0 when valuemaximum has θ (c)

v

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14

xx

Vd

d

d

x

VFx

d

d

BA

rd

From definition of potential energy: dV(x) = -Fdx

From the concept of differential dV =

mgh

mghF

mghhV

kxx

kxF

kxxV

rmMGrr

mMGr

mMG

rr

rVF

r

mMGrV

d

d)( with ground on the reference with force nalGravitatio

d

2/d2/)( with spring a of force Restoring

/)1

(d

d)(

d

d

d

)(d

)( with force nalGravitatio

Examples

2

2

2

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15

vmG With defined as the linear momentum

Gdt

Gdvm

dt

dvmF

)(

Linear momentum

When (i) the total (external) force is zero, or (ii) the collision time t1 t2 is extremely short.

1. In a motion, linear momentum can be conserved,

0dor 0,2

1

t

ttF G

2. Define impulse = change in linear momentum:

1221 GGGddtF

F

time

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16

momentum)linear ofion (conservat 0 ,- Since

)(

ofmoment linear in change

ofmoment linear in change

0 force External

2

1

2

1

2

1

GFF

dtFFGGG

dtFGB

dtFGA

BA

t

tBABA

t

tBB

t

tAA

A B

RF aF

Collision between systems A and B.

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17

Angular Momentum

Take moment about O

Angular momentum about O is :

oo HmrvGrvrmH ˆ sin

mr

v

v

rv

O

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18

Torque = Moment of force about O is defined as :

)d

d to(analogous

d

d

d

)d(

d

d

o

o

t

GF

t

Ht

vmrt

vmr

amrFrM

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19

Example: Prove that the angular momentum of a particle under a central force is conserved.

m

om

v

rF

In polar coordination system :

rr

mmGF

2ˆ0

ttancons

)2 ()2(0)1 () (/

ˆ)2(ˆ

2

2

d

d10: ) implies 2(

22

mrv

θrθrm θrrm rGmm

θθrθrmrθrrmam

θ

o

2

θmr

)θ(mrtr