A spring of force constant $k$ is cut into two parts whose lengths are in the ratio $1 : 2$ . The two parts are now connected in parallel and a block of mass $m$ is suspended at the end of the combined spring. The period of oscillation of the block is:

2 π \sqrt{\frac{2 m}{9 k}}

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2 π \sqrt{\frac{m}{9 k}}

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2 π \sqrt{\frac{2 m}{5 k}}

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2 π \sqrt{\frac{m}{5 k}}

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Solution

The correct option is A $2 π \sqrt{\frac{2 m}{9 k}}$
We know, $k = \frac{E A}{l}$
$k_{1} = \frac{E A}{l / 3} = \frac{3 E A}{l} = 3 k; k_{2} = \frac{E A}{2 l / 3} = \frac{3 E A}{2 l} = \frac{3 k}{2}$
When in parallel $k_{e f f} = k_{1} + k_{2}$
$= \frac{9 k}{2}$
$T = 2 π \sqrt{\frac{m}{k_{e f f}}} = 2 π \sqrt{\frac{2 m}{9 k}}$

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A spring of force constant k is cut into two parts whose lengths are in the ratio 1:2. The two parts are now connected in parallel and a block of mass m is suspended at the end of the combined spring. The period of oscillation of the block is:

The correct option is A 2π√2m9kWe know, k=EAlk1=EAl/3=3EAl=3k;k2=EA2l/3=3EA2l=3k2When in parallel keff=k1+k2 =9k2T=2π√mkeff=2π√2m9k

A spring of force constant $k$ is cut into two parts whose lengths are in the ratio $1 : 2$ . The two parts are now connected in parallel and a block of mass $m$ is suspended at the end of the combined spring. The period of oscillation of the block is: