Two similar springs `P` and `Q` have spring constant `K_(P)` and `K_(Q)` such that `K_(P) gt K_(Q)`. They are stretched, first by the same amount (case a), then the same force (case b). The work done by the spring `W_(P)` and `W_(Q)` are related as, in case (b), respectively

Options

A. `W_(P) = W_(Q) , W_(P) gt W_(Q)`
B. `W_(P)= W_(Q),W_(P) = W_(Q)`
C. `W_(P) gt W_(Q) , W_(Q) gt W_(P)`
D. `W_(P) lt W_(Q) , W_(Q) lt W_(P)`

Answer

C

Explanation

`K_(P) gt K_(Q)`
Case (b) : Same amount of force is applied
`F = K_(P)x_(P)=K_(Q)x_(Q)`
`(x_(P))/(x_(Q)) = (K_(Q))/(K_(P))`
`W_(P) = 1/2 K_(P)x_(P)^(2) , W_(Q) = 1/2 K_(Q)x_(Q)^(2)`
`rArr " " (W_(P))/(W_(Q)) = (K_(P))/(K_(Q)) ((x_(P))/(x_(Q)))^(2) = (K_(Q))/(K_(P) ) lt 1 `
`rArr " " W_(P) lt W_(Q)`
`rArr W_(P) lt W_(Q)`
Case (a) Stretched by the same amount
`F_(P) =K_(P)x " "F_(Q) =K_(Q)x`
`(W_(P))/(W_(Q)) = (1/2K_(P)x^(2))/(1/2K_(Q)x^(2)) = (K_(P))/(K_(Q)) gt 1 `
`rArr " " W_(P) gt W_(Q) `