Why is this Simple Harmonic Motion?

[Iamadam]

Hello =)

I know this doesn't belong here, but I wasn't sure where else to put it - calculus maybe? ANyway, I have the following question:

If the speed of a particle is given by: v = sqrt(5 + 4x - x^2), show that the motion is simple harmonic. [I won't bother with the dot notation, couldn't figure it out in mathtype =S. I'll just use a for acceleration, v for velocity, and x for displacement.]

So, I start off by finding acceleration in terms of x:

But now I'm stuck. I'm not sure why its SHM? I don't know what to do with that constant - if a was equal to -4x, it'd be fine - but that constant of 8 is confusing me.
Help would be appreciated =]
Thanks in advance,

- Adam

 

[Opalg]

... a = 2(4 - 2x).
But now I'm stuck. I'm not sure why its SHM? I don't know what to do with that constant - if a was equal to -4x, it'd be fine - but that constant of 8 is confusing me.

You're practically there at this stage! All you have to do is to make a simple change of variable y=x-2. Then \(\displaystyle a = \ddot{y} = -4y,\) standard SHM equation.

 

[Iamadam]

Ah, thanks for that help =)