# Rotational Dynamics Question

1. Apr 27, 2005

### Hoofbeat

Please could someone check my answers to the following and advise me on any of the bits I'm stuck on. Thanks

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Q. A disk of radius a and mass m is suspended at its centre by a vertical torsion wire which exerts a couple $$\fontsize{5} -c\theta$$ on the disk when it is twisted through an angle $$\fontsize{5} \theta$$ from its equilibrium position. Show that the oscillations of the disk are simple harmonic, and obtain an expression for the period [I've done this proof so that bit's fine]

A wire ring of mass m and radius a/2 is dropped onto the disk and sticks to it. Discuss what happens to (a) the period, (b) the amplitude, (c) the energy of the oscillations for the 2cases where the ring is dropped on (i) at the end of the swing when the disk is instantaenously at rest (ii) at the midpoint of the swing when the disk is moving with is maximum angular velocty. Assume the ring and disk are concentric.
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i) At end of swing
a) The period increase as the moment of inertia has increased
b) The amplitude increases as the velocity remains same as before, but the time period has increased
c) The energy is unchanged as at the end of the swing where v=0, the object only has potential energy.

ii) At midpoint of motion
a) The period increases due to increase in moment of inertia
b) ???
c) PE=0. There is a change in energy as the extra disk increases friction so the KE is lower?

Thanks.

Last edited: Apr 27, 2005
2. Apr 27, 2005

### OlderDan

You need to look at 1b again. The disk is at rest with the torsion wire twisted when the ring gets added. Does that make the wire twist more?

iib. What will happen to the angular velicocity at impact. This is the position of maximum velocity for the subsequent motion.

iic follows form iib. Whe you get b, think about c again

3. Apr 27, 2005

### Hoofbeat

So is 1b) The moment of inertia increases and as no external torques are acting, angular momentum must be conserved. We know $$L=I\omega$$ which is the same as $$L=IV/r$$ so an increase in the inertia will cause the velocity to decrease and thus the amplitude will remain the same (as slower speed and longer time period)?

Last edited: Apr 27, 2005
4. Apr 28, 2005

### OlderDan

Angular momentum is not conserved in this problem. This problem is analogous to a mass on a spring, where the spring bobs up and down, or to a pendulum. In fact this system is called a torsion pendulum. The torsion wire produces a torque in proportion to the angle through which it is twisted. The angular displacement of the disk goes back and forth. There is an anergy associate with the torsion, and there is a back and forth between stored energy and kinetic energy. The equations for the motion of this sytem are very similar to the equations for a mass on a spring.