Simple Harmonic Oscillator Help

1. Jul 16, 2008

noodle_snacks

1. The problem statement, all variables and given/known data

A particle oscillates between the points x = 40mm and x = 160mm with an acceleration a = k(100-x) where k is a constant. The velocity of the particle is 18mm/s when x=100 and zero at x = 40mm and x = 160mm. Determine a) the value of hte constant k, b) the velocity when x = 120mm

2. Relevant equations

$$a = k(100-x)$$

3. The attempt at a solution

This looked like a simple harmonic oscillator to me.

So I went:

$$a = 100k - kx$$

$$\frac{d^2x}{dt^2} = 100k - kx$$
Define:
$$\dot x = \frac{\mathrm{d}x}{\mathrm{d}t}$$
Then Observe:
$$\frac{\mathrm{d}^2 x}{\mathrm{d} t^2} = \ddot x = \frac{\mathrm{d}\dot {x}}{\mathrm{d}t}\frac{\mathrm{d}x}{\mathrm{d}x}=\frac{\mathrm{d}\dot {x}}{\mathrm{d}x}\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{\mathrm{d}\dot{x}}{\mathrm{d}x}\dot {x}$$
Then substitute:
$$\frac{d\dot x}{dx}\dot x = 100k-kx$$

$$d\dot x = (100k-kx)dx$$

$$\int \dot x d\dot x = \int (100k-kx)dx$$

$$\dot x^2 = 50kx - kx^2 + c$$

I got that far in the manipulation, then I got stuck. Where do i go from here or what have I done wrong? My current approach is to solve for the differential then differentiate to get an equation for the velocity. Is there a better approach?

Last edited: Jul 16, 2008
2. Jul 16, 2008

tiny-tim

Welcome to PF!

Hi noodle_snacks! Welcome to PF!

hmm … a bit long-winded …

I'd have started by saying "Let y = x - 100"

Then that gives you y'' = -ky, which you may be able to solve on sight.

If not, then continue y''y' = -kyy', and so on.

It isn't any better … but it is easier!

You got stuck at:
So square-root it, and you get dx/√(....) = constant, and you can use trigonometric substitution to solve that.

Last edited: Jul 16, 2008
3. Jul 16, 2008

Hootenanny

Staff Emeritus
Re: Welcome to PF!

Just to point out here that there is no need to actually solve your final differential equation. The question only asks you to determine the value of k and the value of the velocity for a given displacement, both of which can be done by just plugging numbers into the ODE without actually solving it.