If you shine a light from the side onto an object attached to a rotating circular plate, as suggested on the left side of the original figure, you see a projection of the object’s motion on the vertical axis as a shadow on the wall. This projection reveals a periodic up-and-down motion, with the greatest change occurring at the center.
The relationship between the maximum displacement $y_0$—which corresponds to the distance $r$ from the object to the center of the plate—and the instantaneous position $y(t)$ can be derived directly from the circular motion:
$$
y(t) = y_0 \sin \alpha = y_0 \sin (\omega t)
$$
Plotting this position against the angle $\alpha = \omega t$ yields a graph that represents an oscillation around a central point with amplitude $y_0$. An oscillation that can be described by a sine function is called a harmonic oscillation.
It is important to note that, due to the close relationship between sine and cosine, the motion can also be described using the cosine function. The result is the same wave, but shifted in phase by 90 degrees.
Which of the two functions should be used depends on the initial conditions:
- If the displacement is maximum at time $t = 0$, the motion is best described by the cosine function.
- If the velocity is maximum and the displacement $y(t) = 0$ at $t = 0$, the sine function is appropriate.