R2-1

Physics IReview 2

Review NotesExam 2

Rev. 26-Oct-04 GB

R2-2

Work

Work is a measure of the energy that a force puts into (+) ortakes away from (–) an object as it moves.

We will see that work is a useful way to solve problems wherethe force on an object is a known function of position.

Example: the force of an object connected to an ideal spring:

xkF

where x

is the displacement from equilibrium. (Hooke’s Law)

R2-3

Work for Constant Force

dFW

)cos(dFW

R2-4

Vector Dot Product

F

d

If you know lengths and angle: )cos(dFW

If you know components: zzyyxx dFdFdFW

If in the same direction: dFW

If in opposite directions: dFW

If at right angles: 0W

R2-5

Work for Variable Force

W F x dxx

x

i

f ( )(This is the version for one dimension.)

R2-6

Work-Energy Theorem

N e t w o r k i s d o n e o n a n o b j e c t b y t h e n e t f o r c e :

f

i

x

x

netnet dxFW

K i n e t i c e n e r g y d e f i n e d f o r a n o b j e c t :

2z

2y

2x2

12

2

1 vvvmvmK

W o r k - E n e r g y T h e o r e m : ( w i t h o u t p r o o f )

netif WKK

R2-7

Class #10Take-Away Concepts

1 . W o r k i s a m e a s u r e o f e n e r g y a d d e d t o ( + ) o r t a k e n a w a y ( – ) .

)cos(dFdFW

( c o n s t a n t f o r c e )

f

i

x

x

dxFW ( v a r i a b l e f o r c e , 1 D )

2 . V e c t o r d o t p r o d u c t d e f i n e d .

3 . K i n e t i c E n e r g y : 2z

2y

2x2

12

2

1 vvvmvmK

4 . W o r k - E n e r g y T h e o r e m : netif WKK 5 . P o s i t i v e n e t w o r k m e a n s a n o b j e c t ’ s K . E . i n c r e a s e s ( s p e e d s u p ) .6 . N e g a t i v e n e t w o r k m e a n s a n o b j e c t ’ s K . E . d e c r e a s e s ( s l o w s d o w n ) .7 . Z e r o w o r k m e a n s a n o b j e c t ’ s K . E . s t a y s c o n s t a n t ( c o n s t a n t s p e e d ) .

R2-8

Does work depend on the path?Conservative Forces

For general forces, the w ork does depend on the path that w e take.H ow ever, there are som e forces for w hich w ork does not depend onthe path taken betw een the beginning and ending poin ts.T hese are called conservative forces .

A m athem atically equivalent w ay to put th is is that the w ork done bya conservative force along any closed path is exactly zero.

0xdFcons

(The funny in tegral sym bol m eans a path that closes back on itself.)

R2-9

Conservative Forces Non-Conservative Forces

Examples of Conservative Forces: Gravity Ideal Spring (Hooke’s Law) Electrostatic Force (later in Physics 1)

Examples of Non-Conservative Forces: Human Pushes and Pulls Friction

R2-10

Conservative Forcesand Potential Energy

If we are dealing with a conservative force, we can simplify the processof calculating work by introducing potential energy.

1. Define a point where the potential energy is zero (our choice).

2. Find the work done from that point to any other point in space.

(This is not too hard for most conservative forces.)

3. Define the potential energy at each point as negative the work donefrom the reference point to there. Call this function U.

4 The work done by the conservative force from any point A to anypoint B is then simply W = U(A)–U(B).

R2-11

Two Common Potential Energy Functions in Physics 1

G r a v i t a t i o n a l P o t e n t i a l E n e r g y

hgm)yy(gmU 0g ( y 0 i s o u r c h o i c e t o m a k e t h e p r o b l e m e a s i e r )

S p r i n g P o t e n t i a l E n e r g y2

02

1s )xx(kU

( x 0 i s t h e e q u i l i b r i u m p o s i t i o n a n d k i s t h e s p r i n g c o n s t a n t )

R2-12

Potential Energy, Kinetic Energy, and Conservation of Energy

R e c a l l t h e W o r k - E n e r g y T h e o r e m :

netWK A n d f o r c o n s e r v a t i v e f o r c e s w e h a v e

UW cons I f t h e n o n - c o n s e r v a t i v e f o r c e s a r e z e r o o r n e g l i g i b l e , t h e n

consnet WW P u t t i n g i t t o g e t h e r ,

UK o r 0UK A n o t h e r w a y t o s a y t h i s i s t h e t o t a l e n e r g y , K + U , i s c o n s e r v e d .

R2-13

Example ProblemSkateboarder Going Up a Ramp

hd

v

22

1 vmK 0U

0K hgmU

hgm00vm 22

1

g2v

h2

)sin(g2

v)sin(

hd

2

R2-14

Class #11Take-Away Concepts

1 . M u l t i - d i m e n s i o n a l f o r m o f w o r k i n t e g r a l :

f

i

x

x

xdFW

2 . C o n s e r v a t i v e f o r c e = w o r k d o e s n ’ t d e p e n d o n p a t h .3 . P o t e n t i a l E n e r g y d e f i n e d f o r a c o n s e r v a t i v e f o r c e :

A

0

xdF)A(U

4 . G r a v i t y : hgm)yy(gmU 0g 5 . S p r i n g : 2

02

1s )xx(kU

6 . C o n s e r v a t i o n o f e n e r g y i f o n l y c o n s e r v a t i v e f o r c e s o p e r a t e :UK o r 0UK

R2-15

Is Mechanical EnergyAlways Conserved?

T o t a l M e c h a n i c a l E n e r g y

UKE 0UKE i f o n l y c o n s e r v a t i v e f o r c e s a c t

W h e n N o n - C o n s e r v a t i v e F o r c e s A c t

consnonWUKE T h i s i s e q u i v a l e n t t o

consnoniiff WUKUK N o n - c o n s e r v a t i v e f o r c e s a d d ( + ) o r s u b t r a c t ( – ) e n e r g y .

R2-16

Example of Energy Lost to Friction (Non-Conservative Force)

hd

v

0K f J10295.18.970hgmU f

J22406470vmK2

122

1i 0Ui

A skateboarder with mass = 70 kg starts up a30º incline going 8 m/s. He goes 3 m along theincline and comes to a temporary stop. Whatwas the average force of friction (magnitude)?

m5.13)sin(dh21

R2-17

Example of Energy Lost to Friction (Non-Conservative Force)

hd

v

0K f J10295.18.970hgmU f

J22406470vmK2

122

1i 0Ui

J121122401029UKUKW iifffriction

N7.4033

1211d

WF friction

avg,friction

What do the – signs mean?

R2-18

Elastic and Inelastic Collisions

Momentum is conserved when the external forces are zero or sosmall they can be neglected during the collision. This is often true.

In many collisions a large percentage of the kinetic energy is lost.These are known as inelastic collisions. For example, any collisionin which two objects stick together is always inelastic.

If the kinetic energy after a collision is the same as before, then wehave an elastic collision. During the collision, some of the kineticenergy can convert to potential energy of various kinds, but after thecollision is over all of the kinetic energy is restored.

R2-19

Elastic Collisions inOne Dimension

+X

v2fv1f

v1i v2iI n i t i a l

F i n a l

C o n s e r v a t i o n o f M o m e n t u m :

f22f11i22i11 vmvmvmvm C o n s e r v a t i o n o f E n e r g y :

f22

22

1f1

212

1i2

222

1i1

212

1 vmvmvmvm

T w o e q u a t i o n s , t w o u n k n o w n s ( f i n a l v e l o c i t i e s ) .

R2-20

Elastic Collisions inOne Dimension

+X

v2fv1f

v1i v2iI n i t i a l

F i n a l

i221

2i1

21

21f1 v

mmm2

vmmmm

v

i221

12i1

21

1f2 v

mmmm

vmm

m2v

R2-21

Elastic Collisions inOne Dimension - Example

+X

v2fv1f

v2i

21 m2m 0v i1

1v i2 m / s

32

)1(12

2)0(

1212

v f1

m / s

31

)1(12

21)0(

122

v f2

m / s

Initial: m2 has all (–1) of themom. and KE.

Final: m1 has –4/3 of the mom.and 8/9 of the KE.m2 has +1/3 of the mom.and 1/9 of the KE.

R2-22

Class #12Take-Away Concepts

1 . M o d i f i c a t i o n o f e n e r g y c o n s e r v a t i o n i n c l u d i n g n o n -c o n s e r v a t i v e f o r c e s :

consnonWUKE 2 . N o n - c o n s e r v a t i v e w o r k a d d s ( + ) o r s u b t r a c t s ( – )

e n e r g y f r o m t h e s y s t e m .3 . E l a s t i c c o l l i s i o n p r e s e r v e s K E b e f o r e a n d a f t e r .

( D o n ’ t a s s u m e a l l c o l l i s i o n s a r e e l a s t i c , m o s t a r e n o t . )4 . S p e c i a l e q u a t i o n s f o r 1 D e l a s t i c c o l l i s i o n s .

i221

2i1

21

21f1 v

mm

m2v

mm

mmv

i221

12i1

21

1f2 v

mmmm

vmm

m2v

R2-23

Definitions

A n g u l a r P o s i t i o n :

( i n r a d i a n s )

A n g u l a r D i s p l a c e m e n t : 0

A v e r a g e o r m e a n a n g u l a r v e l o c i t y i s d e f i n e d a s f o l l o w s :

ttt 0

0avg

I n s t a n t a n e o u s a n g u l a r v e l o c i t y o r j u s t “ a n g u l a r v e l o c i t y ” :

tdd

tlim

0t

W a i t a m i n u t e ! H o w c a n a n a n g l e h a v e a v e c t o r d i r e c t i o n ?

R2-24

Direction of Angular Displacement and Angular Velocity

•Use your right hand.

•Curl your fingers in the direction of the rotation.

•Out-stretched thumb points in the direction of the angular velocity.

R2-25

Angular Acceleration

A v e r a g e a n g u l a r a c c e l e r a t i o n i s d e f i n e d a s f o l l o w s :

ttt 0

0avg

I n s t a n t a n e o u s a n g u l a r a c c e l e r a t i o n o r j u s t “ a n g u l a r a c c e l e r a t i o n ” :

2

2

0t tdd

tdd

tlim

T h e e a s i e s t w a y t o g e t t h e d i r e c t i o n o f t h e a n g u l a r a c c e l e r a t i o n i st o d e t e r m i n e t h e d i r e c t i o n o f t h e a n g u l a r v e l o c i t y a n d t h e n … I f t h e o b j e c t i s s p e e d i n g u p , a n g u l a r v e l o c i t y a n d a c c e l e r a t i o n

a r e i n t h e s a m e d i r e c t i o n . I f t h e o b j e c t i s s l o w i n g d o w n , a n g u l a r v e l o c i t y a n d a c c e l e r a t i o n

a r e i n o p p o s i t e d i r e c t i o n s .

R2-26

Equations for Constant

1 . 00 tt

2 . 202

1000 )tt()tt(

3 . )tt)(( 002

10

4 . 202

100 )tt()tt(

5 . 020

2 2

xva

R2-27

Relationships AmongLinear and Angular Variables

MUST express angles in radians.rs rv

ra tangential

rr

rr

va 2

222

lcentripeta

T he rad ia l d irec tion is defined to be +outw ard fro m the center.

lcentripetaradial aa

R2-28

Energy in Rotation

C o n s i d e r t h e k i n e t i c e n e r g y i n a r o t a t i n g o b j e c t . T h e c e n t e r o fm a s s o f t h e o b j e c t i s n o t m o v i n g , b u t e a c h p a r t i c l e ( a t o m ) i n t h eo b j e c t i s m o v i n g a t t h e s a m e a n g u l a r v e l o c i t y ( ) .

2ii

22

12i

2i2

12ii2

1 rmrmvmK

T h e s u m m a t i o n i n t h e f i n a l e x p r e s s i o n o c c u r s o f t e n w h e na n a l y z i n g r o t a t i o n a l m o t i o n . I t i s c a l l e d t h e m o m e n t o f i n e r t i a .

R2-29

Moment of Inertia

F o r a s y s t e m o f d i s c r e t e “ p o i n t ” o b j e c t s :

2ii rmI

F o r a s o l i d o b j e c t , u s e a n i n t e g r a l w h e r e i s t h e d e n s i t y :

dzdydxrI 2

W e m a y a s k y o u t o c a l c u l a t e t h e m o m e n t o f i n e r t i a f o r p o i n t o b j e c t s , b u t w e w i l lg i v e y o u a f o r m u l a f o r a s o l i d o b j e c t o r j u s t g i v e y o u i t s m o m e n t o f i n e r t i a .

I for a solid sphere: 25

2 RMII for a spherical shell: 2

3

2 RMI

R2-30

Correspondence BetweenLinear and Rotational Motion

xvaIm F

22

1 IK

I

You will solve many rotation problemsusing exactly the same techniques youlearned for linear motion problems.

R2-31

Class #13Take-Away Concepts

1 . D e f i n i t i o n s o f r o t a t io n a l q u a n t i t i e s : , , .

2 . C e n t r ip e t a l a n d t a n g e n t i a l a c c e l e r a t io n .

3 . M o m e n t o f i n e r t i a : 2

ii rmI 4 . R o ta t io n a l k in e t i c e n e r g y : 2

2

1 IK 5 . I n t r o d u c t io n to to r q u e :

I6 . C o r r e s p o n d e n c e

x v aIm F

R2-32

Review of Torque

For linear motion, we have “F = m a”. For rotation, we have

I

The symbol “” is torque. We will define it more precisely today.

When the rotation is speeding up, and are in thesame direction. When the rotation is slowing down, and are inopposite directions.

Torque and angular acceleration arealways in the same direction in Physics 1.

R2-33

The Vector Cross Product

We learned how to “multiply” two vectors to get a scalar.That was the “dot” product:

)cos(|b||a|bad

Now we will “multiply” two vectors to get another vector:

bac; )sin(|b||a||c|

The direction comes from the right-hand rule. It is at a right angle

to the plane formed by a and b. In other words, the cross product

is at right angles to both a and b. (3D thinking required!)

R2-34

The Vector Cross Product

R2-35

Torque as a Cross Product

Fr

)sin(|F||r|||

r

i s t h e v e c t o r f r o m t h e a x i s o f r o t a t i o n t o w h e r e t h e f o r c e i s a p p l i e d .

T h e t o r q u e c a n b e z e r o i n t h r e e d i f f e r e n t w a y s :

1 . N o f o r c e i s a p p l i e d ( 0|F|

) .2 . T h e f o r c e i s a p p l i e d a t t h e a x i s o f r o t a t i o n ( 0|r|

) .

3 . F

a n d r

i n t h e s a m e o r o p p o s i t e d i r e c t i o n s ( 0)sin( ) .

R2-36

Angular Momentum of a Particle

vmp

r

center of rotation (defined) A n g u la r m o m e n t u m o f a p a r t i c l eo n c e a c e n t e r i s d e f i n e d :

pr

l

( W h a t i s t h e d i r e c t i o n o f a n g u l a rm o m e n t u m h e r e ? )

Once we define a center (or axis) of rotation, any object with alinear momentum that does not move directly through that pointhas an angular momentum defined relative to the chosen center.

R2-37

Class #14Take-Away Concepts

1. : Speeding up, slowing down. I .

2. Definition of vector cross product:

bac ; )sin(|b||a||c|

3. Torque as a cross product: Fr .

4. Angular momentum of a particle: pr

l .

R2-38

How Does Angular Momentum of a Particle Change with Time?

Take the time derivative of angular momentum:

tdpd

rptdrd

)pr(td

dtd

d

l

Find each term separately:

0pvptdrd

(Why?)

netnetFrtdpd

r (Why?)

so

nettdd l

(Newton’s 2nd Law for angular momentum.)

R2-39

Angular Momentum of a Particle:Does It Change if = 0?

vmp = 1 kg m/s (+X dir.)(0,0)

(0,–3) (4,–3)

X

Y

The figure at the left shows the sameparticle at two different times. No forces(or torques) act on the particle.Is its angular momentum constant?(Check magnitudes at the two times.)

Blue angle: = 90ºl = r p sin() = (3) (1) sin(90º) = 3 kg m2/s

Red angle: = arctan(3/4) = 36.87ºl = r p sin() = (5) (1) sin(36.87º) = 3 kg m2/s

[r sin()] is the component of r at a right angle to p

. It is

constant.It is also the distance at closest approach to the center.

r

(blue)

r(red)

R2-40

Conservation of Angular Momentum

Take (for example) two rotating objects that interact.

1onext2from1on1

tdd l

2onext1from2on2

tdd l

The total angular momentum is the sum of 1 and 2:

2onext1onext21

tdd

tdd

tdLd

ll

(Why?)

If there are no external torques, then

0tdLd

R2-41

Class #15Take-Away Concepts

1. Angular momentum of a particle (review): pr

l .2. Newton’s 2nd Law for angular momentum:

nettdd l

3. Conservation of angular momentum (no ext. torque):

0tdLd

R2-42

Formula Sheet Organization

L i n e a r K i n e m a t i c s

1 . 00 ttavv

2 . 202

1000 )tt(a)tt(vxx

3 . )tt)(vv(xx 002

10

4 . 202

100 )tt(a)tt(vxx

5 . 020

2 xxa2vv

R2-43

Formula Sheet OrganizationN e w t o n ’ s 2 n d L a w a n d L i n e a r M o m e n t u m

6 . amFF net

1 0 . vmp

1 1 .td

pdFF net

1 2 . pdtFJ

1 3 . ipP

1 4 . extFtd

Pd

1 5 . imM

1 6 . iicm xmM

1x iicm ym

M

1y

1 7 . cmvMP

R2-44

Formula Sheet Organization

W o r k a n d E n e r g y ( L i n e a r M o t i o n )

1 8 . yyxx baba)cos(baba

1 9 . dFW

2 0 . xdFW

2 1 . )vv(mvmK 2y

2x2

12

2

1

2 2 . netif WKK

2 3 . xdFU cons

2 4 . )yy(gmU 0g

2 5 . 202

1s )xx(kU

2 6 . consnonWUK

R2-45

Formula Sheet Organization

R o t a t i o n a l K i n e m a t i c s

3 0 . 00 tt

3 1 . 202

1000 )tt()tt(

3 2 . )tt)(( 002

10

3 3 . 202

100 )tt()tt(

3 4 . 020

2 2

R2-46

Formula Sheet OrganizationR o t a t io n a l M o t io n / L in e a r M o t io n

7 .v

r2T

8 . rr

va 2

2

lcentripeta

9 . rmr

vmF 2

2

lcentripeta

2 7 . rs 2 8 . rv gentialtan

2 9 . ra gentialtan

R2-47

Formula Sheet OrganizationNewton’s 2nd Law and Angular Mom.

35. )sin(baba

36. 2ii rmI

39. Fr

40.td

LdI

41. pr

l

42. iL l

43.

IL

R2-48

Formula Sheet Organization

Work and Energy (Rotational)

37. 2

2

1rot IK

38. dW

R2-49

Formula Sheet Organization

C o l l i s i o n s

4 4 x . after,x,22after,x,11before,x,22before,x,11 vmvmvmvm

4 4 y . after,y,22after,y,11before,y,22before,y,11 vmvmvmvm

4 5 a . i,221

2i,1

21

21f,1 v

mm

m2v

mm

mmv

4 5 b . i,221

12i,1

21

1f,2 v

mm

mmv

mm

m2v