Question

A solid sphere of mass M rolls without slipping and presses a spring of spring constant K as shown in the figure. Then the compression in the spring will be

• A. $V\sqrt{\frac{2M}{3K}}$
• B. $V\sqrt{\frac{2M}{3K}}$
• C. $V\sqrt{\frac{5K}{7M}}$
• D. $V\sqrt{\frac{7M}{5K}}$

Answer 1

The correct option is D $V\sqrt{\frac{7M}{5K}}$

In case of pure rolling, ratio of rotational to translational kinetic energy for a solid sphere $=\frac{K.E_{rotational}}{K.E_{translational}}=\frac{2}{5}$

$\Rightarrow K{E}_{total}=\frac{7}{5}K{E}_{translational}=\frac{7}{5}\left(\frac{1}{2}M{V}^2\right)$

If maximum compression $=x_{max}$, by conservation of energy,
$\frac{1}{2}K{x}_{max}^2=\frac{7}{5}\times \frac{1}{2}M{V}^2$
$\Rightarrow x_{max}=V\sqrt{\frac{7M}{5K}}$