Simple Harmonic Motion

## Key Ideas

There are three conditions for something to be in simple harmonic motion. All are equivalent.

The object’s position–time graph is a sine or cosine graph.

The restoring force on the object is proportional to its displacement from equilibrium.

The energy vs. position graph is parabolic.

The mass on a spring is the most common example of simple harmonic motion.

The pendulum is in simple harmonic motion for small amplitudes.

# Amplitude, Period, and Frequency

Simple harmonic motion is the study of

**oscillations**.An oscillation is the motion of an object that regularly repeats itself over the same path.

Objects undergo oscillation when they experience a restoring force: a force that restores an object to the equilibrium position.

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.

The time it takes for an object to pass through one cycle is the period, abbreviated T.

# 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:

m is the mass of the object on the spring

k is the “spring constant.”

To find the period, all we need to know is the mass.

When dealing with a vertical spring, it is best to define the rest position as x = 0 in the equation for the potential energy of the spring.

If we do this, then gravitational potential energy can be ignored.

# Pendulums

## Simple Pendulums

the formula for the period of a simple pendulum is:

L is the length of the pendulum

g is the acceleration attributable to gravity

the period of a pendulum does not depend on the mass of whatever is hanging on the end of the pendulum.

To calculate the period of a pendulum, we must know the length of the string.

## Compound Pendulum

If the hanging mass has a size approaching the length of the string, the period of this “compound” pendulum is

I is the rotational inertia of the hanging mass

m is the hanging mass

d is the distance from the center of mass to the top of the string.

# Simple Harmonic Motion

## Key Ideas

There are three conditions for something to be in simple harmonic motion. All are equivalent.

The object’s position–time graph is a sine or cosine graph.

The restoring force on the object is proportional to its displacement from equilibrium.

The energy vs. position graph is parabolic.

The mass on a spring is the most common example of simple harmonic motion.

The pendulum is in simple harmonic motion for small amplitudes.

# Amplitude, Period, and Frequency

Simple harmonic motion is the study of

**oscillations**.An oscillation is the motion of an object that regularly repeats itself over the same path.

Objects undergo oscillation when they experience a restoring force: a force that restores an object to the equilibrium position.

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.

The time it takes for an object to pass through one cycle is the period, abbreviated T.

# 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:

m is the mass of the object on the spring

k is the “spring constant.”

To find the period, all we need to know is the mass.

When dealing with a vertical spring, it is best to define the rest position as x = 0 in the equation for the potential energy of the spring.

If we do this, then gravitational potential energy can be ignored.

# Pendulums

## Simple Pendulums

the formula for the period of a simple pendulum is:

L is the length of the pendulum

g is the acceleration attributable to gravity

the period of a pendulum does not depend on the mass of whatever is hanging on the end of the pendulum.

To calculate the period of a pendulum, we must know the length of the string.

## Compound Pendulum

If the hanging mass has a size approaching the length of the string, the period of this “compound” pendulum is

I is the rotational inertia of the hanging mass

m is the hanging mass

d is the distance from the center of mass to the top of the string.