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Potential Energy Formulae

Two similar s...

Question

Two similar springs P and Q have spring constants $K_{P}$ and $K_{Q}$ , such that $K_{P} > K_{Q}$ . They are stretched, first by the same amount (case a), then by the same force (case b). The work done by the springs W_Pand W_Q are related as, in case (a) and case (b), respectively :

A

W_P< W_Q; W_Q< W_P

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B

W_P = W_Q; W_P> W_Q

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C

W_P = W_Q; W_P = W_Q

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D

W_P> W_Q; W_Q> W_P

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Solution

The correct option is D
W_P> W_Q; W_Q> W_P

Case (a) : w = $\frac{1}{2} k x^{2}$

$w_{P} = \frac{1}{2} K p x^{2}$
$W_{Q} = \frac{1}{2} k_{Q} x^{2}$
$∴ k_{P} > k_{Q}, w_{P} > w_{Q}$

Case (b) : w = $\frac{1}{2}$ Fx

$F = k_{P} x_{P} = k_{Q} w_{Q}$
$∴ \frac{x_{P}}{x_{Q}} = \frac{k_{Q}}{k_{P}}$
$∴ \frac{w_{P}}{w_{Q}} = \frac{x_{P}}{x_{Q}} = \frac{k_{Q}}{k_{P}}$
$∵ k_{Q} < k_{P}, w_{P} < w_{Q}$

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