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Position probability distribution for a pendulum

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Position probability distribution for a pendulum. Consider the classical motion of a pendulum bob which, for small amplitudes of oscillation, moves effectively as a harmonic oscillator along a horizontal axis according to the equation.

x(t) = Asinwt

The probability that the bob will be found within a small distance delta x at x in random observations is proportional to the time it spends in this region during each swing. Obtain a mathematical expression for this probability as a function of x, assuming delta x<<A.

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Solution Summary

We derive the position probability distribution of a classical pendulum (harmonic oscillator) undergoing small oscillations about equilibrium.

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We are given a classical pendulum with small amplitude, which moves effectively as a harmonic oscillator along a horizontal axis according to the equation:

x(t) = Asinwt


** Please see the attached file for the complete solution **

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