Question

A uniform rod $AB$ length $L$ and mass $m$ is suspended freely at $A$ and hangs vertically at rest when a particle of same mass $m$ is fired horizontally with speed $V$ to strike the rod at its midpoint. If the particle is brought to rest after the impact. Then the impulsive reaction at $A$ is :

• A. $\frac{mv}{4}$
• B. $\frac{mv}{2}$
• C. $mv$
• D. None of these

Answer 1

Answer: (A)

$\frac{mv}{4}$

Writing the impulse equation for particle $J=m(v-0)$...(i)
for rod $J-R_A=m{v}_{centerofmass}$...(ii)
Assuming the rod started rotational motion with angular velocity $\omega$
$v_{centerofmass}=\omega \times l/2$

Writing angular impulse equation about point A
$J\times \frac{l}{2}=I_A\omega$...(iii)
Solving three equations we get, $R_A=\frac{J}{4}=\frac{mv}{4}$