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## Homework Statement

This is Problem 6.34 from Kleppner:

A marble of radius b rolls back and forth in a shallow dish of radius R. Find the frequency of small oscillations. R>>b.

Apparently, the answer is [itex]\omega=\sqrt{\frac{5g}{7R}}[/itex].

## Homework Equations

## The Attempt at a Solution

Ok so I have my work so far shown in the uploaded picture. I chose my origin to be fixed to the center of the dish. You can see my expression for L and the problem is that term [itex]b^{2}\dot{\theta }[/itex]. If it were [itex](R - b)^{2}\dot{\varphi }[/itex] then I would have the answer but it isn't. I tried working with the geometry as much as possible e.g. considering areas of sectors swept out by the two different angles and seeing if they could be equal but I couldn't get anything useful. Any help would be appreciated.

https://fbcdn-sphotos-h-a.akamaihd.net/hphotos-ak-snc6/v/181191_324228377687326_892524912_n.jpg?oh=5761c9f6c173a7c7a5b8f99566ea0d3c&oe=50FF13AD&__gda__=1358899830_0b1d66b83972727376fa8fc0c558f7ce