A rod of mass 𝑚 and length 𝐿, pivoted at one of its ends, is hanging vertically. A bullet of the same mass moving at speed 𝑣 strikes the rod horizontally at a distance 𝑥 from its pivoted end and gets embedded in it. The combined system now rotates with angular speed $$\omega$$ about the pivot. The maximum angular speed $$\omega_M$$ is achieved for $$x = x_M$$. Then

A

$$\omega = \frac{3vx}{L^2 + 3x^2}$$

B

$$\omega = \frac{12 vx}{L^2 + 12x^2}$$

C

$$x_M = \frac{L}{\sqrt{3}}$$

D

$$\omega_M = \frac{v}{2L}\sqrt{3}$$