Two small balls A and B , each of mass m , are joined rigidly to...

Two small balls

A

and

B

, each of mass

m

, are joined rigidly to the ends of a light rod of length

L

as shown. The rod is clamped at its centre so that it can rotate freely about a horizontal axis through the clamp. The system is held at rest in the horizontal position. A particle

P

of mass

m

is dropped from a height

h

onto the ball

B

and sticks to it. (a) Find the angular momentum and the angular speed of the system just after the collision. (b) Find the minimum value of

h

so that the system makes a full rotation after the collision.
Two small balls A and B , each of mass m , are joined rigidly to the ends of a light rod of length L as shown. The rod

Two small balls A and B , each of mass m , are joined rigidly to the ends of a light rod of length L as shown. The rod

Options

A(a) $\dfrac{mL\sqrt{gh}}{\sqrt{2}}$ , $\sqrt{\dfrac{8gh}{3L}}$ (b) $\dfrac{3L}{2}$
B(a) $mL\sqrt{gh}$ , $\sqrt{\dfrac{2gh}{L}}$ (b) $L$
C(a) $\dfrac{mL\sqrt{gh}}{2}$ , $\sqrt{\dfrac{4gh}{3L}}$ (b) $\dfrac{L}{2}$
D(a) $\dfrac{mL\sqrt{gh}}{\sqrt{2}}$ , $\sqrt{\dfrac{6gh}{L}}$ (b) $2L$

Correct answer

A. (a) $\dfrac{mL\sqrt{gh}}{\sqrt{2}}$ , $\sqrt{\dfrac{8gh}{3L}}$ (b) $\dfrac{3L}{2}$

Step-by-step solution

(a)

\dfrac{mL\sqrt{gh}}{\sqrt{2}}

\sqrt{\dfrac{8gh}{3L}}

(b)

\dfrac{3L}{2}

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