Class 28 - Rolling, Torque and Angular Class 28 - Rolling, Torque and Angular MomentumMomentum

Chapter 11 - Friday October 29thChapter 11 - Friday October 29th

Reading: pages 275 thru 281 (chapter 11) in Reading: pages 275 thru 281 (chapter 11) in HRWHRWRead and understand the sample problemsRead and understand the sample problemsAssigned problems from chapter 11 (due at Assigned problems from chapter 11 (due at 11pm on Sunday November 7th): 11pm on Sunday November 7th):

2, 6, 8, 12, 22, 24, 32, 38, 40, 50, 54, 642, 6, 8, 12, 22, 24, 32, 38, 40, 50, 54, 64

•Review of rolling motion

•Torque and angular momentum

•Newton's second law in angular form

•Conservation of angular momentum

•Demos and sample problems

Rolling motion as rotation and Rolling motion as rotation and translationtranslation

s RThe wheel moves with speed ds/dt

com

dv R

dt

Rolling motion as rotation and Rolling motion as rotation and translationtranslation

s RThe wheel moves with speed ds/dt

com

dv R

dt

The kinetic energy of rolling

2 212

2 2 21 12 2

2 21 12 2

P P com

com

com com r t

K I I I MR

K I MR

K I Mv K K

R x

P

sf

gF

singF

cosgF

N

Rolling down a rampRolling down a ramp•However, we do not really have to compute However, we do not really have to compute ffss (see section (see section 12-3).12-3).

•We can, instead, analyze the motion about We can, instead, analyze the motion about PP, in which , in which case, case, FFggsinsin is the only force component with a moment arm is the only force component with a moment arm about about PP.. Use:Use: torque = torque = II

sing PR F I

coma R

Thus:Thus:2 sin P comMR g I a

2P comI I MR

2

sin

1 /comcom

ga

I MR

Torque and angular momentumTorque and angular momentum

definitionr F

•Torque was discussed in the previous chapter; cross Torque was discussed in the previous chapter; cross products are discussed in chapter 3 (section 3-7) and at products are discussed in chapter 3 (section 3-7) and at the end of this presentation; torque also discussed in this the end of this presentation; torque also discussed in this chapter (section 7).chapter (section 7). is defined as:l l r p m r v Angular momentum

•Here, Here, pp is the linear momentum is the linear momentum mv mv of the object.of the object.

sinl mvr

rp rmv

r p r mv

•SI unit is Kg.mSI unit is Kg.m22/s./s.

Torque and angular momentumTorque and angular momentum

definitionr F

•Here, Here, pp is the linear momentum is the linear momentum mv mv of the object.of the object.

sinl mvr

rp rmv

r p r mv

•SI unit is Kg.mSI unit is Kg.m22/s./s.

is defined as:l l r p m r v Angular momentum

•Torque was discussed in the previous chapter; cross Torque was discussed in the previous chapter; cross products are discussed in chapter 3 (section 3-7) and at products are discussed in chapter 3 (section 3-7) and at the end of this presentation; torque also discussed in this the end of this presentation; torque also discussed in this chapter (section 7).chapter (section 7).

Newton's second law in angular formNewton's second law in angular form

net

dpF

dt

Linear formLinear form

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

For a system of many particles, the total angular momentum is:For a system of many particles, the total angular momentum is:

1 2 31

n

n ii

L l l l l l

,1 1

n ni

net i neti i

dL dl

dt dt

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

net

dl

dt

angular formangular form

No surprise:

Angular momentum of a rigid body about a Angular momentum of a rigid body about a fixed axisfixed axisWe are interested in the component of We are interested in the component of

angular momentum parallel to the axis of angular momentum parallel to the axis of rotation:rotation:

1 1

2

n n

z iz i i ii i

L l m v r vr dm

r r dm r dm I

In fact:In fact: L I

a constantL

i fL L

i i f fI I

f i

i f

I

I

Conservation of angular momentumConservation of angular momentum

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

It follows from Newton's second law that:It follows from Newton's second law that:

Conservation of angular momentumConservation of angular momentum

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

It follows from Newton's second law that:It follows from Newton's second law that:

What happens to kinetic energy?What happens to kinetic energy?

2 22 21 1 1

2 2 22i i i i

f f f f i i if f f

I I IK I I I K

I I I

•Thus, if you increase Thus, if you increase by reducing by reducing II, you , you end up increasing end up increasing KK..

•Therefore, you must be doing some work.Therefore, you must be doing some work.

•This is a very unusual form of work that you This is a very unusual form of work that you do when you move mass radially in a do when you move mass radially in a rotating frame. rotating frame.

•The frame is accelerating, so Newton's laws The frame is accelerating, so Newton's laws do not hold in this frame do not hold in this frame

a constantL

i fL L

i i f fI I

f i

i f

I

I

More on conservation of angular momentumMore on conservation of angular momentum

The vector product, or cross The vector product, or cross productproduct, where sina b c c ab

a b b a

Direction of to both and c a b

ˆ ˆ ˆ ˆ ˆ ˆi i j j k k 0

ˆ ˆ ˆ ˆ ˆ ˆi j k j i k

ˆ ˆ ˆ ˆ ˆ ˆj k i k j i

ˆ ˆ ˆ ˆ ˆ ˆk i j i k j

i

jk j+ v e

i

jk j-v e

ˆ ˆ ˆ ˆ ˆ ˆi j k i j kx y z x y za b a a a b b b

ˆ ˆ ˆ ˆ ˆi j i j kx y x y x ya b a b a b

ˆ ˆ ˆi j ky z y z z x z x x y y xa b a b b a a b b a a b a b