A weight W is attached suddenly, without initial velocity, to the lower end of a light vertical elastic wire. The same weight would produce a static extension x if applied gradually. Neglect the mass of the wire and assume perfectly elastic behaviour. What is the maximum extension after the sudden attachment?
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Question

A weight W is attached suddenly, without initial velocity, to the lower end of a light vertical elastic wire. The same weight would produce a static extension x if applied gradually. Neglect the mass of the wire and assume perfectly elastic behaviour. What is the maximum extension after the sudden attachment?

Options

G1 Physics · Accepted Answer

Let the effective spring constant of the wire be k. Under gradual loading, W = kx. When the load is suddenly attached, consider the instant of maximum extension X. At that instant the speed is zero. Work done by gravity: W X. Elastic energy stored in the wire: U = ½kX². Energy conservation gives WX = ½kX². For X ≠ 0, W = ½kX. Hence, X = 2W/k. Since x = W/k, we obtain X = 2x. Correct option: D.