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A mass m constrained to move on a frictionless horizontal surface is attached to a frictionless peg by an ideal, massless spring with spring constant k. The unstretched length of the spring is L_1, as shown in Figure 1. When the mass moves in a circle about the peg with constant angular velocity w, the length of the spring is L_2 as shown in Figure 2. Express your answers in terms of some or all of the quantities m, k, w and L_1, and any necessary constants.

Assume the total energy of the system in Figure 1 is zero. Determine the total energy of the rotating system in Figure 2.

Explanation / Answer

When moving in circle,

the centripetal force necessary to move in circle must be provided by the spring force

Now, spring force, Fs = k*x

where x = extension of spring = L2 - L1

So, Fs = k*(L2 - L1)

Now, centripetal force, Fc = m*W^2*R

where R = radius of circle = L2

So, Fc = m*W^2*L2

For circular motion, Fc = Fs

So, m*W^2*L2 =  k*(L2 - L1) ----------- (1)

So, m*W^2/k = (L2-L1)/L2 = 1 - L1/L2

So, L1/L2 = 1 - mW^2/k

So, L2 = L1/(1 - mW^2/k) = k*L1/(k - mW^2) <----------answer

b)

Total energy = Spring Potential energy + Kinetic energy of mass = Up + KE

Up = 0.5*k*x^2 = 0.5*k*(L2-L1)^2

KE = 0.5*m*W^2*R^2 = 0.5*m*W^2*(L2)^2

So, TE =  0.5*k*(L2-L1)^2 + 0.5*m*W^2*(L2)^2 <------------answer

You can plug in the value of L2 in terms of L1 to simplify more if you want. But you havent asked about it so have left it...

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