Question

,A block of mass m is attached to two unstretched springs of spring constants k₁ and k₂ as shown in the figure (8-E9). The block is displaced towards the right through a distance x and is released. Find the speed of the block as it passes through the mean position shown.

Figure

Answer 1

The body is displaced x towards the right.
Let v be the velocity of the body at its mean position.
Applying the law of conservation of energy,

$\frac{1}{2}m{\nu }^2=\frac{1}{2}{k}_1{x}^2+\frac{1}{2}{k}_2{x}^2$



$$\begin{gathered} \Rightarrow m{\nu }^2=x^2\left(k_1+k_2\right) \\ \Rightarrow {\nu }^2=\frac{x^2\left(k_1+k_2\right)}{m} \\ \Rightarrow \nu =x\sqrt{\frac{k_1+k_2}{m}} \end{gathered}$$