A mass m rests on a frictionless horizontal table and is connected to rigid supp

Question

A mass m rests on a frictionless horizontal table and is connected to rigid supports via 2 identical springs each of relaxed length L0 and spring constant k. Each spring is stretched to a length L considerable greater than L0. Horizontal displacements of m from its equilibrium position are labeled x and y. (Diagram insert didn't work, all it really tells you is that x is on the axis of the springs and y is perpendicular to the springs) a) Write down the differential equation of motion (i.e., Newton's Law) governing small oscillations in the x direction. 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 of the oscillations along x and y. d) If at t=0 the mass m is released from the point x = y = A0 with zero velocity, what are its x and y coordinates at any later time t? --------------------------- So, I started with just ma = -2kx for a, leading me to the differential 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 know how I could write any of this in L and L0. Also, b doesn't really make sense to me because if y is small when compared to L, it seems like there shouldn't be any oscillation 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 somebody help me? It has something to do with springs already being stretched, but it seems they are making an assumption somewhere that I don't understand. A mass m rests on a frictionless horizontal table and is connected to rigid supports via 2 identical springs each of relaxed length L0 and spring constant k. Each spring is stretched to a length L considerable greater than L0. Horizontal displacements of m from its equilibrium position are labeled x and y. (Diagram insert didn't work, all it really tells you is that x is on the axis of the springs and y is perpendicular to the springs) a) Write down the differential equation of motion (i.e., Newton's Law) governing small oscillations in the x direction. 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 of the oscillations along x and y. d) If at t=0 the mass m is released from the point x = y = A0 with zero velocity, what are its x and y coordinates at any later time t? --------------------------- So, I started with just ma = -2kx for a, leading me to the differential 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 know how I could write any of this in L and L0. Also, b doesn't really make sense to me because if y is small when compared to L, it seems like there shouldn't be any oscillation 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 somebody help me? It has something to do with springs already being stretched, but it seems they are making an assumption somewhere that I don't understand.

Explanation / Answer

Part B for y involves knowing that there is a tension exerted by each spring. Since the springs are identical and there is no x displacement, the tension exerted on both sides of the mass is equal. Force acting on y becomes 2Tsin(theta) = -2k(l'-l0)y/l' = -2ky(1-l0/l')
l' = sqrt(l^2 - y^2); since y<<l, l' ~ sqrt(l^2) ~ l
Thus Force = -2ky(1 - l0/l).

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