A spring with stiffness k and relaxed length L_0 stands vertically on a table. A

Question

A spring with stiffness k and relaxed length L_0 stands vertically on a table. A mass M sits on the spring in static equilibrium. The quantities k, L_0 and M are known: the compressed length L, shown in the figure, is unknown. Determine the compressed length of the spring L in terms of known quantities and constants. Using your hand, you compress the spring so that the spring now has a length of L/3 and you hold the spring motionless at this position. Calculate the work done by your hand. Briefly explain in words, why the sign of the work you calculated is reasonable. You let go of the block and watch it shoot straight up into the air. Find the maximum height reached by the block in terms of known quantities and constants.

Explanation / Answer

(a) At equilibrium position, the weight of the block is balanced by the spring force due to compression of the spring.

Mg = k(Lo - L)

=> L = Lo - Mg/k

(b) Work done by the hand is the change in the elastic potential energy of the spring.

W = kL2/2 - k(L/3)2/2 = 4kL2/9 = 4k(Lo - Mg/k)2/9

Since the elastic potential energy of the spring is increased, work done by the hand is positive. The work done by the hand is stored as the elastic potential energy of the spring.

(c) The elastic potential energy of the spring is converted to the gravitational potential energy of the block at its maximum height.

k(Lo - L/3)2/2 = MgH

=> 2MgH = k[Lo - (Lo - Mg/k)/3]2

=> 2MgH = k[2Lo + Mg/k]2/9

=> 18MgH = (2kLo + Mg)2/k

=> H = (2kLo + Mg)2 / 18Mgk

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