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Question: A man enters the pendulum clock shop with large number of clocks and takes a photograph. He finds that most of the pendulums were at the turning points and only a few were captured crossing the mid point. Why is it so?

Soln:

If we project the pendulum motion on a horizontal line it can be modeled as

x=a.sin(wt+phi)

where 'a' is the amplitude and 'w' is the angular frequency of the motion. Now consider the case we have lots of pendulum clocks hung on the wall of a shop. Then the inital phase will be a random variable whose value falls between (-pi to pi). Now if we apply the formula for function of single random variable we would get for x as

f(x)=1/(pi.sqrt(a^2-x^2)) -a<x<a

From the previous equation we observe that the density function diverges for the turning points and hence the pendulum should be spending most of the time there. Thats why naturally most of the photos show the pendulum at the extremas.

Thank you