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 an angular speed $\omega$ about the pivot. The maximum angular speed ${\omega }_{\mathrm{M}}$ is achieved for $x=x_{\mathrm{M}}$. Then

• Correct answer
  A

  $\omega =\frac{3vx}{L^2+3x^2}$
• B

  $\omega =\frac{12vx}{L^2+12x^2}$
• Correct answer
  C

  $x_M=\frac{L}{\sqrt{3}}$
• Correct answer
  D

  ${\omega }_M=\frac{v}{2L}\sqrt{3}$

Hence, the correct answer is options: 'A', 'C', 'D'.