# Simple Harmonic Motion Differential Equation

1. Oct 1, 2008

### Matuku

1. The problem statement, all variables and given/known data
A particle of mass m moves in one dimension under the action of a force given by -kx where x is the displacement of the body at time t, and k is a positive constant. Using F=ma write down a differential equation for x, and verify that its solution is x=Acos($$\omega$$t+$$\phi$$), where $$\omega$$2=k/m (omega squared, that is). If the body starts from rest at the point x=A at time t=0, find an expression for x at later times.

2. Relevant equations

3. The attempt at a solution
I think the differential equation they're looking for is,
a=-kx/m

As a=d2x/dt2

But from here I can't see where to go; integration of course leads to the wrong formula.

2. Oct 1, 2008

### Kurdt

Staff Emeritus
You have the differential equation so just substitute the solution in and show that both sides are equal.

3. Oct 1, 2008

### Matuku

So differentiate the solution given to us to get it in terms of acceleration and then just compare that with the a=-kx/m?

4. Oct 1, 2008

### Kurdt

Staff Emeritus
Thats correct.

5. Oct 1, 2008

### Matuku

I'm sorry I'm still sightly confused; I now have:

$$a=-A\omega^{2}cos(\omega t+\phi)=\frac{-kx}{m}$$

Which implies that $$x=Acos(\omega t+\phi)$$ but doesn't really show why? Is this what you intended or am I missing something?

6. Oct 1, 2008

### Kurdt

Staff Emeritus
You're also told what omega is in the question, and you haven't substituted for x on the right hand side.

7. Oct 1, 2008

### Matuku

Oh of course it is! We're told what x is!