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Question
Find the elastic potential energy stored in each spring shown in figure when the block is in equilibrium. Also find the time period of vertical oscillation of the block.
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Solution
All three spring attached to the mass M are in series.
k1, k2, k3 are the spring constants.
Let k be the resultant spring constant.
\[\frac{1}{k} = \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3}\]
\[ \Rightarrow k = \frac{k_1 k_2 k_3}{k_1 k_2 + k_2 k_3 + k_3 k_1}\]
\[\text { Time period }\left( T \right)\text{ is given by, }\]
\[T = 2\pi\sqrt{\frac{M}{k}}\]
\[ = 2\sqrt{\frac{M\left( k_1 k_2 + k_2 k_3 + k_3 k_1 \right)}{k_1 k_2 k_3}}\]
\[ = 2\sqrt{M\left( \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} \right)}\]
As force is equal to the weight of the body,
F = weight = Mg
Let x1, x2, and x3 be the displacements of the springs having spring constants k1, k2 andk3 respectively.
For spring k1,
\[x_1 = \frac{Mg}{k_1}\]
\[\text { Similarly }, x_2 = \frac{Mg}{k_2}\]
\[\text { and } x_3 = \frac{Mg}{k_3}\]
\[ \therefore {PE}_1 = \frac{1}{2} k_1 x_1^2 \]
\[ = \frac{1}{2} k_1 \left( \frac{Mg}{k_1} \right)^2 \]
\[ = \frac{1}{2} k_1 \frac{M^2 g^2}{k_1^2}\]
\[ = \frac{1}{2}\frac{M^2 g^2}{k_1} = \frac{M^2 g^2}{2 k_1}\]
\[\text { Similarly }, {PE}_2 = \frac{M^2 g^2}{2 k_2}\]
\[\text { and } {PE}_3 = \frac{M^2 g^2}{2 k_3}\]
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