# A problem about harmonic oscillators

A ball with mass $m$ and radius $r$ rolls without sliding inside a cylinder with radius $R (R>>r)$, with $\theta <<1$. Find the angular frequency $\omega$

What I Know: There are two movements involved: the rotation of the center of mass around the center of the cylinder $C$ and the rotation around the center of mass. These two movements are coupled because there is no sliding. So, if $\theta_{1}(t)$ is the angular displacement associated to the rotation around the center of mass and $\theta_{2}(t)$ is the angular displacement associated to the rotation of the center of mass, then:

$R\dot\theta_{2} = r\dot\theta_{1}$

If the movement were merely the rotation of the center of mass, then: $ma_{\theta} = -mg\sin(\theta) \therefore mR\ddot\theta = -mg\sin(\theta)$ (the tangent axis is oriented to increasing values of $\theta$) This would give $\ddot\theta + \frac{g}{R} \theta = 0$, considering $\sin \theta \approx \theta$. In this case, we would simply have $\omega = \sqrt{\frac{g}{R}}$.

I don't know how to account for the fact that the ball is also rolling. Could someone give me a hint to solve the original problem, considering the existence of two movements?

• This site is going to be of help for you. Also, think what distance makes travels the ball when it makes a complete rotation in its own center-of-mass. Take a point of the top of the ball and think what distance along the cylinder bottom travels the ball until the point returns to the top of the ball. Mar 8, 2015 at 22:56

• using exactly the same method you used in your earlier answer, I found that $\ddot\theta_{1} = \frac{5g}{7r}\theta_{1}$, where $\theta_{1}$ is the angular displacement associated with the rotation around the center of mass. As $\theta_{1}r = \theta_{2}R$ (because there is no sliding), I would get $\ddot\theta_{2} = \frac{5g}{7r}\theta_{2}$ and $\omega = \sqrt{\frac{5g}{7r}}$. Is this correct? Mar 9, 2015 at 2:13
• It doesn't look right - your frequency must depend on R since that defines the "potential well". You made a mistake going from $\theta_1$ to $\theta_2$ (your equation looked right but you didn't use it). Mar 9, 2015 at 3:04