Question

A mass m rests on a frictionless horizontal table and isconnected to rigid supports via 2 identical springs each of relaxedlength L0 and spring constantk. Each spring is stretched to a length L considerable greater thanL0. Horizontal displacementsof m from its equilibrium position are labeled x and y.

(Diagram insert didn't work, all it really tells you is that xis on the axis of the springs and y is perpendicular to thesprings)

a) Write down the differential equation of motion (i.e.,Newton's Law) governing small oscillations in the xdirection.

b) Same as a but in the y direction (assume that y <<L).

c) In terms of L and L0, calculate the ratio of the periods ofthe oscillations along x and y.

d) If at t=0 the mass m is released from the point x = y = A0with zero velocity, what are its x and y coordinates at any latertime t?

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So, I started with just ma = -2kx for a, leading me to thedifferential equation x''(t) + (2k/m) x = 0.

Then I did basically the same thing with y.

However, when I get to c, there's my problem - I don't knowhow I could write any of this in L and L0.

Also, b doesn't really make sense to me because if y is smallwhen compared to L, it seems like there shouldn't be anyoscillation in the y direction.

The answer the back of the book shows for c and d are:

c) Tx/Ty = (1 - L0 / L)1/2

d) x(t) = A0cos( (2k/m)1/2 t)

y(t) = A0cos( (2k(L-L0)/mL)1/2 t)

I have no idea how they got to these answers. Can somebodyhelp me? It has something to do with springs already beingstretched, but it seems they are making an assumption somewherethat I don't understand.