The graphs of Figure 15.3 and the position function $x\left(t\right)=A\text{cos}\omega t$ are for an oscillation in which the object just happened to be at $x_0=A$ at $t=0$. But you will recall that $t=0$ is an arbitrary choice, the instant of time when you or someone else starts a stopwatch. What if you had started the stopwatch when the object was at $x_0=-A$, or when the object was somewhere in the middle of an oscillation? In other words, what if the oscillator had different initial conditions? The position graph would still show an oscillation, but neither Figure 15.3 nor $x\left(t\right)=A\text{cos}\omega t$ would describe the motion correctly.

To learn how to describe the oscillation for other initial conditions it will help to turn to a topic you studied in Chapter 4—circular motion. There’s a very close connection between simple harmonic motion and circular motion.

Imagine you have a turntable with a small ball glued to the edge. FIGURE 15.4a shows how to make a “shadow movie” of the ball by projecting a light past the ball and onto a screen. The ball’s shadow oscillates back and forth as the turntable rotates. This is certainly periodic motion, with the same period as the turntable, but is it simple harmonic motion?

FIGURE 15.4 A projection of the circular motion of a rotating ball matches the simple harmonic motion of an object on a spring.

To find out, you could place a real object on a real spring directly below the shadow, as shown in FIGURE 15.4b. If you did so, and if you adjusted the turntable to have the same period as the spring, you would find that the shadow’s motion exactly matches the simple harmonic motion of the object on the spring. Uniform circular motion projected onto one dimension is simple harmonic motion.

To understand this, consider the particle in FIGURE 15.5. It is in uniform circular motion, moving counterclockwise in a circle with radius A. As in Chapter 4, we can locate the particle by the angle $\varphi$ measured counterclockwise (ccw) from the x-axis. Projecting the ball’s shadow onto a screen in Figure 15.4 is equivalent to observing just the x-component of the particle’s motion. Figure 15.5 shows that the x-component, when the particle is at angle $\varphi$, is

FIGURE 15.5 A particle in uniform circular motion with radius $A$ and angular velocity $\omega$.

Recall that the particle’s angular velocity, in $\text{rad}\text{/}\text{s}$, is

This is the rate at which the angle $\varphi$ is increasing. If the particle starts from ${\varphi }_0=0$ at $t=0$, its angle at a later time t is simply

As $\varphi$ increases, the particle’s x-component is

This is identical to Equation 15.4 for the position of a mass on a spring! Thus the x-component of a particle in uniform circular motion is simple harmonic motion.

The names and units can be a bit confusing until you get used to them. It may help to notice that cycle and oscillation are not true units. Unlike the “standard meter” or the “standard kilogram,” to which you could compare a length or a mass, there is no “standard cycle” to which you can compare an oscillation. Cycles and oscillations are simply counted events. Thus the frequency $f$ has units of hertz, where $1\text{Hz}=1\text{s}{}^{-1}$. We may say “cycles per second” just to be clear, but the actual units are only “per second.”

The radian is the SI unit of angle. However, the radian is a defined unit. Further, its definition as a ratio of two lengths $(\theta =\text{s}\text{/}r)$ makes it a pure number without dimensions. As we noted in Chapter 4, the unit of angle, be it radians or degrees, is really just a name to remind us that we’re dealing with an angle. The $2\pi$ in the equation $\omega =2\pi f$ (and in similar situations), which is stated without units, means $2\pi \text{rad}\text{/}\text{cycle}$. When multiplied by the frequency $f$ in $\text{cycles}\text{/}\text{s}$, it gives the frequency in $\text{rad}\text{/}\text{s}$. That is why, in this context, $\omega$ is called the angular frequency.