Question

An un deformed spring of spring constant k is connected to a bead of mass m as shown in figure. Bead can move along long rigid rod
without friction. A particle of mass m moving with velocity v in the vertical plane containing spring and rod strikes bead at an angle ${45}^{\circ }$ with horizontal and sticks to bead. Choose correct alternative representing maximum elongation of spring if all particles are in same horizontal plane.

• A. $\left[\sqrt{\frac{m}{k}}\right]V$
• B. $\frac{1}{2}\left[\sqrt{\frac{m}{k}}\right]V$
• C. $2\left[\sqrt{\frac{m}{k}}\right]V$
• D. $\frac{1}{4}\left[\sqrt{\frac{m}{k}}\right]V$

Answer 1

The correct option is B $\frac{1}{2}\left[\sqrt{\frac{m}{k}}\right]V$

Conserving moment of system along rod.
$\Rightarrow mvcos{45}^{\circ }=2m{v}_1$
$v_1=\frac{v}{2\sqrt{2}}$
$-\frac{1}{2}k{x}^2=-\frac{1}{2}\left(2m\right){\left(\frac{v}{2\sqrt{2}}\right)}^2$
$\Rightarrow x=\frac{1}{2}\left(\sqrt{\frac{m}{k}}\right)v$