Simple Harmonic Motion

From: Alison Tenk
Date: 17 Feb 1999

Subject: Show that a particle performs SHM

The position of a particle from point O at time t is given by
x = 5cos(2t). Show that the particle performs simple harmonic motion.

I answered this question by saying that cos is an oscillating function
and so the particle oscillates to and fro, which is what SHM is all about,
but my teacher said that was not enough. What should the right answer be?

Maths Help suggests:

To formally demonstrate simple harmonic motion, you must show that
the relationship

is true for some value of w (sorry, can't type the Greek "omega" here!)
Remember that "x-double-dot" stands for acceleration.

We are given the displacement x at time t.
Differentiating this with respect to t gives the velocity, and
differentiating again gives the acceleration. Thus we obtain

displacement x = 5cos(2t)
the velocity v = -10sin(2t)
acceleration a = -20cos(2t)

Looking at the expressions above for displacement and acceleration,
we see that acceleration = - 4 times displacement
This is the required format for SHM (see box above), with w = 2.

I think this is what your teacher is looking for.

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