Question

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

Answer 1

Case (a) `(W_(P))/(W_(Q)) = ((1)/(2) K_(P) x^(2))/((1)/(2) K_(Q) x^(2)) = (K_(P))/(K_(Q)), K_(P) gt K_(Q)`,
hence, `W_(P) gt W_(Q)`
Case (b) `F = X_(P) x_(P) = K_(Q) X_(Q)`,
`W = (1)/(2) Kx^(2) = (1)/(2). (F)/(x) x^(2) = (1)/(2) Fx = (F^(2))/(2K)`
`(W_(P))/(W_(Q)) = (F^(2) // 2 K_(P))/(F^(2) // 2K_(Q)) = (K_(Q))/(K_(P)), K_(Q) lt K_(P)`
hence, `W_(P) lt W_(Q) implies W_(Q) gt W_(P)`