Column I lists the various modes of oscillations of masses connected to springs. Column II lists the corresponding frequencies of oscillations when executing S.H.M. Match them properly.

	Column I		Column II
A		p)	$\frac{1}{2\pi }\sqrt{\frac{3k}{2m}}$
B		q)	$\frac{1}{2\pi }\sqrt{\frac{2k}{m}}$
C		r)	$\frac{1}{2\pi }\sqrt{\frac{k}{3m}}$
D		s)	$\frac{1}{2\pi }\sqrt{\frac{3k}{m}}$

 

a) Motion is simple harmonic

$\frac{d^{2}x}{d{t}^2}=\frac{-k}{m_1{m}_2}(m_2+2m_1).x$

${\omega }^2=k(\frac{1}{m_1}+\frac{2}{m_2})=k(\frac{1}{m}+\frac{1}{2m}).x$

s

$\omega =\sqrt{\frac{2k}{m}}$

$\therefore$frequency $=\frac{1}{2\pi }\sqrt{\frac{2k}{m}}$

b) Angular frequency $\omega =\sqrt{\frac{k(m+2m)2m^2}{2m^2}}=\sqrt{\frac{3k}{2m}}$

$\therefore$frequency $=\frac{1}{2\pi }\sqrt{\frac{3k}{2m}}$

c) Here effective spring constant $=3k$

$\therefore$frequency $=\frac{1}{2\pi }\sqrt{\frac{3k}{m}}$

d) Effective spring constant $\left(K\right)$ is given by $\frac{1}{K}=\frac{1}{k}+\frac{1}{k}=\frac{3}{k}$

$K=\frac{k}{3}$

$\therefore$frequency $=\frac{1}{2\pi }\sqrt{\frac{k}{3m}}$