Where does the particle spend most of its time?

An interesting question to ask in simple harmonic motion is 'Where does the particle spend most of its time?' This is a classical analogy of a question about the wavefunction for a quantum mechanical oscillator: where is an interaction most probable? The displacement of the classical simple harmonic oscillator as a function of time is given by a sine function, as shown in the graph. To answer this question you can divide this into an inner half (-0.5m < x < 0.5m) and an outer half (-1m < x < -0.5m or 0.5m < x < 1m), take random samples or 'snapshots' of the oscillator as a function of time, and count how many times you catch the particle in the inner or outer halves.

If you press the next button, you will see that an easy way to do this is to take a video of the oscillator (in this case a 'car' on a linear air-track connected to the ends by springs), and then watch that video frame by frame to obtain your random 'snapshots' of the motion. You'll notice that as this proceeds, the ratio of counts in the outer half to counts in the inner half (called o/i) converges towards a value of 1.8. This takes a number of samples to achieve as you need to build up the statistics to get a stable answer.

In the final slide you'll see that you can obtain the ratio o/i from the graph of the sinusoidal motion as exactly 1.80. This is obtained by drawing horizontal lines at x = 0.5 and x = -0.5m, and then tracing the points where these intercept the sine curve down onto the time axis. If you measure the widths of the yellow regions (particle in the outer half) to the blue regions (particle in the inner half), the ratio of these widths is 1.8.

You can also see this via the direct relation between circular motion (essentially simple harmonic motion in two dimensions) and simple 1D harmonic motion. The lengths of the circular segments in the outer halves of the motion (see yellow arrows around the circle) are 1.8 times those of the circular segments for the inner halves of the motion (see blue arrows around the circle), and again you arrive at the same answer. Incidentally, the best way to demonstrate the equivalence of simple harmonic and uniform circular motions is to put a candle on a record turntable, turn out the lights and view it from above (in which case you see circular motion) and from the side (in which case you see 1D harmonic motion). This can be quite useful for making sense of concepts like angular frequency when its transferred from circular motion to harmonic motion.

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