Consider a circular disk suspended from a wire fixed to a ceiling. If the disk is rotated, the wire will twist. When the disk is released, the twisted wire exerts a restoring force on the disk, causing it to rotate past its equilibrium point, twisting the wire the other direction, as shown below. This system is called a torsional oscillator. It has been found experimentally that the torque exerted on the disk is proportional to the angular displacement of the disk, or:

where κ is a proportionality constant, a property of the wire. Note the similarity to our spring equation F = - kx. Since τ = Iα for any rotational motion we can state that

If we substitute m for I, k for κ, and x for θ we can see that this is the exact same differential equation we had for our spring system. Thus we may skip to the final solution, describing the angular displacement of the disk as a function of time:

where θ_m is defined as the maximum angular displacement and ω is the angular frequency given by ω = . Note: It is important not to confuse angular frequency and angular velocity. ω in this case refers to the angular frequency of the oscillation, and cannot be used for angular velocity.

From our expression for angular frequency we can derive that

This equation for the period of a torsional oscillator has a significant experimental use. Suppose a body of unknown moment of inertia is placed on a wire of known constant κ. The period of oscillation can be measured, and the moment of inertia of the body can be determined experimentally. This is quite useful, as the rotational inertia of most bodies cannot be easily determined using the traditional calculus-based method.

From our examination of the torsional oscillator we have derived that its motion is simple harmonic. This oscillator can almost be seen as the rotational analogue of the mass-spring system: just as with the mass-spring we substituted θ for x, I for m and κ for k. Not all simple harmonic oscillators have such close correlation.

Another common oscillation is that of the simple pendulum. The classic pendulum consists of a particle suspended from a light cord. When the particle is pulled to one side and released, it swings back past the equilibrium point and oscillates between two maximum angular displacements. It is clear that the motion is periodic--we want to see if it is simple harmonic.

We do so by drawing a free body diagram and examining the forces on the pendulum at any given time. The two forces acting on the pendulum at any given time are tension from the rope and gravity. At the equilibrium point the two are antiparallel and cancel exactly, satisfying our condition that there must be no net force at the equilibrium point. When the pendulum is displaced by an angle θ, the gravitational force must be resolved into radial and tangential components. The radial component, mg cosθ, cancels with the tension, leaving net tangential force;

In this case the restoring force is not proportional to the angular displacement θ, but is rather proportional to the sine of the angular displacement, sinθ. Strictly speaking, then, the pendulum does not engage in simple harmonic motion. However, most pendulums function at very small angles. If the angle is small we may make the approximation sinθ θ. With this approximation we can rewrite our force expression:

This equation does predict simple harmonic motion, as force is proportional to angular displacement. We can simplify by noticing that the linear displacement of the particle corresponding to an angle of θ is given by x = Lθ. Substituting this in, we see that:

Thus we have an equation in the same form as our mass-spring equation; in this case k = . We can skip the calculus and simply state the period of the pendulum:

pendulum

Note that the period, and thus the frequency, of the pendulum is independent of the mass of the particle on the cord. It only depends on the length of the pendulum and the gravitational constant. Keep in mind, also, that this is only an approximation. If the angle exceeds more than fifteen degrees or so, the approximation breaks down.

The torsional oscillator and the pendulum are two easy examples of simple harmonic motion. This type of motion, described by the same equations we have derived, comes up in molecular theory, electricity and magnetism, and even astronomy. The same method we applied in this section can be applied to any situation in which harmonic motion is involved.

Through our study of simple harmonic oscillations we have used sine and cosine functions, and talked about angular frequency. It seems natural that there should be some connection between simple harmonic motion and uniform circular motion. In fact, there is an astonishingly simple connection that can be easily seen.

Consider a particle traveling in a circle of radius R centered about the origin, shown below: What is the x coordinate of the particle as it goes around the circle? The particle is shown at point Q, at which it is inclined an angle of θ from the x-axis. Thus the position of the particle at that point is given by:

However, if the particle is traveling with a constant angular velocity ω, then we can express θ as: θ = ωt. In addition, the maximum value that x can take is at the point (R,0), so we can state that x_m = R. Substituting these expressions into our equation,

This is the exact form as our equation for displacement of a simple harmonic oscillator. The similarity leads us to a conclusion about the relation between simple harmonic motion and circular motion:

This is an astonishing statement. We can see this relation through the following example. Place a mass on a spring such that its equilibrium point is at the point x = 0. Displace the mass until it is at the point (R,0). At the same time that you release the mass, set a particle in uniform circular motion from the point (R,0). If the two systems have the same value for ω, then the x coordinate of the position of the mass on the spring and the particle will be exactly the same. This relation is a powerful application of the concepts of simple harmonic motion, and serves to increase our understanding about oscillations.