Simple Pendulums for AP Physics B & C

Practice problems for these concepts can be found at: Simple Harmonic Motion Practice Problems for AP Physics B & C

Problems that involve simple pendulums—in other words, basic, run-of-the-mill, grandfather clock-style pendulums—are actually really similar to problems that involve springs. For example, the formula for the period of a simple pendulum is this:

Looks kind of like the period of a mass on a spring, right? In this equation, L is the length of the pendulum, and g is the acceleration attributable to gravity (about 10 m/s²). Of course, if your pendulum happens to be swinging on another planet, g will have a different value.¹

One interesting thing about this equation: The period of a pendulum does not depend on the mass of whatever is hanging on the end of the pendulum. So if you had a pendulum of length L with a peanut attached to the end, and another pendulum of length L with an elephant attached to the end, both pendulums would have the same period in the absence of air resistance.

To calculate the period of this pendulum, we must know the length of the string. We can calculate this using conservation of energy. Then, we'll convert the period to a frequency.

Before the string is released, all of the bowling ball's energy is in the form of gravitational PE. If we define the zero of potential to be at the ball's lowest point, then at that point all the bowling ball's energy is in the form of KE. We will use a subscript "a" to represent values before the bowling ball is released and "b" to represent values when the bowling ball is at its lowest point.

The height of the bowling ball before it is released, h_a, can be calculated using trigonometry.

So, getting back to our previous equation, we have

0 + mg(L – L cos θ) = 1/2mv_b² + 0.

We know θ and we know v_b, so we can solve for L.

L – L cos θ = (1/g) 1/2v_b²

L (1 – cos θ) = (1/g) 1/2v_b²

L = 13.2 m

Now that we know L, we can find the frequency.

Practice problems for these concepts can be found at: Simple Harmonic Motion Practice Problems for AP Physics B & C