Explanation

(b) At maximum compression the solid cylinder will stop
So loss in K.E. of cylinder = gain in P.E. of spring
$\Rightarrow \frac{1}{2}mv^{2} + \frac{1}{2}I\omega^{2} = \frac{1}{2}kx^{2}$
$\Rightarrow \frac{1}{2}mv^{2} + \frac{1}{2}\frac{mR^{2}}{2}\left( \frac{v}{R} \right)^{2} = \frac{1}{2}kx^{2}$
$\Rightarrow \frac{3}{4}mv^{2} = \frac{1}{2}kx^{2}$
$\Rightarrow \frac{3}{4} \times 3 \times (4)^{2} = \frac{1}{2} \times 200 \times x^{2}$
$\Rightarrow \frac{36}{100} = x^{2}$
$\Rightarrow x = 0.6\ m$