2008 Physics 2111 Fundamentals of Physics Chapter 11 1 Fundamentals of Physics Chapter 12 Rolling,...

2008 Physics 2111 Fundamentals of Physics

Chapter 11 1

Fundamentals of Physics

Chapter 12 Rolling, Torque & Angular Momentum

1. Rolling2. The Kinetic Energy of Rolling

Rolling as Pure Rotation3. The Forces of Rolling

Friction & RollingRolling Down a Ramp

4. The Yoyo5. Torque Revisited6. Angular Momentum7. Newton’s 2nd Law in Angular Momentum Form8. Angular Momentum of a System of Particles9. Angular Momentum of a Rigid Body Rotating About a Fixed Axis10. Conservation of Angular Momentum

Review & SummaryQuestionsExercises & Problems


Chapter 11 2

Rolling without Slipping

Rs

Units: (m/s) (rad/s) (m)

Rvcom

Racom

Units: (m) (radians) (m)

R


Chapter 11 3

Rolling


Chapter 11 4

Example: Value of x for rolling without slipping?

F m a I

x F

I m R 25

2

a R Rolling without slipping:

x > 2/5 R topspin

x = 0 no initial spin

x R 2

5Independent of m and F!


Chapter 11 5

The Forces of Rolling

No Initial Rotation: Topspin:

Frictional force reduces the speed &

increases the angular speed until:

Frictional force accelerates the ball in the direction of motion:

increasing v

decreasing

Rolling with slipping initially

Rolling without slipping

v Rcom


Chapter 11 6

Translation & Rotation

Add: Linear velocity due to rotation

Linear velocity of the center-of-mass

pure rotation + pure translation = rolling motion

To get: Linear velocity of each point on a rolling wheel.

See p247 in HRW.


Chapter 11 7

The Kinetic Energy of Rolling

The instantaneous axis of rotation is at Point P:

A rolling object has two types of kinetic energy: a rotational KE due to its rotation about its center of mass and a translational KE due to the translation of its center of mass.

K I

I I M R

v

R

K I M v

K K K

P

P com

com

com com

R T

12

12

12

2

2

2 2

K I P 1

22

comvR


Chapter 11 8

The Forces of Rolling

Friction & Rolling: friction is required!

No slipping (aka “smooth rolling”) Racom

fs acts on the wheel at P, opposing its tending to slide.

Wheel is rotating about point P.

No work is done by fs!

The wheel does not slow down.


Chapter 11 9

Rolling down a Ramp

F ma

f M g M aS com

sin

I

R f IS com com

a R

> 0 counter clockwise

a < 0 down the ramp

ag

I

M R

comcom

sin

12

Any body rolling down a ramp!


Chapter 11 10

Uniform Ball Rolling Down

ag

I

M R

comcom

sin

12

I M R 25

2

a gcom 5

7sin

v Rcom

v g hcom 107

12

2 12

2I M v M g hcom com

Conservation of energy:

M = 6.00 kg

Radius R

= 300

h = 1.20 m

v = ?

fs = ?


Chapter 11 11

ag

I

M R

comcom

sin

12

221 RMIcylinder

2RMI ring

252 RMI sphere

ringcylindersphere aaa

Hoop, Disk and Sphere rolling down a ramp:


Chapter 11 12

Wheel has constant acceleration & no slipping

M kg10

R m0 3.

a ms0 6.

amfF

amF

S

net

IRf

I

S

net

Ra

26.0

4

mkgI

NfS

fS = ? I = ?

Sf


Chapter 11 13

Ball rolls down the hill

x

x = ?

hgmIvmHgm 2212

21

Rv 2

52 RmI

smv 5.7

Energy:

Projectile: mx 8.4

H = 6.0 m

h = 2.0 m


Chapter 11 14

Loop the Loop

top atNwhenh 0?

R6 h whenQatN ?

252 rmI

rRgmIvmhgm 22212

21 Energy:

rR Newton:

R

vmamgmN

2

Rh 7.2 :answer

gmN 750 :answer


Chapter 11 15

Torque “Revisited”

r x F

r F sin

Fr Fr


Chapter 11 16

l r x p m r x v

l m r v

sin

Angular Momentum

pmomentumwithparticleaConsider

:isintPotorespectwith

momentumangularIts

Oprl


Chapter 11 17

All of the particles have the same momentum.

Largest Magnitude of Angular Momentum about O ???

Negative Angular Momentum???

Zero Angular Momentum???

1 and 3

2 and 3

5


Chapter 11 18

Angular Momentum

b) Torque = ?

a) Angular momentum = ?

l r x p m r x v

l m r v

sin

s

mkgl

2

12

out of the page

r x F

r F sin

mN0.3 out of the page

m = 2.0 kg

r = 3.0 m

v = 4.0 m/s

F = 2.0 N


Chapter 11 19

Newton’s 2nd Law in Angular Momentum Form

The vector sum of all the torques acting on a particle is equal to the time rate of change of the angular momentum of that particle.

l m r x v

dldt

m rdvdt

drdt

v

dldt

m r a r ma

dldt

r Fnet

0

dldt net

l

Time derivative of the angular momentum:


Chapter 11 20

Checkpoint

Largest Torque about Point O???

Positive Torque about Point O???

3F

Zero Torque about Point O???

1F

2F

4F


Chapter 11 21

Angular Momentum of a System of Particles

The net external torque acting on a system of particles is equal to the time rate of change of the system’s total angular momentum.

Internal torques sum to zero:

1 2 1 2 1 2 1 2

2 1 1 2

1 2 1 2 2 1

r F r F

F F

r r F

, ,

, ,

,

dLdt net

n

iilL

1

n

iinet

n

i

i

dt

Ld

dt

dl

dt

Ld

1,

1


Chapter 11 22

Angular Momentum About a Fixed Axis

L r x p m r x v mr I 2

v r


Chapter 11 23

A Rigid Body Rotating About a Fixed Axis

iiiiii vrmprl

iiii vrml iii vandrl

2

sin

sin

iiiz

iiiiz

iiiiz

iiz

rml

vrml

vrml

ll

Angular momentum of a mass element wrt the origin:

Component of angular momentum in z-direction:

iiii rrv


Chapter 11 24

A Rigid Body Rotating About a Fixed Axis

L I

iiiiz rml 2

Total angular momentum in z-direction:

z

n

iiiz

n

iii

n

iizz

IrmL

rmlL

1

2

1

2

1


Chapter 11 25

Checkpoint : same radius same mass same force

Largest Angular Momentum about Center???

Largest Angular Speed after a time interval???

Newton’s 2nd Law: constantRFdt

Ldnet

All the angular momenta are the same.

L I Ismallestlargest

ringcylindersphere III


Chapter 11 26

Conservation of Angular Momentum

The Spinning Volunteer:

If the net external torque acting on a system is zero, the angular momentum of the system remains constant, no matter what changes take place within the system.

L I i i

L I f f

I If i

f i Spins faster!


Chapter 11 27


If the component of the net external torque on a system along a certain axis is zero, then the component of angular momentum of the system along that axis remains constant, no matter what changes take place within the system.

L I constant

The Diver:


Chapter 11 28

L L Lb wh

The system = stool + person + spinning wheel

L Lwh



Chapter 11 29

A train starts running on a wheel; A train starts running on a wheel; = ? = ?

mM

0

0

WTG

WT

IvRm

LLL

Rvv

vvv

TG

WGTWTG

2RMIW

RMm

vm

RMRvRm

02

0 fi LL

move. not does train R

vmM

move. not does heelw

mM

0

v


Chapter 11 30

The sun goes out and then shrinks!

if

fi

ffii

fi

r

R

rMRM

II

LL

2

2212

21

2

T

revolutionminutes

2

8

62

3

1096.6

1037.625

f

if

T

m

mdays

R

rTT

The sun will become a white dwarf.

Sun goes out

Shrinks size Earth

Period ?

2008 Physics 2111 Fundamentals of Physics Chapter 11 1 Fundamentals of Physics Chapter 12 Rolling,...

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