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Initially spring $S$ are in natural length. An application of external varying force $F$ causes the block to move slowly distance $x$ towards wall on smooth floor :

Column-I

Column-II
A
Work done by $S_{2}$ on block
P
Zero
B
Work done by $S_{2}$ on $S_{1}$
Q
$- \frac{1}{2} (\frac{k_{1} k_{2}}{k_{1} + k_{2}}) x^{2}$
C
Work done by $F$ on block
R
$\frac{1}{2} (\frac{k_{1} k_{2}}{k_{1} + k_{2}}) x^{2}$
D
Work done by $S_{1}$ on wall
S
$\frac{1}{2} \frac{k_{1} k_{2}^{2} x^{2}}{{(k_{1} + k_{2})}^{2}}$

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A-S, B-Q, C-P, D-R

A-Q, B-S, C-R, D-P

A-S, B-R, C-Q, D-P

A-Q, B-P, C-S, D-R

answer is B.

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Detailed Solution

$K_{e q} = \frac{K_{1} K_{2}}{K_{1} + K_{2}}$
$F_{1} = K_{1} x_{1} F_{2} = K_{2} x_{2}$
$x_{2} = \frac{K_{1} x_{1}}{K_{2}}, x_{0} = (\frac{K_{1} + K_{2}}{K_{2}}) x_{1}$
$K_{1} x_{1} = K_{2} x_{2} = K x_{0}$
Work done by $F$ on block = $\frac{1}{2} K_{e q} x_{0}^{2}$
Work done by $S_{2}$ on block = $- \frac{1}{2} \frac{K_{1} (K_{1} + K_{2}) {x_{1}}^{2}}{K_{2}}$

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