Hi everyone, not a homework problem, just something I was thinking about when I found in a book.(adsbygoogle = window.adsbygoogle || []).push({});

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

A hollow Cylinder is rolling down an incline of angle theta. Inside the cylinder is a smaller solid cylinder rolling freely inside it. (Refer to diagram)

Given an x coordinate from the top of the slope which the cylinders were released, and:

Large Cylinder has mass M, moment of inertia I, radius R

Small cylinder has mass m, moment of inertia i, radius r

(p.s. I labeled the small mass m wongly in the diagram!)

The small cylinder is allowed to roll freely in the large cylinder, and we call the angle it makes from vertical phi.

What is theTotal Kinetic Energyof the system?

2. Relevant equations

v=rw

K=(1/2)Iw^2 > where w is angular velocity

3. The attempt at a solution

The Kinetic Energy for the Large Cylinder seems straightforward:

[tex]

K=\frac{1}{2}\left(M\dot{x}^2+\frac{I\dot{x}^2}{R^2}\right)

[/tex]

which is the translational KE and the rotational KE.

However, I am not sure about about the smaller cylinder. I believe there are 3 parts to its KE, namely the translational KE, the KE associated with it doing a circular motion on the rim of the large cylinder (I use the parallel axis theorem), and its own rotational KE.

[tex]

K=\frac{1}{2}\left(m\dot{x}^2+\left(i+m\left(R-r\right)^2\right)\dot{\phi}^2+\frac{i\dot{x}^2}{r^2}\right)

[/tex]

However, I am unsure about this because the small cylinder is also rotating somewhat around the large cylinder which is itself rolling down the slope.

Any advice or help is appreciated! Thank you!

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# Homework Help: Kinetic Energy of two Rolling Objects on a Slope

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