Simple Harmonic Motion for AP Physics B & C

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

Amplitude, Period, and Frequency

Simple harmonic motion is the study of oscillations. An oscillation is motion of an object that regularly repeats itself over the same path. For example, a pendulum in a grandfather clock undergoes oscillation: it travels back and forth, back and forth, back and forth … Another term for oscillation is "periodic motion."

Objects undergo oscillation when they experience a restoring force. This is a force that restores an object to the equilibrium position. In the case of a grandfather clock, the pendulum's equilibrium position—the position where it would be if it weren't moving—is when it's hanging straight down. When it's swinging, gravity exerts a restoring force: as the pendulum swings up in its arc, the force of gravity pulls on the pendulum, so that it eventually swings back down and passes through its equilibrium position. Of course, it only remains in its equilibrium position for an instant, and then it swings back up the other way. A restoring force doesn't need to bring an object to rest in its equilibrium position; it just needs to make that object pass through an equilibrium position.

For conservation of energy,  the equation for the force exerted by a spring, F = kx. This force is a restoring force: it tries to pull or push whatever is on the end of the spring back to the spring's equilibrium position. So if the spring is stretched out, the restoring force tries to squish it back in, and if the spring is compressed, the restoring force tries to stretch it back out. Some books present this equation as F = –kx. The negative sign simply signifies that this is a restoring force.

One repetition of periodic motion is called a cycle. For the pendulum of a grandfather clock, one cycle is equal to one back-and-forth swing.

The maximum displacement from the equilibrium position during a cycle is the amplitude. In Figure 17.1, the equilibrium position is denoted by "0," and the maximum displacement of the object on the end of the spring is denoted by "A."

The time it takes for an object to pass through one cycle is the period, abbreviated "T." Going back to the grandfather clock example, the period of the pendulum is the time it takes to go back and forth once: one second. Period is related to frequency, which is the number of cycles per second. The frequency of the pendulum of the grandfather clock is f = 1 cycle/s, where "f " is the standard abbreviation for frequency; the unit of frequency, the cycle per second, is called a hertz, abbreviated Hz. Period and frequency are related by this equation:

Vibrating Mass on a Spring

A mass attached to the end of a spring will oscillate in simple harmonic motion. The period of the oscillation is found by this equation:

In this equation, m is the mass of the object on the spring, and k is the "spring constant." As far as equations go, this is one of the more difficult ones to memorize, but once you have committed it to memory, it becomes very simple to use.

Let's think about how to solve this problem methodically. We need to find two values, a period and a speed. Period should be pretty easy—all we need to know is the mass of the block (which we're given) and the spring constant, and then we can plug into the formula. What about the speed? That's going to be a conservation of energy problem—potential energy in the stretched-out spring gets converted to kinetic energy—and here again, to calculate the potential energy, we need to know the spring constant. So let's start by calculating that.

First, we draw our free-body diagram of the block.

We'll call "up" the positive direction. Before the mass is oscillating, the block is in equilibrium, so we can set F^s equal to mg. (Remember to convert centimeters to meters!)

kx = mg

k (0.20 m) = (10 kg)(10 m/s²)

k = 500 N/m

Now that we have solved for k, we can go on to the rest of the problem. The period of oscillation can be found be plugging into our formula.

To compute the velocity at the equilibrium position, we can now use conservation of energy.

KE_a + PE_a = KE_b + PE_b

When dealing with a vertical spring, it is best to define the rest position as x = 0 in the equation for potential energy of the spring. If we do this, then gravitational potential energy can be ignored. Yes, gravity still acts on the mass, and the mass changes gravitational potential energy. So what we're really doing is taking gravity into account in the spring potential energy formula by redefining the x = 0 position, where the spring is stretched out, as the resting spot rather than where the spring is unstretched.

In the equation above, we have used a subscript "a" to represent values when the spring is stretched out the extra 5 cm, and "b" to represent values at the rest position.

When the spring is stretched out the extra 5 cm, the block has no kinetic energy because it is being held in place. So, the KE term on the left side of the equation will equal 0. At this point, all of the block's energy is entirely in the form of potential energy. (The equation for the PE of a spring is 1/2kx², remember?) And at the equilibrium position, the block's energy will be entirely in the form of kinetic energy. Solving, we have

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