= i angular momentumangular momentum - u of t physicsameyerth/phy131f14/lect22c.pdf · torque = i...

Physics 131: Lecture 22

Today’s Agenda

Rotational dynamics Torque Torque = I Angular MomentumAngular Momentum

Physics 201: Lecture 10, Pg 1

An Unfair RaceA frictionless block and a rolling (without slipping) disk are

released at the top of identical inclined planes. Which will reach the bottom first?

H(a) Frictionless block(b) Rolling Disk(c) The same time

H


Clicker Question 1:A frictionless block and a rolling (without slipping) disk are


(A) The disk loses energy to friction as it rolls, but the box is frictionless and so it speeds up more quickly and gets to the bottom first.

(B) The potential energy of the disk is converted into translational and rotational kinetic energy, so the translational speed grows more slowly than that of the box, which has no rotational energy., gy

(C)The net forces on the two objects are equal, but the force on the disk gets partially used up in creating the torque necessary to make it roll.

(D) The net forces on the two objects are equal, but the force on the disk is not directed parallel to the ramp, and so does not create as great an acceleration d th


down the ramp.

An Unfair RaceA frictionless block and a rolling (without slipping) disk areA frictionless block and a rolling (without slipping) disk are


H2

21 MVKTOT

Doesn’t friction take away energy?

H22 I

21

21 MVKTOT

There is only one source of energy, gravitational potential energy. For the disk that is rotating it must spend this energy to rotate


For the disk that is rotating it must spend this energy to rotate and to move down the incline, so it will have less energy to move down the incline and thus will move slower.

An Unfair RaceA frictionless block and a rolling (without slipping) disk areA frictionless block and a rolling (without slipping) disk are


H

HH


Linear and Angularg

Linear AngularDisplacement s Velocity v yAcceleration a Inertia m IInertia m IK ½ m v2 ½ I 2

N2L F = ma = IMomentum p = mv L = Iw

Today


p

Angular momentum of a rigid bodyAngular momentum of a rigid bodyabout a fixed axis:

Here we have a disk with moment of inertia and angular velocity

It has angular momentum of It has angular momentum of Rotation is positive (CounterClockwise)

IL

z

IL

Angular momentum is a vector!


Angular Momentumg

The magnitude of the angular velocityangular velocity vector is .

The angular velocity t i t lvector points along

the axis of rotation in the direction given bythe direction given by the right-hand rule as illustrated.


External/Internal TorquesExternal/Internal Torques

Ld

dtLd

net

dtLd

netextnet

inti

dt

dLdext

net

0z

dtnet

Ldext


dtnet f

Newton’s second law for rotations

Conservation of Angular Momentum Newton s second law for rotations

tL

ext dd

When the external torques on an system sum to zero the total angular momentum of the system will

td

zero, the total angular momentum of the system will be conserved

The angular momentum of an isolated system is


The angular momentum of an isolated system is conserved

Conservation of Linear Momentum

The angular momentum of an isolated system is conserved The angular momentum of an isolated system is conserved

if LL

II

object 1 object 2 object 1 object 2

if II

If + If = Ii + Ii

j


Demonstrations: Angular momentum

What happens to your angular momentum as you What happens to your angular momentum as you pull in your arms? Does it increase, decrease or stay the same? What happens to your angular velocity as you pull in your arms?velocity as you pull in your arms?

i f

Ii Ifi f


Example: Rotating Table...There are no external torques acting on the student-stool system, so angular momentum will be conserved.system, so angular momentum will be conserved. Initially: Li = Iii

Finally: Lf = If f 1II if

I fi

i f

Ii Ifi f


Clicker Question 2: What happens to your kinetic energy as you pull in your

arms?arms? (a) increases(b) decreases

No external torques, ang mom is conserved( )

(c) stays the samei f i fi f

Ii If

i f i

I

f

Ii f Ii If


Clicker Question 3: A student sits on a barstool holding a bike wheel. The wheel is initially spinning CCW in the horizontal plane (as viewed from above) L= 25 kg m2/s She now turns the bike wheel over. What happens?(a) She starts to spin CCW.(b) She starts to spin CW.(c) Nothing


Clicker Question 4: Suppose you are standing on the center of a merry-go-roundSuppose you are standing on the center of a merry go round

that is at rest. You are holding a spinning bicycle wheel over your head so that its rotation axis is pointing upward. The wheel is rotating counterclockwise when observed from above. (Foris rotating counterclockwise when observed from above. (For this problem, neglect any air resistance or friction between the merry-go-round and its foundation.)

Suppose you now grab the edge of the wheel with your handSuppose you now grab the edge of the wheel with your hand, stopping it from spinning. What happens to the merry-go-round?

(A) It remains at rest.(B) It begins to rotate CCW (as observed from above).( ) g ( )(C) It begins to rotate CW (as observed from above).


Rotating Neutron Starg

A star is dying. Its core collapses such that its radius goes from 6 000 km to about 40 km Assume the coregoes from 6,000 km to about 40 km. Assume the core does not lose mass.

If it were rotating at an angular speed of 0.12 rad/s, what is its speed now?

How many times perHow many times per second does this revolve?


Clicker Question 5: A star is dying. Its core collapses such that its radius

goes from 6 000 km to about 40 km If it was rotating atgoes from 6, 000 km to about 40 km. If it was rotating at an angular speed of 0.12 rad/s, what is its speed now? Assume no outside torques act and the core does not lose masslose mass.

(a) 2700 rad/s( )(b) 4600 rad/s(c) 436 rad/s(d) 18 rad/s(e) 0.12 rad/s


Clicker Question 6:

Two astronauts a distance dI apart of mass M are spinning with an angular velocity of around an axis about theirwith an angular velocity of around an axis about their center. They pull on the rope so they are now a distance of dF apart. What is the initial angular momentum of the astronauts?astronauts?

(a) 2 M (dI )2 M M( ) ( I )(b) M (dI )2 (c) ½ M (dI )2

M M

(d) ¼ M (dI )2 (e) 4 M (dI )2

dI


Clicker Question 6: Two astronauts 35 m apart of mass 75 kg are spinning with

an angular velocity of 45 rad/s around an axis about theiran angular velocity of 45 rad/s around an axis about their center. They pull on the rope so they are now a distance of 15 m apart. What angular speed do they rotate with now?now?

(a) 345 rad/s( )(b) 300 rad/s(c) 245 rad/s(d) 45 rad/s(e) 90 rad/s


Clicker Question 7:

IF LL 22 11 22

21

21

IFF MdMd

22IFF dd

2

I

F dd

Fd


Clicker Question 8:You want to measure the moment of inertia for a rotating oval object. Initially, this object has an angular speed of 35 rad/s. You drop a ring of mass 30 kg and radius 0 2 m on top of theYou drop a ring of mass 30 kg and radius 0.2 m on top of the object. The ring and object apply frictional forces on each other and eventually come to have the same angular speed of 24 rad/s.

What is the moment of inertia of the object?(a) 4.67 kg m2

( ) 2

ring

(b) 2.62 kg m2

(c) 3.45 kg m2

(d) 8.92 kg m2( ) g(e) 1.23 kg m2 i f


initial finalNo external torques, ang mom is conserved

Clicker Question 8:We can find the angular momentum:

And then set them equal:

ioi IL fof mrIL 2ioi fof

2

fmr foio mrII 2 62.2

fi

fo

mrI

kg m2

i


initial finalf

= i angular momentumangular momentum - u of t physicsameyerth/phy131f14/lect22c.pdf · torque = i...

Documents