Proof - Sinusoidal Motion

In this section we prove

1. that a linear F-x graph gives sinusoidal motion,
2. that the period of the motion is and
3. that the period is independent of the amplitude.

The object moves along the circle at constant speed, but even though its overall speed is constant, the x and y components of its velocity are continuously changing, as shown by the unequal spacing of the points when projected onto the line below. Projected onto the line, its motion is the same as that of an object experiencing a force F=-kx.

The basic idea of the proof can be understood by imagining that you are watching a child on a merry-go-round from far away. Because you are in the same horizontal plane as her motion, she appears to be moving from side to side along a line. Circular motion viewed edge-on doesn't just look like any kind of back-and-forth motion, it looks like motion with a sinusoidal x-t graph, because the sine and cosine functions can be defined as the x and y coordinates of a point at angle q on the unit circle. The idea of the proof, then, is to show that an object acted on by a force that varies as F=- kx has motion that is identical to circular motion projected down to one dimension. The equation

will also fall out nicely at the end.

For an object performing uniform circular motion, we have

|a| = v²/r .

The x component of the acceleration is therefore

where θ is the angle measured counterclockwise from the x axis. Applying Newton's second law,

Since our goal is an equation involving the period, it is natural to eliminate the variable v = circumference/T = 2πr/T, giving

The quantity r cos θ is the same as x, so we have

Since everything is constant in this equation except for x, we have proven that motion with force proportional to x is the same as circular motion projected onto a line, and therefore that a force proportional to x gives sinusoidal motion. Finally, we identify the constant factor of 4π ² m /T ² with k, and solving for T gives the desired equation for the period,

Since this equation is independent of r, T is independent of the amplitude.

The moons of Jupiter.