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Number Theory/Pendulum Wall Clock.

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Question
Dear Prof Scott

Is it possible to design and Construct a Pendulum clock where the pendulum can rotate 180 degree?.

Thanks
Prashant

Answer
No, for the pendulum swings slightly less every time.  I mean, it may start out trying to do 180 by being straight out on one side, but it will quickly lose that potential that when it swings to the other side.  I think the swing rate is determined by a multiple of cos(x) where x is the angle the pendulum swings from straight up and down.  As x approaches 90, the cos(x) approaches 0.  As x stays near 0, the cos(x) stays very close to 1.  As can be seen in calculus, if f(x) = cos(x), f'(x) = -sin(x).  With x near 0, -sin(x) is even closer to 0, so the cos(x) curve is almost level at this point.

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Scott A Wilson

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