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Let $O$ be a fixed point of the body and suppose that $O, \underline{e_1},\underline{e_2}, \underline{e_3}$ form principal axes for the body with principal moments of inertia $A, B, C $. The body is free to rotate about the point $O$ about the axis $\underline{e_3}$ which are both fixed. The $\underline{e_3}$ axis is horizontal. Suppose that the centre of mass lies a distance $h$ along the positive $\underline{e_1}$ axis and let $\theta$ denote the angle between $\underline{e_1}$ and the downward vertical.

Deduce that, $$\ddot{\theta} = -\frac{Mgh}{C}sin{\theta}.$$

Using an appropriate approximation solve this equation for a small $\theta$.

Obviously a set shape is needed in order to find out the moment f inertia values for $A, B, C$. So i am unsure how to tackle this question without those.

Any ideas?

Connor Davies
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  • Typically you use the shape to figure out the moments of inertia. Here you are already given the moments of inertia, so you don't need the shape. So I'm not sure what your question is. –  Jun 08 '17 at 01:16
  • The equation of motion you have stated follows immediately from taking moments about $O$. For small $\theta$, $\sin\theta\simeq\theta$, leading to the SHM equation. – David Quinn Jun 08 '17 at 05:38

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