Question

Match the Column $I$ with Column $II$.

Column I		Column I
A		p	$T=2\pi \sqrt{\frac{m\left(k_1+k_2\right)}{k_1{k}_2}}$
B		q	$T=2\pi \sqrt{\frac{2m}{k}}$
C		r	$T=2\pi \sqrt{\frac{m}{2k}}$
D		s	$T=2\pi \sqrt{\frac{m}{k_1+k_2}}$

• A. $A-p,B-q,C-s,D-r$
• B. $A-s,B-r,C-p,D-q$
• C. $A-r,B-p,C-s,D-q$
• D. $A-p,B-r,C-q,D-s$

Answer 1

Answer: (B)

In figure $A$, two springs are connected in parallel. The effective spring constant is, $k_{eff}=k_1+k_2$
$\therefore T=2\pi \sqrt{\frac{m}{k_1+k_2}};A-s$
In figure $B$, two identical springs are connected in parallel. The effective spring constant is
$k_{eff}=k+k=2k\therefore T=2\pi \sqrt{\frac{m}{2k}};B-s$
In figure $C$, two springs are connected in series. The effective spring constant is
$\frac{1}{k_{eff}}=\frac{1}{k_1}+\frac{1}{k_2}=\frac{k_2+k_1}{k_1{k}_2}$ or $k_{eff}=\frac{k_1{k}_2}{k_1+k_2}$
$\therefore T=2\pi \sqrt{\frac{m\left(k_1+k_2\right)}{k_1{k}_2}};C-p$
In figure $D$, two identical springs are connected in series. The effective spring constant is
$\frac{1}{k_{eff}}=\frac{\left(k\right)\left(k\right)}{k+k}=\frac{k}{2}$
$\therefore T=2\pi \sqrt{\frac{2m}{k}};D-q$