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.

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is A

$k_1,k_2,k_3$ are in series,

$\frac{1}{k}=\frac{1}{k_1}+\frac{1}{k_2}+\frac{1}{k_3}\Rightarrow k=\frac{k_1{k}_2{k}_3}{k_1{k}_2+k_2{k}_3+k_1{k}_3}$

Time period $T=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{m\left(k_1{k}_2+k_2{k}_3+k_1{k}_3\right)}{k_1{k}_2{k}_3}}=2\pi \sqrt{m⟮\frac{1}{k_1}+\frac{1}{k_2}+\frac{1}{k_3}⟯}$

Now, Force=weight=mg

$\therefore At{k}_{1}spring,x_1=\frac{mg}{k_1}$

Similarly $x_2=\frac{mg}{k_2}and{x}_3=\frac{mg}{k_3}$

$\therefore P{E}_1=\left(\frac{1}{2}\right)k_1{x}_1^2=\frac{1}{2}{k}_1⟮\frac{Mg}{k_1}{⟯}^2=\frac{1}{2}{k}_1\frac{m^2{g}^2}{k_1^2}=\frac{m^2{g}^2}{2k_1}$

Similarly $P{E}_2=\frac{m^2{g}^2}{2k_2}andP{E}_3=\frac{m^2{g}^2}{2k_3}$