Harmonic motion

## Big Picture

Simple harmonic motion (SHM), also known as oscillatory motion, describes the motion of objects that move back and forth along a sine or cosine curve due to a restoring force. In addition, SHM can model the movements or the periodic change of states of all kinds of objects, such as the vibration of springs. When amplitude is not constant over time and there is a net force, other than the restoring force, acting on the system, the object is not considered to be in SHM.

## Key Terms

Simple Harmonic Motion (SHM): A symmetrical back and forth motion where the distance traveled in each cycle is constant.

Period: The time it takes for an object in SHM to return to its original state (complete one cycle). SI units: s

Frequency: The number of times an object in SHM returns to its original state (completes a cycle) in a second. SI units: Hz (1 cycle per second).

Amplitude: The greatest distance an object moves away from its equilibrium position. SI units: m

Restoring Force: A force acting on an object that moves it back to its equilibrium position. The force’s magnitude and direction varies with its position. The direction will always be toward the equilibrium position and the magnitude generally increases the farther the object is from equilibrium. SI units: N

## Simple Harmonic Motion

Simple harmonic motion (SHM) is a special type of back-and-forth motion. Systems that go through SHM:

• do not lose any energy (friction assumed to be zero)
• do not have constant acceleration, so we cannot use the three equations for linear motion to describe SHM
• have a period (T) and a frequency (f) (equal to the inverse of period: $$f = \frac{1}{T}$$
• have an amplitude (A) that is constant - it is half the total distance traveled in a period
• have zero restoring force at the equilibrium point
• have maximum kinetic energy and speed at the equilibrium point

One common example of a system that experiences SHM is a mass attached to a spring.

• The restoring force is the spring force.
• The spring force is described by Hooke’s law: $$F = -kΔx$$, where $$Δx$$ is the displacement.
• -k is the spring constant
• The spring potential energy is: $$E_{sp} = \frac{1}{2}k \Delta x^2$$

To the right is a diagram of a spring-mass system in SHM.

Image Credit: CK-12 Foundation, CC-BY-NC-SA 3.0

Another system that experiences SHM is a pendulum. SHM only correctly approximates the motion of a pendulum when the amplitude is much less than the length of the pendulum.

• The restoring force is gravity.
• The period is determined entirely by the length of the pendulum and the acceleration of gravity at that location.

## Other Types of Harmonic Motion

In systems going through harmonic motion (not simple harmonic motion), there is a net force other than the restoring force, so the amplitude is not constant. The system is now in either damped or driven harmonic motion:

• Damped harmonic motion: A frictional force causes the amplitude to decease over time.
• Driven harmonic motion: An external force acts on an object during its oscillation.

## Important Equations

$$T = \frac{1}{f}$$

T - period

f - frequency

Period for mass-spring system in SHM:

$$T = 2 \pi \sqrt{\frac{m}{k}}$$

m - mass

k - spring constant

Period for pendulum in SHM:

$$T = 2 \pi \sqrt{\frac{l}{g}}$$

l - length

g - gravity

$$x(t) = x_o + A \cos (2 \pi f(t-t_o))$$

x - position

A - amplitude

$$v(t) = -2 \pi A \sin(2 \pi f(t-t_o))$$

v - velocity

t - time