# Sinusoidal functions

1. Jan 30, 2006

### lomantak

A mass is supported by a spring so that it rests 0.5 m above a table top. The mass is pulled down 0.4 m and released at time t=0. This creates a periodic up-and-down motion, called simple harmonic motion. It takes 1.2 s for the mass to return to the low position each time.

Could someone please give me an equation of the sinusoidal function when the height of the mass above the table top is a function of time for the first 2.0 s?

Thanks.

2. Jan 31, 2006

### Tide

You know the amplitude and you know the frequency (via the period). What more do you need? :)

3. Feb 6, 2006

### scholzie

Your amplitude is given to you (the amount pulled down). The period, 1.2s, is the amount of time it will take to go all the way up, and then back down again.

Remember your unit circle? In this case, the low point corresponds to $\cos{\pi}$. It makes one full revolution around the unit circle in 1.2 seconds. Since the entire circle ($2\pi$) is travelled once, then $\omega = \frac{2\pi}{1.2sec}$ where $\omega$ is called the angular frequency. Basically, $\omega$ is the angular rate of change in $t$ seconds.

So, if cosine is at a maximum of 1 and a minimum of -1, then you need to multiply it by the amplitude at its maximum and minimum to get the max and min amplitudes, right?

$$y = A\cos{(\omega t)}$$
But, it's 0.5m above the table, so everything is shifted up h = 0.5m
$$y = A\cos{(\omega t)} + h$$

I'll leave you to do the actual calculations, but hopefully you see how this works. Also note that cosine and sine are the same function, but cosine is shifted by pi/2 radians. You could also write that as y = Asin( wt + pi/2 ) + h

Last edited: Feb 6, 2006