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.

Answer 1

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 x_1, x₂, and x₃ be the displacements of the springs having spring constants k₁, k₂ andk₃ respectively.
​For spring k_1,

\[x_1  = \frac{Mg}{k_1}\] 

\[\text { Similarly },    x_2  = \frac{Mg}{k_2}\] 

\[\text { and }  x_3  = \frac{Mg}{k_3}\] 

\[ \therefore  {PE}_1  = \frac{1}{2} k_1  x_1^2 \] 

\[                     = \frac{1}{2} k_1  \left( \frac{Mg}{k_1} \right)^2 \] 

\[                   = \frac{1}{2} k_1 \frac{M^2 g^2}{k_1^2}\] 

\[                   = \frac{1}{2}\frac{M^2 g^2}{k_1} = \frac{M^2 g^2}{2 k_1}\] 

\[\text { Similarly },    {PE}_2  = \frac{M^2 g^2}{2 k_2}\] 

\[\text { and } {PE}_3  = \frac{M^2 g^2}{2 k_3}\]