Suppose that a spring is oscillating up and down with vertical position given by $u(t) = \sin(t)$. If you pick a large number of random $t$ to look at the position, then prove that the PDF is inversely proportional to the velocity of the spring.
2026-03-31 15:43:35.1774971815
Probability denisity function of simple harmonic
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This is basically a direct corollary of the statement that the probability of finding your particle at any interval $[u,u+du]$ is proportional to the time it spends within this interval (in each period of the oscillation) and the fact that this time is inverse proportional to the velocity of the particle at this interval (which can be taken to be constant as $dx\rightarrow 0$.