## Mathematical Recreations

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#### The Simple Harmonic Oscillator

The problem for December 2001 involves some very basic physics. If your physics is a bit rusty, then you may find this discussion of the simple harmonic oscillator helpful.

Consider a weight of mass M suspended by a spring with spring constant k, such that the force F exerted by the spring is given by

F  =  –k x.

Assume the spring is massless, and ignore air friction.

The position of the mass is given by the coordinate x. At x = 0 the spring exerts no force on the mass. If the mass is displaced in either direction, the spring exerts a restoring force tending to pull the mass back toward x = 0.

The force will tend to accelerate the mass according to Newton's Third Law of Motion:

F  =  m a

Acceleration is the second derivative of position with respect to time. That is, velocity is the rate of change of position and acceleration is the rate of change of velocity. Keep in mind that velocity and acceleration are signed quantities. Physicists like to reserve the word "speed" for the magnitude of the velocity.

Combining the two equations, we get a simple differential equation:

 d2 x d t2
=  –
 k m
x

This has the familiar solution:

x  =  A cos (ωt + φ),

where ω = (k/m)1/2, and A and φ are arbitrary constants. The system undergoes one oscillation when ωt increases by 2π, so the frequency of the oscillation is ω/(2π).

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Thanks,
Steve