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