Speed of Cylinder & Hoop Down Ramp: Analysis

[missnuss]

Homework Statement

Consider both a cylinder of radius Rc and mass mc and a hoop of radius Rh and mass mh. If both are at the top of a ramp of length L and at an angle theta what are the objects speed at the bottom of the ramp?

Homework Equations

The Attempt at a Solution

 

[issacnewton]

You have to show your work before people help you. That IS the modus operandi of this
forum. Check thread started by me for example...

 

[missnuss]

Sorry. I didn't post what I had tried to do because I have no clue where to even start. I'm not looking for the answer, I just wanted some help.

 

[physman55]

Use conservation of energy. Are the cylinders "allowed" to roll down the ramp? If so, you have to add in a rotational kinetic energy term into your equation.

 

[issacnewton]

If L is the length of the ramp, then the bottom point of any symmetric object would be
at height $L\sin \theta$. Using conservation of energy, the potential energy
at the top should be equal to the sum of translational and rotational kinetic energies
at the bottom. For an object with moment of inertia about its center of mass I,

$$mg(L \sin \theta)= \frac{1}{2}mv^2 + \frac{1}{2}I \omega^2$$

for pure rolling motion, $\omega = v/R$ so

$$mg(L \sin \theta)= \frac{1}{2}mv^2 + \frac{1}{2}I \frac{v^2}{R^2}$$

solving for v, we have

$$v = \left[\frac{2gL \sin \theta}{1+(I/mR^2)}\right]^{1/2}$$

from here, you can derive special cases for different bodies, in your case, a solid
cylinder (or is it hollow) and a hoop.

 

[missnuss]

Thank you so very much