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. `delta=(K_(2))/(K_(1)`
B. `delta=(K_(1))/(K_(2)`
C. `delta=(K_(2))/(K_(1)`
D. `delta= (K_(1))/(K_(2))`

Answer

B, C

Explanation

Let x be the extension produced in both the springs.
Then, force on spring `P,F_(1) = K_(1)x`
Force on spring Q, `F_(2) = K_(2)x `
Work done, `W_(1) = (1)/(2)K_(1)x^(2)`
`therefore " "delta = (W_(1))/(W_(2)) = (K_(1))/(K_(2))`
When the springs are stretched by same force F,
Extension produced in `P, x_(1) = (F)/(K_(1))`
Extension produced in `Q, x_(2) = (F)/(K_(2))`
`therefore (x_(1))/(x_(2)) = (K_(2))/(K_(1))`
Work done, `W_(1) = (1)/(2)K_(1)x_(1)^(2)`
`W_(2) = (1)/(2)K_(2)x^(2)^(2)`
`therefore delta. = (W_(1))/(W_(2)) = (K_(1))/(K_(2))* (x_(1)^(2))/(x_(2)^(2))`
`rArr = (K_(1))/(K_(2)) xx ((K_(2))/(K_(1)))^(2) = (K_(2))/(K_(1))`