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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.
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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