Simple Harmonic Motion

Motion that repeats itself is periodic motion. A particular kind of periodic motion is known as simple harmonic motion. When an object is disturbed from equilibrium, its motion is probably simple harmonic motion. Here are some examples of periodic motion that approximate simple harmonic motion:

A particular and useful kind of periodic motion is simple harmonic motion (SHM). Our prototype for SHM is a mass attached to a spring.


The restoring force exerted by the spring is F = - k x,

F = - k x

So the net force on our Simple Harmonic Oscillator (SHO) is

Fnet = - k x = ma = m d2x/dt2

d2x/dt2 = - (k/m) x


a = - (k/m) x

for our prototypical SHO.

To quote The Mechanical Universe video, "we can use the time-honored technique of guessing" and "guess" a solution of

x = A sin (C t)

and then check to see if this is, indeed, a solution and find what the constant C must be.

dx/dt = A C cos (C t)

d2x/dt2 = - A C2 sin (C t)

But that means

d2x/dt2 = - C2 [ x ]

and we already know

d2x/dt2 = - (k/m) x

So that requires

C2 = k/m


C = SQRT(k/m)

Instead of just calling this C, this is labeled , a lower case Greek "omega" and is known as the angular frequency. That means we can write our solution as

x = A sin ( t)

The amplitude is the maximum distance the mass moves from its equilibrium position. It moves as far on one side as it does on the other. The amplitude is A in our solution,

x = A sin ( t)

The time that it takes to make one complete repetition or cycle is called the period of the motion. We will usually measure the period in seconds.

Frequency is the number of cycles per second that an oscillator goes through. Frequency is measured in "hertz" which means cycles per second.

The angular frequency is measured in radians per second. To convert that to periods per second or cycles per second, we need

2 f =

because there are 2 radians in one period or one cycle. f is just "the frequency" and is given the units of hertz (Hz) which means cycles per second.

Period and frequency are closely connected; they contain the same information.

T = 1/f

f = 1/T

For this spring-and-mass simple harmonic oscillator, we find the period is given by

Of course, if the position x(t) does not start at zero when the time t starts at zero, the equation above,

x(t) = A sin ( t)

will have (or cause) problems! We can handle this change in time by writing this as

x(t) = A sin ( t + )

is the lower case Greek letter "phi" and is usually called the "phase angle" or the "phase shift". It is simply related to where the SHO is when time t is zero.

In terms of the ordinary frequency f, we can write the equation for the position of a SHO as

x(t) = A sin (2 f t)


x(t) = A sin (2 f t + )

An interesting and important characteristic of all simple harmonic motion is that the period remains the same even for motions with quite different amplitudes. The period is independent of the amplitude.

Circular motion is another example of motion whose period is independent of the amplitude. Two riders on a merry-go-round have the same period even tho' one sits near the edge and has a large amplitude while the other sits near the axis of rotation and has a small amplitude.

Vertical Spring
Return to ToC, Simple Harmonic Motion
(c) Doug Davis, 2001; all rights reserved