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.
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 (SHM) is a special type of back-and-forth motion. Systems that go through SHM:
One common example of a system that experiences SHM is a mass attached to a spring.
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.
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:
\(T = \frac{1}{f}\)
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