Angular Momentum & Torque for Systems of Particles

Lecturer: Professor Stephen T. Thornton

Reading Quiz

A particle is located in the xy-plane at a location x = 1 and y = 1 and is moving parallel to the +y axis. A force is exerted on the particle along the +x axis. L and are in what directions about the origin?

t

A) L and are along the +z axis.

B) L and are along the -z axis.

C) L is along the +z axis; is along the –z axis.

D) L is along the -z axis; is along the +z axis.

E) L is along the +y axis; is along the +x axis.

t

tt

t

t

·v

FO

x

y

x

y

z

·r

p

L r p= ´

F

t

Answer: C

·

Last Time

Angular momentum

Vector (cross) products

Torque again with vectors

Today

Angular momentum and torque

system of particles

rigid objects

Unbalanced torque

Kepler’s 2nd law

Atwood Machine. An Atwood machine consists of two masses, and connected by a cord that passes over a pulley free to rotate about a fixed axis. The pulley is a solid cylinder of radius and mass 0.80 kg. (a) Determine the acceleration a of each mass. (b) What percentage of error in a would be made if the moment of inertia of the pulley were ignored? Ignore friction in the pulley bearings.

A 7.0 kgm B 8.2 kgm

0 0.40 mR

System of Particles

The angular momentum of a system of particles can change only if there is an external torque—torques due to internal forces cancel.

This equation is valid in any inertial reference frame. It is also valid about a point uniformly moving in an inertial frame of reference. We are starting to get very technical!

System of Particles

The equation above is not valid in general about a point accelerating in an inertial frame of reference.

But the center of mass is special! The equation is true even for an accelerating center of mass of a system of particles or for a rigid object:

Angular Momentum for a Rigid Object

For a rigid object, we can show that its angular momentum when rotating around a particular axis is given by:

Add up all the particles. If L is along a symmetry axis (z here) through CM, particles on one side of symmetry axis cancel L on the other side.

iL

rotating

axisd Idt

I LL ω

So we finally have these equations for a rigid object.

The values must be calculated about

1) Origin or axis fixed in an inertial frame.

or

2) An origin at the CM or about an axis passing through the CM.

If we do not have this, then things get real complicated! We have reached our limit here!!

Torque and Angular Momentum Vectors

t L τ

r

ddt

L

ω

Torque

Gravity and Extended Objects

Gravitational torque acts at the center of mass, as if all mass were concentrated there:

Torque

Gravity and Extended Objects

Gravitational torque acts at the center of mass, as if all mass were concentrated there.

Do the Falling Rigid Body demo again.

Conceptual QuizYou are looking at a bicycle wheel along its axis. The wheel rotates CCW and is supported by a string attached to the rear of the handle. When the wheel is released, the end of the handle closest to you will A)  move upB)  move to the leftC)  move to the rightD)  move down

Do bicycle wheel demo.

Answer: CMove to the right. The picture below is looking from above.

iL

L fL

Torque

iLL

fL

r

Lt

iL

fL

Top view

Conceptual QuizA man sits at rest on a frictionless rotating stool. He holds a rotating bicycle wheel that has an angular momentum L directed up. When he flips the wheel over, so that it has L directed down, the angular momentum of the system (man + stool + wheel) is  A)   zero.B)   L, up.C)   L, down.D)   2L, up.E) 2L, down.

Answer: B

Angular momentum has to be conserved. There is no torque to change it.

Do experiment.

Angular Momentum and Torque for a Rigid Object

A system that is rotationally imbalanced will not have its angular momentum and angular velocity vectors in the same direction. A torque is required to keep an unbalanced system rotating.

inp

and inL

An unbalanced car wheel will cause problems on your wheel bearings. We need to keep our wheels well balanced, dynamically not just statically.

There is no torque so L is constant, and Kepler’s second law states that each planet moves so that a line from the Sun to the planet sweeps out equal areas in equal times.

1( sin )

21

sin but sin2

constant2

dA r v dt

dArv L m mrv

dtdA L

dt m

r v

Kepler’s 2nd Law

Conceptual QuizConceptual Quiz

You are holding a spinning bicycle wheel while standing on a stationary turntable. If you suddenly flip the wheel over so that it is spinning in the opposite direction, the turntable will:

A) remain stationaryA) remain stationary

B) start to spin in the same B) start to spin in the same direction as before flippingdirection as before flipping

C) to spin in the same direction C) to spin in the same direction as after flippingas after flipping

Conceptual QuizConceptual Quiz

You are holding a spinning bicycle wheel while standing on a stationary turntable. If you suddenly flip the wheel over so that it is spinning in the opposite direction, the turntable will:

The total angular momentum of the system is L upward, and it is conserved. So if the wheel has −L downward, you and the table must have +2L upward.

A) remain stationaryA) remain stationary

B) start to spin in the same B) start to spin in the same direction as before flippingdirection as before flipping

C) start to spin in the same C) start to spin in the same direction as after flippingdirection as after flipping

Conceptual QuizConceptual Quiz

A) disk 1A) disk 1

B) disk 2B) disk 2

C) not enough info C) not enough info

Two different spinning disks have the same angular momentum, but disk 1 has more kinetic energy than disk 2.

Which one has the bigger moment of inertia?

LL

Disk 1Disk 2

See hint on next slide.

Conceptual QuizConceptual QuizTwo different spinning disks have the same angular momentum, but disk 1 has more kinetic energy than disk 2.

Which one has the bigger moment of inertia?

A) disk 1A) disk 1

B) disk 2B) disk 2

C) not enough info C) not enough info

LL

Disk 1Disk 2

KE = I 2 = L2 (2 I) (used L = I ).

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Conceptual QuizConceptual QuizTwo different spinning disks have the same angular momentum, but disk 1 has more kinetic energy than disk 2.

Which one has the bigger moment of inertia?

A) disk 1A) disk 1

B) disk 2B) disk 2

C) not enough info C) not enough info

LL

Disk 1Disk 2

KE = I 2 = L2 (2 I) (used L = I ).

Because L is the same, bigger I means smaller KE.

12 /