## SAT Physics Subject Test

**Chapter 7 ****Oscillations**

**PENDULUMS**

You may see a question about pendulums on the SAT Physics Subject Test, so we’ll cover them in this chapter. A **simple pendulum** consists of a weight of mass *m* (called the *bob*) attached to a massless rod that swings, without friction, about the vertical equilibrium position. The restoring force is provided by gravity and as the figure below shows, the magnitude of the restoring force when the bob is at an angle *θ* to the vertical is given by the equation

*F*_{restoring} = *mg* sin *θ*

Although the displacement of the pendulum is measured by the angle that it makes with the vertical, instead of by its linear distance from the equilibrium position (as was the case for the spring–block oscillator), the simple pendulum has many of the same important features as the spring–block oscillator. For example

• displacement is zero at the equilibrium position.

• at the endpoints of the oscillation region (where *θ* = ± *θ*_{max}), the restoring force and the tangential acceleration (*a*_{t}) have their greatest magnitudes, the speed of the pendulum is zero, and the potential energy is maximized.

• as the pendulum passes through the equilibrium position, its kinetic energy and speed are maximized.

Despite these similarities, there is one important difference. Simple harmonic motion results from a restoring force that has a strength that’s proportional to the displacement. The magnitude of the restoring force on a pendulum is *mg* sin *θ*, which is *not* proportional to the displacement *θ*. Strictly speaking, the motion of a simple pendulum is not really simple harmonic. However, if *θ* is small, then sin *θ* ≈ *θ* (measured in radians), so in this case, the magnitude of the restoring force is approximately *mgθ*, which *is* proportional to *θ*. This means that if *θ*_{max} is small, the motion can be treated as simple harmonic.

If the restoring force is given by *mgθ*, rather than *mg* sin *θ*, then the frequency and period of the oscillations depend only on the length of the pendulum and the value of the gravitational acceleration, according to the following equations.

and

Notice that neither frequency nor period depends on the amplitude (the maximum angular displacement, *θ*_{max}); this is a characteristic feature of simple harmonic motion. Also notice that neither depends on the mass of the weight.