# Rotational mechanics, angular momentum revision

1. May 10, 2012

### Nono713

1. The problem statement, all variables and given/known data

The problem is attached as an image. Note this is from a past exam.

2. Relevant equations

Conservation of angular momentum.
Rotational kinetic energy.

3. The attempt at a solution

a) The moment of inertia of the man and stool is given as 3 kg m^2, and the dumbells can be considered point masses, so we can just add them all up:

$$I_\mathrm{initial} = I_\mathrm{man} + 2I_\mathrm{dumbell~~~ away} = 3 + 2(3 \times 1^2) = 9$$

$$I_\mathrm{final} = I_\mathrm{man} + 2I_\mathrm{dumbell ~~~pulled~~~ in} = 3 + 2(3 \times 0.25^2) = 3.375$$

b) The initial angular velocity of the man is $$1.5$$, and its initial moment of inertia is $$9$$, so the system's angular momentum is $$L = 1.5 \times 9 = 13.5$$. From conservation of angular momentum, the system must have the same angular momentum after the dumbells have been pulled in, so $$L = I_\mathrm{final} \omega_\mathrm{final}$$. So $$\omega_\mathrm{final} = \frac{L}{I_\mathrm{final}} = \frac{13.5}{3.375} = 4$$.

c) Using the rotational kinetic energy formula:

$$K_\mathrm{initial} = \frac{1}{2} I_\mathrm{initial} \omega^2_\mathrm{initial} = \frac{1}{2} \times 9 \times 1.5^2 = 10.125 J$$

$$K_\mathrm{final} = \frac{1}{2} I_\mathrm{final} \omega^2_\mathrm{final} = \frac{1}{2} \times 3.375 \times 4^2 = 27 J$$

I am note sure I got it right, shouldn't kinetic energy be conserved? Or does some of it go into potential energy because of the increased radius? I mean clearly the guy is going to spin faster, so the extra kinetic energy must be coming from somewhere.

PS: imagine the correct units are in there, I'm just too lazy to type them up in LaTeX :tongue:

#### Attached Files:

• ###### Problem.png
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2. May 10, 2012

### Staff: Mentor

Where did the energy come from to move the dumbbells?

3. May 10, 2012

### Nono713

Does it come from the work done by the man to pull the dumbells in closer? (chemical energy stored in the man's muscles I suppose) - or maybe I'm overthinking it?

4. May 10, 2012

### Staff: Mentor

No, not overthinking; That's correct. So, since mechanical energy is being added by a system-internal source, the kinetic energy will not be conserved.

5. May 10, 2012

### Nono713

Right, that makes sense! Thanks a lot!