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In figure $(A),$ mass ' $2 m$ ' is fixed on mass ' $m$ ' which is attached to two springs of spring constant $k$. In figure $(B),$ mass ' $m$ ' is attached to two spring of spring constant ' $k$ ' and ' $2 k$ '. If mass ' $m$ ' in $(A)$ and $(B)$ are displaced by distance ' $x$ ' horizontally and then released, then time period $T_{1}$ and $T_{2}$ corresponding to $(A)$ and $(B)$ respectively follow the relation.
$\frac{T_{1}}{T_{2}}=\frac{3}{\sqrt{2}}$
$\frac{ T _{1}}{ T _{2}}=\sqrt{\frac{3}{2}}$
$\frac{ T _{1}}{ T _{2}}=\sqrt{\frac{2}{3}}$
$\frac{ T _{1}}{ T _{2}}=\frac{\sqrt{2}}{3}$
Solution
$T _{1}=2 \pi \sqrt{\frac{3 m }{2 k }}$
$T _{2}=2 \pi \sqrt{\frac{ m }{3 k }}$
$\frac{ T _{1}}{ T _{2}}=\frac{2 \pi \sqrt{\frac{3 m }{2 k }}}{2 \pi \sqrt{\frac{ m }{3 k }}}=\frac{3}{\sqrt{2}}$
