## Oscillatory Motion - Determining equation of motion

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

A particle with a mass of 0.5 kg is attached to a horizontal spring with a force constant of 50 N/m. At the moment t = 0, the particle has a maximum speed of 20 m/s and is moving to the left.
(a) Determine the particle's equation of motion.
(b) Where in the motion is the potential energy three times the kinetic energy?
(c) Find the minimum time interval required for the particle to move from x=0 to x=1.00m.
(d) Find the length of a simple pendulum with the same period.

2. Relevant equations
(a)
w=sqrt(k/m)

v_max = Aw

(b)
3(.5*m*w^2*A^2*sin^2(wt+phi))= .5*k*A^2*cos^2(wt+phi)

3. The attempt at a solution

I searched he forums here for the same question and found this thread:
http://www.physicsforums.com/archive.../t-231270.html

I think I understand how to find omega and the maximum velocity (though the signs may be incorrect), but I don't understand how to solve for phi. The only thing I could think of was to set

v=-wAsin(wt+phi) to -20=10*(-2)sin(phi) for t=0.
This returned a phi=3pi/2 or -pi/2, unless I'm doing something wrong.
I also tried to set x(0)=0, so
0=(-2)cos(phi) which results in phi=pi/2.
These answers disagree with what the linked thread found for phi and what my professor's answer had for phi.

I'm just having a hard time understand phi, which seems like such a simple concept. If anyone could steer me in the right direction, I think I could finish this problem. Thanks for any help!
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Recognitions:
Homework Help
 v=-wAsin(wt+phi) to -20=10*(-2)sin(phi) for t=0.
If -20=10*(-2)sin(phi), then sin(phi)=1, so phi can be pi/2. This is consistent with what you got using x(0)=0.
 Recognitions: Homework Help Just a thought: if the speed is maximum as the particle passes through the equilibrium point at t=0, why not make the velocity function a cosine and do away with the phase constant? Integrate once to find the position function (which will then be a sine function, again with no phase constant). Should make life easier.

## Oscillatory Motion - Determining equation of motion

If you're going to do more with oscillators (or waves) I recommend purchasing A. P. French's Vibrations and Waves It's inexpensive -- likely free down-loadable -- there are MIT lectures based on it.

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