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)=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

B

Explanation

(a) Both P and Q are stretched by the same amount and `K_(P) gt K_(Q) i.e. x_(1)=x_(2)=x`
`therefore (W_(P))/(W_(Q))=((1)/(2)K_(P)x^(2))/((1)/(2)K_(Q)x^(2))=(K_(P))/(K_(Q)) therefore W_(P) gt W_(Q)`
(b) In the second case
Same force is used i.e. `F_(1)=F_(2)=F`
`because F=Kx therefore x =(F)/(K)`
`W=(1)/(2)Kx^(2)=(1)/(2)K(F^(2))/(K^(2))=(1)/(2)(F^(2))/(K)`
`therefore (W_(P))/(W_(Q))=(1)/(2)(F^(2))/(K_(P))xx(2K_(Q))/(F^(2))=(K_(Q))/(K_(P))`
But `K_(P) gt K_(Q) therefore W_(Q) gt W_(P)`
Thus `W_(P) gt W_(Q) and W_(Q) gt W_(P)`