Angular velocity - different ans by conserv E and momentum

1. Dec 19, 2007

catkin

[SOLVED] Angular velocity - different ans by conserv E and momentum

1. The problem statement, all variables and given/known data
This is Advanced Physics by Adams and Allday, spread 3.31, question 3.

A pirouetting skater halves his moment of inertia by pulling in his arms and legs closer to his axis of rotation.
a) By what factor does his angular velocity increase?

2. Relevant equations
Angular momentum $L = I \omega$

Rotational kinetic energy $R.K.E. = 0.5 I {\omega}^2$

3. The attempt at a solution
I think this problem is soluable by either conservation of momentum or conservation of energy but I get different answers using these methods.

Using subscript 1 to denote the skater's initial state and subscript 2 to denote the skater's final state,

by conservation of energy

$$L_1 = L_2$$

$$I_1 {\omega}_1 = I_2 {\omega}_2$$

$$I_1 {\omega}_1 = 0.5 I_1 {\omega}_2$$

$$\frac {{\omega}_2} {{\omega}_1} = \frac {I_1} {0.5 I_1}$$

$$\frac {{\omega}_2} {{\omega}_1} = 2$$

by conservation of energy

$$R.K.E._1 = R.K.E._2$$

$$0.5 I_1 {\omega_1}^2 = 0.5 I_2 {\omega_2}^2$$

$$I_1 {\omega_1}^2 = (0.5 I_1) {\omega_2}^2$$

$${\omega_1}^2 = 0.5 {\omega_2}^2$$

$$\frac {{\omega_2}^2} {{\omega_1}^2} = 2$$

$$\frac {\omega_2} {\omega_1} = \sqrt{2}$$

What am I doing wrong?

2. Dec 19, 2007

Hootenanny

Staff Emeritus
You cannot apply conservation of angular kinetic energy here since kinetic energy is not conserved.

3. Dec 19, 2007

catkin

Thanks Hootenanny. OK. If it is not conserved then where does it go?

4. Dec 19, 2007

Hootenanny

Staff Emeritus
How does the skater change his moment of inertia?

5. Dec 19, 2007

catkin

Ah ha! Many thanks.

As he pulls elements of his mass toward the axis he is doing work against the centripetal force.

6. Dec 19, 2007

Hootenanny

Staff Emeritus
Now, how can that possibly be? In what direction does the centripetal force act?

7. Dec 19, 2007

catkin

Thanks for sticking with me on this :-)

The centripetal force acts toward the axis (so it provides the centripetal acceleration that makes the elements of mass move in a circle).

Step by step (I think I went too fast) ...

"How does the skater change his moment of inertia?" By "compacting" his body, that is by bringing the outer parts (masses) closer to the axis. OK?

If his arms are initially horizontal, the tension in his wrists is providing the centripetal force on his hands. OK?

And now I'm stuck.

8. Dec 19, 2007

Hootenanny

Staff Emeritus
No worries
Spot on
Sounds good
Yup
So to bring his arms in, the skater must exert a force on his arms so...

9. Dec 19, 2007

catkin

It's not that he's working against the centripetal force but that he's providing the centripetal force ...

As he pulls elements of his mass toward the axis he is providing the centripetal force. This is force and movement in the direction of the force so work is done. The energy represented by this work goes into the R.K.E.

OK now?

Last edited: Dec 19, 2007
10. Dec 19, 2007

Hootenanny

Staff Emeritus
Spot on

11. Dec 19, 2007

catkin

And spot on help, thanks! :)

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