Describing SHM

Lets define a few quantities first.

Imagine a mass bouncing up and down on a spring or a rubber band. This is a pretty good approximation to vertical SHM.

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Notice that the vertical displacement of the mass varies sinusoidally with time. In other words, if we plotted the vertical displacement (lets call it y) against time we would get a sine wave.

We can write an equation for the displacement as a function of time.

It is of the form y = A sin ω t where A is the amplitude and ω is a constant related to the frequency. With this equation we can calculate the displacement of the object at any time (if we know A and ω and the displacement starts as a sine wave at t=0))

In fact if measured in radians per second, ω = 2 π f  and ω = 2 π / T

You can measure it in degrees per second in which case ω = 360 f  and ω = 360 / T which you might choose to do if you haven't done radians in maths yet.

Consider these examples

1. A mass on a spring would oscillate freely with a frequency of 3 Hz. If the mass were lifted from its equilibrium position 5 cm the released calculate its displacement at 0.1s, 0.2s and 0.3s after its release.

Looking at the graph you should see that the equation has the form y = A cos ω t  where A = 5 cm    and    ω = 2 π f  =  6 π rads /s

for t = 0.1s   y = 5 cos 0.6 π = -1.54cm         for t = 0.2s  y = 5 cos 1.2 π = -4.05cm           for t = 0.3s   y = 5 cos 1.8 π  = 4.04cm

Remember - Make sure your calculator is in radian mode

2. The depth of water in a harbour varies sinusoidally with a period of 12 hours. At low tide it is 5m and at high tide it is 9m. If, on a particular day, low tide is at midnight, what will the depth be at 5 a.m.?

Looking at the graph above you should see that the equation we need is of the form y = - A cos ω t. i.e. a negative cosine.

A = 2m (as the equilibrium depth would be 7m) and ω = 2 ω / T  =  π / 6 radians per hour.

at t = 5 we get y = -2 cos 5π / 6 = 1.73. Remember we need to add 7 to get the depth which gives 8.73m.