Simple Harmonic Motion (SHM)

"If an elderly but distinguished scientist says that something is possible he is almost certainly right, but if he says that it is impossible he is very probably wrong"
Arthur C. Clarke

• Oscillatory motion is Simple Harmonic Motion if the magnitude of the restoring force (F_R ) is linearly proportional to the magnitude of the displacement (x) from equilibrium.  This relationship is known as Hooke's Law after the seventeenth century English physicist Robert Hooke . 

F_R  =  - k x

where k is the constant of proportionality, commonly called the "spring" constant, even for SHM not involving springs. The negative sign indicates that the restoring force and the displacement are measured in opposite directions.

• Using Newton’s second law for the motion of a mass (m) attached to a spring on a horizontal surface, we find,

This is a differential equation with general solution,

where A and φ are constants and , the angular frequency, is given by

Since the solution for x is a cosine function it repeats every 2π so that the period of the oscillations, T, is

and the frequency, f is

  Frequency is measured in Hertz (Hz),  1 Hz = 1 s^-1

• But what about the constants A and φ ? Their values are typically determined by the initial conditions, that is the values of x and dx/dt at t = 0. Substituting the given initial values into the general solution we find that A is the AMPLITUDE , the maximum value of the displacement. φ is called the PHASE CONSTANT.

• Note the behavior (at right) of the displacement (x), velocity(dx/dt) and acceleration versus time. These graphs are drawn for φ = 0 .

Note that in this diagram the displacement is labelled 'y' rather than 'x'.

• Frequency (and period) is independent of the amplitude of the oscillations.

• SHM is the simplest type of oscillatory motion. Oscillatory motion can be expected so long as the restoring force is opposite to the displacement. For example the restoring force could be proportional to x³ rather than x; in this case the differential equation obtained is much harder to solve. However, for small amplitude displacements all oscillatory motion is approximately simple harmonic.

• ENERGY

When an object undergoes SHM the total energy of the system is made up of kinetic and potential energies the relative amounts of which oscillate with the frequency of the motion.

For example, in the case of a mass on a spring, kinetic energy (K) is converted to and from ELASTIC potential energy (U). The dependence of K and U on both time and displacement are indicated below, where we have used the solution for the displacement as a function of time.

Note that the total energy E, is constant,

as demonstrated in the below figure for energy versus time.

"The prime minister held a meeting with the cabinet today. He also spoke to the bookcase and argued with the chest of drawers."

Ronnie Barker

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu