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In the following questions, match column $-I$ with column $-II$ and choose the correct options
$A-P,\,B-Q,\,C-R,\,D-S$
$A-Q,\, B-P,\, C-R,\, D-S$
$A-Q,\, B-P,\, C-S,\, D-R$
$A-P, \,B-Q,\, C-S,\, D-R$
Solution
In figure $(A),$ the effective spring constant, $\mathrm{K}=\mathrm{k}_{1}+\mathrm{k}_{2}$
In figure $(B),$ the effective spring constant
$\frac{1}{K}=\frac{1}{k_{1}}+\frac{1}{k_{2}}$ or $K=\frac{k_{1} k_{2}}{\left(k_{1}+k_{2}\right)}$
In figure $(C),$ the effective spring constant, $\mathrm{K}=\mathrm{k}+\mathrm{k}=2 \mathrm{k}$
In figure $(D),$ let the effective spring constant be $K$ If mass $\mathrm{m}$ be pulled down by distnace $y,$ the springs $A$ and $B$ will be stretched by $y'$ $\left(=y \cos 45^{\circ}\right)$ Total restoring force is $\mathrm{F}=-\mathrm{Ky}=\mathrm{F}_{\mathrm{A}} \cos 45^{\circ}+\mathrm{F}_{\mathrm{b}}$
$\cos 45^{\circ}=2\left(\mathrm{ky}^{\prime}\right) \cos 45^{\circ}=2 \mathrm{k}\left(\mathrm{y} \cos 45^{\circ}\right) \cos 45^{\circ}=\mathrm{ky}$
$\mathrm{K}=\mathrm{k}$ Use time period $\mathrm{T}=2 \pi \sqrt{\frac{m}{K}}$
