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} $
$\therefore T= 2\pi\sqrt{\frac{2m}{k} } ; D - q$