Question

Match the combination of springs shown in Column I with their respective time periods in Column II.

Column I		Column II
A		i	$T=2\pi \sqrt{\frac{m\left(k_1+k_2\right)}{k_1{k}_2}}$
B		ii	$T=2\pi \sqrt{\frac{2m}{k}}$
C		iii	$T=2\pi \sqrt{\frac{m}{2k}}$
D		iv	$T=2\pi \sqrt{\frac{m}{k_1+k_2}}$

• A. a-(i), b-(ii), c-(iv), d-(iii)
• B. a-(iv), b-(iii), c-(i), d-(ii )
• C. a-(iii), b-(i), c-(iv), d-(ii)
• D. a-(i), b-(iii), c-(ii), d-(iv)

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 spring are connected in parallel. The effective spring constant is
$k_{eff}=k=k=2k\therefore T=2\pi \sqrt{\frac{m}{2k}};B-r$
In figure $C,$ two identical spring 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 spring are connected in series. The effective spring constant is
$k_{eff}=\frac{(k)(k)}{k+k}=\frac{k}{2}\therefore T=2\pi \sqrt{\frac{2m}{k}};D-q$