# Simple harmonic motion diff. equation

1. Nov 18, 2009

### Breedlove

1. The problem statement, all variables and given/known data
A mass of 1 slug is suspended from a spring whose spring constant is 9 lb/ft. The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of $$\sqrt{3}$$ ft/s. Find the times at which the mass is heading downward at a velocity of 3 ft/s.

2. Relevant equations
x(double prime) + k/m x=0
(k/m)^(1/2)=w

3. The attempt at a solution
So first, m=1, k=9 and the initial conditions: x(0)=-1 and x(prime)(0)=-$$\sqrt{3}$$
x(double prime)+9x=0 which has roots (plus or minus)3i. The solution then looks like

x(t)=C1cos(3t)+C2sin(3t) and then plugging in initial conditions I get C1=-1, C2=-$$\sqrt{3}$$/3

Then converting it to another form where A=$$\sqrt{C1^2+C2^2}$$ and phi=arctan(C1/C2)
x=Asin(wt+phi)
x=2$$\sqrt{3}$$/3 sin(3t+4pi/3)
Taking the derivative to get the velocity:
x(prime)=6$$\sqrt{3}$$cos(3t+4pi/3)
setting x(prime)=3 and solving for t yields a negative value for t, t=-.97

I know I didn't really show a lot of work in a form that is easy to read or understand, but where am I going wrong here? Thanks for any help you can provide!

Last edited: Nov 18, 2009
2. Nov 19, 2009

### clamtrox

Cosine function gets a given value in more than one point; maybe you should solve the first positive value of t where it holds.