There is an easy way to produce simple harmonic motion by using uniform circular motion. Figure 2 shows one way of using this method. A ball is attached to a uniformly rotating vertical turntable, and its shadow is projected on the floor as shown. The shadow undergoes simple harmonic motion. Hooke’s law usually describes uniform circular motions (ω$\omega$ constant) rather than systems that have large visible displacements. So observing the projection of uniform circular motion, as in Figure 2, is often easier than observing a precise large-scale simple harmonic oscillator. If studied in sufficient depth, simple harmonic motion produced in this manner can give considerable insight into many aspects of oscillations and waves and is very useful mathematically. In our brief treatment, we shall indicate some of the major features of this relationship and how they might be useful.

Figure 3 shows the basic relationship between uniform circular motion and simple harmonic motion. The point P travels around the circle at constant angular velocity \(\omega\)$$. The point P is analogous to an object on the merry-go-round. The projection of the position of P onto a fixed axis undergoes simple harmonic motion and is analogous to the shadow of the object. At the time shown in the figure, the projection has position x$x$ and moves to the left with velocity \(v\). The velocity of the point P around the circle equals \(\overline{v}_{\mathrm{max}}\)$$.The projection of \(\overline{v}_{\mathrm{max}}\)$$ on the \(x-\mathrm{axis}\) is the velocity \(v\)$$ of the simple harmonic motion along the \(x-\mathrm{axis}\).

To see that the projection undergoes simple harmonic motion, note that its position \(x\) is given by

where \(\theta = \omega t, \omega\) is the constant angular velocity, and \(X\)$$ is the radius of the circular path. Thus,

The angular velocity \(\omega\)$$ is in radians per unit time; in this case \(2 \pi\) radians is the time for one revolution \(T\). That is, \(\omega = 2 \pi /T\). Substituting this expression for \(\omega\)$$, we see that the position \(x\)$$ is given by:

This expression is the same one we had for the position of a simple harmonic oscillator in Simple Harmonic Motion: A Special Periodic Motion. If we make a graph of position versus time as in Figure 4, we see again the wavelike character (typical of simple harmonic motion) of the projection of uniform circular motion onto the \(x\)-axis.

Now let us use Figure 4 to do some further analysis of uniform circular motion as it relates to simple harmonic motion. The triangle formed by the velocities in the figure and the triangle formed by the displacements \((X, x, \thinspace \mathrm{and} \thinspace \sqrt{X^2-x^2})\) are similar right triangles. Taking ratios of similar sides, we see that

This expression for the speed of a simple harmonic oscillator is exactly the same as the equation obtained from conservation of energy considerations in Energy and the Simple Harmonic Oscillator. You can begin to see that it is possible to get all of the characteristics of simple harmonic motion from an analysis of the projection of uniform circular motion.

Finally, let us consider the period \(T\)$$ of the motion of the projection. This period is the time it takes the point P to complete one revolution. That time is the circumference of the circle \(2 \pi X\) divided by the velocity around the circle, \(v_{\mathrm{max}}\)${\text{max}}_{}$. Thus, the period \(T\)$$ is

We know from conservation of energy considerations that

Solving this equation for \(X/v_{\mathrm{max}}\)$$ gives

Substituting this expression into the equation for \(T\)$$ yields

Thus, the period of the motion is the same as for a simple harmonic oscillator. We have determined the period for any simple harmonic oscillator using the relationship between uniform circular motion and simple harmonic motion.

Some modules occasionally refer to the connection between uniform circular motion and simple harmonic motion. Moreover, if you carry your study of physics and its applications to greater depths, you will find this relationship useful. It can, for example, help to analyze how waves add when they are superimposed.

A guitar string stops oscillating a few seconds after being plucked. To keep a child happy on a swing, you must keep pushing. Although we can often make friction and other non-conservative forces negligibly small, completely undamped motion is rare. In fact, we may even want to damp oscillations, such as with car shock absorbers.

For a system that has a small amount of damping, the period and frequency are nearly the same as for simple harmonic motion, but the amplitude gradually decreases as shown in Figure 1. This occurs because the non-conservative damping force removes energy from the system, usually in the form of thermal energy. In general, energy removal by non-conservative forces is described as 

where \(W_{\mathrm{nc}}\)${\text{nc}}_{}$ is work done by a non-conservative force (here the damping force). For a damped harmonic oscillator, \(W_{\mathrm{nc}}\) is negative because it removes mechanical energy (KE + PE) from the system.

If you gradually increase the amount of damping in a system, the period and frequency begin to be affected, because damping opposes and hence slows the back and forth motion. (The net force is smaller in both directions.) If there is very large damping, the system does not even oscillate—it slowly moves toward equilibrium. Figure 3 shows the displacement of a harmonic oscillator for different amounts of damping. When we want to damp out oscillations, such as in the suspension of a car, we may want the system to return to equilibrium as quickly as possible Critical damping is defined as the condition in which the damping of an oscillator results in it returning as quickly as possible to its equilibrium position The critically damped system may overshoot the equilibrium position, but if it does, it will do so only once. Critical damping is represented by Curve A in Figure 3. With less-than critical damping, the system will return to equilibrium faster but will overshoot and cross over one or more times. Such a system is underdamped; its displacement is represented by the curve in Figure 2. Curve B in Figure 3 represents an overdamped system. As with critical damping, it too may overshoot the equilibrium position, but will reach equilibrium over a longer period of time. 

Critical damping is often desired, because such a system returns to equilibrium rapidly and remains at equilibrium as well. In addition, a constant force applied to a critically damped system moves the system to a new equilibrium position in the shortest time possible without overshooting or oscillating about the new position. For example, when you stand on bathroom scales that have a needle gauge, the needle moves to its equilibrium position without oscillating. It would be quite inconvenient if the needle oscillated about the new equilibrium position for a long time before settling. Damping forces can vary greatly in character. Friction, for example, is sometimes independent of velocity (as assumed in most places in this text). But many damping forces depend on velocity—sometimes in complex ways, sometimes simply being proportional to velocity.