In the two cases shown below identical ladders are leaning against frictionless walls. In which case...

In the two cases shown below identical ladders are leaning against frictionless walls. In which case is the force of friction between the ladder and the ground bigger?

A) Case 1 B) Case 2 C) Same

Catching up from Yesterday

Case 1 Case 2Mechanics Lecture 18, Slide 1

In the two cases shown below identical ladders are leaning against frictionless walls. In which case is the force of friction between the ladder and the ground bigger?

A) Case 1 B) Case 2 C) Same

Case 1 Case 2A) Because the bottom of the ladder is further away from the wall.

B) The angle is steeper, which means there is more normal force and thus more friction.

C) Both have same mass.

Mechanics Lecture 18, Slide 2

CheckPoint

CheckPoint

Suppose you hang one end of a beam from the ceiling by a rope and the bottom of the beam rests on a frictionless sheet of ice. The center of mass of the beam is marked with an black spot. Which of the following configurations best represents the equilibrium condition of this setup?

A) B) C)


CheckPointWhich of the following configurations best represents the equilibrium condition of this setup?

A) B) C)

If the tension has any horizontal component, the beam will accelerate in the horizontal direction.

Objects tend to attempt to minimize their potential energy as much as possible. In Case C, the center of mass is lowest.


DemoDemo

footprint

footprint

Stability & Potential Energy


CM DemosCM Demos

Physics 211Lecture 19

Today’s Concepts:Angular Momentum


Linear and Rotation


New Cast of Characters with Old Theme

Remember


Angular Momentum

We have shown that for a system of particles

What is the rotational version of this?

Momentum is conserved ifEXT

dpF

dt

0EXTF

The rotational analogue of force F is torque r F

Define the rotational analogue of momentum p to be Angular Momentum: L r p

For a symmetric solid object L I


Torque & Angular Momentum

where and

Total angular momentum is conserved

EXT

dL

dt

L r p

L I ext EXTr F

In the absence of external torques 0EXT

dL

dt


Example – Disk dropped on Disk


0, initialbottom

oinitialtop ,

ffinalbottomfinaltop ,,

Friction is an “internal force” because it is “internal” to the “system”. Friction is between two parts of the “system”.Friction may change the Kinetic Energy.But friction, being an internal force, does not change the Angular Momentum (nor the linear momentum in linear situations).

DemoDemo

Example – Disk dropped on Disk

02

1 2 oinitial MRL fffinal MRMRL 22

2

1

2

1

fo MRMR 22

2

1 02

1 f


0, initialbottom

oinitialtop ,


2

2afterafter

LK

I

2

2beforebefore

LK

I same

x2

What about Kinetic Energy?

221

2 2

LK I

I


0, initialbottom

oinitialtop ,


Case 1 Case 2

A) Case 1B) Case 2C) Same

In both cases shown below a solid disk of mass M, radius R, and initial angular velocity o is dropped onto an initially stationary second disk having the same radius. In Case 2 the mass of the bottom disk is twice as big as in Case 1. If there are no external torques acting on either system, in which case is the final kinetic energy of the system bigger?

Same initial L

CheckPoint


0, initialbottom

oinitialtop ,

M

M

0, initialbottom

oinitialtop ,

2M

M

Case 1 Case 2

A) Case 1B) Case 2C) Same

In which case is the final kinetic energy of the system bigger?

A) L is equal but bigger mass in case 2

B) Case 2 has greater mass

C) Conservation of angular momentum

CheckPoint

2

2

LK

I


M 2MM M

Demo: wheel rim dropDemo: wheel rim drop

Your Comments


This doesn’t mean anything.This doesn’t mean anything.

Sort of . . .Sort of . . .Work is done by friction so KE is Work is done by friction so KE is notnot conserved. conserved.

Okay; now Okay; now whatwhat does that mean? does that mean?

Find the Find the finalfinal KE in both cases. KE in both cases.

True but what about the True but what about the finalfinal KE? KE?

That’s for omega but what about the That’s for omega but what about the finalfinal KE? KE?True but what about the True but what about the finalfinal KE? KE?

True but what about the True but what about the finalfinal KE? KE?? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ?

axis

Point Particle Moving in a Straight Line

L R p

pRL

11

1

)sin( 11 pRpRL

22

2

)sin( 22 pRpD pD pD

3

)sin( 33 pRpRL

33


2R

3R

1R

p

p

p

D

Direction given by right hand rule(out of the page in this case)

L R p

L mvD pD


axis

p

D

Demo: train on bike wheelDemo: train on bike wheel

R

M

mv

Before

Top View

Disk Kid

Playground Example

mvRLBefore )2

1( 22 mRMRILAfter

Before AfterL L mMR

v

21

1


After

Disk Kid

Algebra Details

mvRLBefore

Before AfterL L mMR

v

21

1


)2

1( 22 mRMRILAfter

The magnitude of the angular momentum of a freely rotating disk around its center is L. You toss a heavy block onto the disk along the direction shown. Friction acts between the disk and the block so that eventually the block is at rest on the disk and rotates with it. What is the magnitude of the final angular momentum of the disk-block system:

A) LB) LC) L

Instead of a block, imagine it’s a kid hopping onto a merry go round

CheckPoint


Top View

The block provides no angular momentum since its perpendicular distance from the axis of rotation is 0. No external forces act on this systems, so angular momentum is conserved.

What is the magnitude of the final angular momentum of the disk-block system:

A) LB) LC) L

CheckPoint


Top View

The block provides no angular momentum since its perpendicular distance from the axis of rotation is 0. No external forces act on this systems, so angular momentum is conserved.

Your Comments


Zero!>!>! Zero!>!>! HowHow can that be? can that be?

Will it be conserved in the Will it be conserved in the nextnext CheckPoint? CheckPoint?Good! Altho’ I wouldn’t call it an external force.Good! Altho’ I wouldn’t call it an external force.

Good! The block’s Good! The block’s perpendicularperpendicular distance from the axis of rotation is zero. . distance from the axis of rotation is zero. .

But the block is But the block is NOTNOT at rest. at rest.

What is the final angular momentum?What is the final angular momentum?What does that mean for the final angular momentum?What does that mean for the final angular momentum?

What about the What about the nextnext CheckPoint?CheckPoint?

! ! ! ! !! ! ! ! !

okayokay

Tell me more.Tell me more.

What about the block?What about the block?Tell me more.Tell me more.

Top View

ACT


The magnitude of the angular momentum of a freely rotating disk around its center is L. You toss a heavy block onto the disk along the direction shown. Friction acts between the disk and the block so that eventually the block is at rest on the disk and rotates with it. Is the total angular momentum of the disk-block system conserved during this?

A) Yes

B) No

Friction decreases the angular momentum of the disk

But the TOTAL angular momentum of the block+disk is CONSTANT!

Top View

CheckPoint


The magnitude of the angular momentum of a freely rotating disk around its center is L. You toss a heavy block onto the disk along the direction shown. Friction acts between the disk and the block so that eventually the block is at rest on the disk and rotates with it. What is the magnitude of the final angular momentum of the disk-block system:

Instead of a block, imagine it’s a kid hopping onto a merry go round

A) LB) LC) L

CheckPoint

What is the magnitude of the final angular momentum of the disk-block system:

B) L is conserved so Linitial=Lfinal

A) Since the block has angular momentum to begin with, that is added into L.

C) The block adds momentum going in the opposite direction, so the total momentum is smaller than the momentum of the disk.

A) LB) LC) L

Top View


Your Comments


Tell me more.Tell me more.

Smaller than what? In the previous CheckPoint, the lever arm was Smaller than what? In the previous CheckPoint, the lever arm was zerozero..

Okay but tell me more.Okay but tell me more.Good.Good.

What about the angular momentum of the What about the angular momentum of the blockblock??

No, not this time.No, not this time.

Any better by now?Any better by now?

Why bigger? Angular momentum has a Why bigger? Angular momentum has a directiondirection connected to it. connected to it.

What about the block?What about the block?

What about the block?What about the block?Bravo!Bravo!

Bravo!Bravo!

ACT

A student holding a heavy ball sits on the outer edge a merry go round which is initially rotating counterclockwise. Which way should she throw the ball so that she stops the rotation?

A) To her leftB) To her rightC) Radially outward

top view: initial final

A B

C


ACT

A student is riding on the outside edge of a merry-go-round rotating about a frictionless pivot. She holds a heavy ball at rest in her hand. If she releases the ball, the angular velocity of the merry-go-round will:

A) Increase B) Decrease C) Stay the same


m2

m10

21

1

2diskI m R


odiskinitial IL

m2

m1o

21

1

2diskI m R


2

2

1odiskinitial IK

( )final disk rod fL I I

f

Solve for f

m2

m10

0initial diskL I

22

1

12rodI m Lfinal initialL L


final initialL L


f

M

21( )2final disk rod fK I I


f

M

Just like

for linear momentum

average

PF

t

for angular momentumaverage

L

t

0( )rod rod fL I 0( )rod faverage

I

t



Conservation IdeasConservation Ideas

Internal forces do not change the momentum.Internal forces do not change the momentum.Internal torques do not change the angular Internal torques do not change the angular momentum.momentum.

However, internal forces or internal torques However, internal forces or internal torques can still do work and, thus, can still do work and, thus, change the Kinetic Energy.change the Kinetic Energy.

Heads Up! for next preLectureHeads Up! for next preLecture


In the two cases shown below identical ladders are leaning against frictionless walls. In which case...

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