# Probability density function of a pendulum displacement

#### venkatmn

Hi, I need a verification for this question. Can some one help me?

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

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#### Shooting Star

Homework Helper
The logic is OK. In the original question, no mathematics was asked for. A physicist would simply say that the speed in the region near the extrema is slower than the speed in the middle, and so the bob spends more time near the extrema per cycle, and hence the probability is more of finding it near the end points than near the middle.

#### venkatmn

Thanks for the verification but I am looking for a mathematical model describing the pendulum motion.........