Simple Harmonic Motion Brain Teaser

[cj]

I saw this in an old, junior-level, classical mechanics
textbook and haven't been able to figure it out.

A particle undergoing simple harmonic motion has a velocity:

\frac{dx_1}{dt}

when the displacement is:

x_1

and a velocity

\frac{dx_2}{dt}

when the displacement is:

x_2

What is the angular frequency and the amplitude of the motion in terms of the given quantities?

I know the solution to the SHM wave equation is:

\begin{equation}<br /> x(t) = A \cdot sin( \omega t + \phi )\end{equation}

And that:

\begin{equation}<br /> dx(t)/dt = A \omega \cdot cos( \omega t + \phi )\end{equation}

But can't see how to express omega or A in these terms.

 

[rayjohn01]

If x = A.sin(w.t) then

x1 = A sin(w.t1)
x2 = A.sin(w.t2)
dx/dt 1= A.w.cos(w.t1)
dx/dt 2 = A.w.cos(w.t2)

I am not going to do it but there appears to be enough information , to eliminate t1,t2 and get A and w.
For instance a) and c) can eliminate t1 , b) and d) eliminate t2 .

 

[Doc Al]

But can't see how to express omega or A in these terms.

Apply what you know. I'll get you started. You are given: At time t_1 the displacement is x_1 and the speed is v_1. (I didn't like your notation, so I changed it. )

So... just plug into your SHM equations:
x_1 = A sin(\omega t_1)
v_1 = A \omega cos(\omega t_1)
Combine these equations to get a relationship between \omega and A.

Now do the same for time t_2, and then you should be able to solve for \omega and A.