Angular momentum

•  CAPA Set #10 is due today

Web page: http://www.colorado.edu/physics/phys1110/phys1110_sp12/

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Pulling the yo-yo Yo-yo with inner radius R1 and outer radius R2 is on a table. String is pulled with force F as shown. There is enough static friction to prevent the yo-yo from sliding. Which way does the yo-yo move?

Rolling without slipping: and

so

Substituting gives

Rearranging:

but you need to make sure you get the signs right!

R2

Pulling the yo-yo

Applying rolling without slipping: and

Using the extended free body diagram and Newton’s 2nd law we found the equations:

After some math: (which is to the left)

then F exerts no torque and

Clicker question 1 Set frequency to BA

Two 0.2 kg spools of thread are on a frictionless table. Each is pulled with 0.4 N of force. A is free to rotate (and let out string) while in B the string is attached to the spindle. After 2 seconds, which will be moving faster (largest center of mass speed)?

A.  A B.  B C.  same speed

A

B

Top view

Clicker question 1 Set frequency to BA

Two 0.2 kg spools of thread are on a frictionless table. Each is pulled with 0.4 N of force. A is free to rotate (and let out string) while in B the string is attached to the spindle. After 2 seconds, which will be moving faster (largest center of mass speed)?

A.  A B.  B C.  same speed

A

B

Top view

Clicker question 2 Set frequency to BA

Two 0.2 kg spools of thread are on a frictionless table. Each is pulled with 0.4 N of force. A is free to rotate (and let out string) while in B the string is attached to the spindle. After 2 seconds, which one will have the most kinetic energy?

A.  A B.  B C.  same kinetic energy

A

B

Top view

Clicker question 2 Set frequency to BA

Two 0.2 kg spools of thread are on a frictionless table. Each is pulled with 0.4 N of force. A is free to rotate (and let out string) while in B the string is attached to the spindle. After 2 seconds, which one will have the most kinetic energy?

A.  A B.  B C.  same kinetic energy

A

B

Top view

Spool of thread Two 0.2 kg spools on frictionless table are pulled with F=.4 N. A is free to rotate while the B string is attached. After 2 seconds what’s going on?

A

B

Only 1 force in x direction so gives

velocity of

for a center of mass

Translational kinetic energy

Distance spool travels:

Work done by force on spool B:

For spool B, all work goes into translational kinetic energy

For A: and so

Spool of thread A

B

For spool A, extra 3.2 J of work goes into rotational kinetic energy

Previous slide used . What about the rotational part? For B no torque because r and F are parallel.

Let’s set

After 2 seconds Rotational KE

Amount of string unwound: Work done unwinding string:

Question about frictionless surface For spool A (R=2 cm) we learned the following (after 2 seconds):

A.  Yes B.  No A

Top view

Spool moves 4 m Spool rotates 400 rad

Is this an example of rotating without slipping?

If this were rolling without slipping then

None of these are true. There is no friction between table and spool causing rotation without slipping.

The thread coming off the spool is sort of rolling without slipping

Quick word on work and power In the linear case we have

The equivalent for angular motion is

The force causing the torque must be tangential to the rotation. That is, torque and angular displacement must be in the same (or opposite) direction.

Remember power is work per time: For linear motion also have For rotational motion also have

Angular momentum

Angular momentum:

We again have another angular analog to the linear case. Linear momentum gives an idea about how hard it would be to slow something down. Angular momentum tells us how hard it is to slow down something which is spinning. Linear momentum:

Newton’s 2nd law becomes Can also talk about the angular momentum of a particle (like a satellite orbiting the Earth). In this case, angular momentum is so

Clicker question 3 Set frequency to BA

A planet in elliptical orbit about the Sun is in the position shown. With the origin located at the Sun, the torque on the planet… A.  is 0 B.  is in the +z direction C.  is in the –z direction D.  is in the x-y plane E.  is none of these

planet

S x

y

z

Clicker question 3 Set frequency to BA

A planet in elliptical orbit about the Sun is in the position shown. With the origin located at the Sun, the torque on the planet… A.  is 0 B.  is in the +z direction C.  is in the –z direction D.  is in the x-y plane E.  is none of these For any central force like gravity, the force vector is parallel (or antiparallel) to the vector so no torque can be applied.

Thus, a central force cannot affect the angular momentum

planet

S x

y

z

Conservation of angular momentum Previously found if no external force acts on a system, momentum is constant. We called this conservation of momentum.

Similarly, if no external torque acts, angular momentum is constant. This is conservation of angular momentum.

Since , if then so is constant

Just like internal forces inside a system cannot produce a net force on the system due to Newton’s 3rd law, internal torques cannot produce a net torque so only external torques can change the total angular momentum of the system.

Conservation of angular momentum

Similarly, for angular momentum conservation problems we will usually use the equation or

To solve problems in which momentum is conserved we generally used the equation or

The spinning professor Consider a person with arms outstretched holding weights and the same person with arms tucked in holding weights. Could estimate moment of inertia for both situations using a cylinder for the body, rods for the arms, and point masses for the weights.

Clearly so if there is an initial angular velocity then since there is no external torque so

Note if angular momentum is conserved but then rotational kinetic energy is not conserved:

Where does the work to change the kinetic energy come from? Force over distance is required to pull the weight in.

Clicker question 4 Set frequency to BA

A disk of mass M and radius R is rotating with angular velocity ω0. Consider the effect on the angular velocity if a small object of mass m is dropped onto the disk. If the object is dropped at a radius of R the final angular velocity is ω1 and if the object is dropped at a radius of R/2 the final angular velocity is ω2. What is the relationship between the various angular velocities? A.  B.  C.  D.  E. 

Clicker question 4 Set frequency to BA

A disk of mass M and radius R is rotating with angular velocity ω0. Consider the effect on the angular velocity if a small object of mass m is dropped onto the disk. If the object is dropped at a radius of R the final angular velocity is ω1 and if the object is dropped at a radius of R/2 the final angular velocity is ω2. What is the relationship between the various angular velocities? A.  B.  C.  D.  E. 

Conservation of angular momentum gives us:

so

Since there is no external torque, the overall angular momentum of the system remains the same which is how we solved the problem.

Note that if we just consider the disk, the angular momentum decreased. Where did the torque come from?

More on clicker question 4

When the mass lands, friction acts to cause an acceleration of the mass and the equal and opposite force slows the disk.

Need to carefully determine what is part of the system on which conservation of momentum can be applied.

New types of collision problems If an object comes in with some initial angular momentum then we need to account for that as well.

Initial angular momentum about the disk axis is

Angular momentum is conserved in the collision.

If the axis is fixed in place then momentum is not conserved (it is conserved when you consider the Earth as part of the system).

If the axis is not fixed then momentum is also conserved.

Angular momentum is a vector If I get on the rotating platform, initially at rest and start spinning a bicycle wheel clockwise whose axis points up, what will happen?

I will start rotating in the opposite direction.

Initial angular momentum is 0 and no external torque is applied so final angular momentum must also be 0.

Bicycle wheel has angular momentum pointing down (clockwise with vertical axis upward) Something else in the system must have angular momentum pointing in the opposite direction to have a total angular momentum of 0. (“Something else” is a rotating professor.)

Clicker question 5 Set frequency to BA

Three identical wheels are all spinning with the same angular velocity . The total angular momentum of the 3-wheel system has magnitude L. One of the three wheels is flipped upside-down, while the magnitude of its angular velocity remains constant. The new angular momentum of the 3-wheel system has magnitude… A.  L (same as before) B.  L/3 C.  2L/3 D.  L/2 E.  Some other value

Clicker question 5 Set frequency to BA

Three identical wheels are all spinning with the same angular velocity . The total angular momentum of the 3-wheel system has magnitude L. One of the three wheels is flipped upside-down, while the magnitude of its angular velocity remains constant. The new angular momentum of the 3-wheel system has magnitude… A.  L (same as before) B.  L/3 C.  2L/3 D.  L/2 E.  Some other value Angular momentum is a vector.

Initial: Final: