# Homework Help: Rigid Body Motion - cylinder on a plane

1. Apr 7, 2010

### Salviati

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

Pretty picture:

I am currently stuck on #1.

2. Relevant equations
Not exactly sure but...
For #1, I assume the relevant equations are those relating the (i) gravitational force to momentum and (ii)torque due to gravity to angular momentum:

(i) F = dP/dt
(ii) K = dM/dt, where K = r x F

3. The attempt at a solution
I don't know how to begin. I figured maybe if I find r of the center of mass, take the absolute value of the derivative with respect to time then I'll know the speed...but where do I place the coordinate system then? Or I thought, if I use the equations listed above to find the momentum and angular momentum, I could do something with those to find the speed. I just don't know, quite confused right now.

Last edited: Apr 7, 2010
2. Apr 7, 2010

### Salviati

Oookay, here's a more serious attempt at an answer:

The instantaneous axis of rotation for this problem is the axis where the cylinder is in contact with the plane. So, the motion of the cylinder can be described as a rotation around this axis (not sure how to imagine this, but ok). This simplifies the problem because the distance of the center of mass from this axis is:
$$\sqrt{a^2+R^2-2aRcos\phi}$$, where $$\phi$$ is the angle between the perpendicular from the instantaneous axis of rotation and the center of mass. Therefore, the velocity of the cylinder is:
$$V = \dot{\phi}$$$$\sqrt{a^2+R^2-2aRcos\phi}$$.

The immediate problem that I see with this answer is that it doesn't incorporate the moment of inertia, I, like it says it should in the question.

3. Apr 7, 2010

### vela

Staff Emeritus
Your answer looks fine. Just make sure it gives you sensible results for easy-to-check cases. I'm not sure why the problem mentioned I since it's not a geometrical parameter.