Physics Puzzler: Swinging Physics: The Answers
This works even if the pendulum bobs are large, so long as both bobs swing in exactly the same way, that is, along the same arc. And it even works if the angle of the swing is large. If the individual pieces didn’t swing in an arc of the same radius, gluing them together would result in a swing around some other arc, with a different period. This reminds us that the R in the formula for a simple pendulum is the radius of the arc of the bob’s motion, not simply the suspension length. A mass sliding on a frictionless track curved in the same circular arc would have the same period. Textbooks somewhat mislead by using L in the formula for the period, implying the length of the supporting string. The pendulum with a string support is only a special case.
If you would like more rigorous proof, you can derive the period starting with conservation of energy. //
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