Question

A rod $A B$ of mass $M$ and length $8 l$ lies on a smooth horizontal surface. A particle, of mass $' m'$ and velocity $v_{0},$ strikes the rod perpendicular to its length, as shown in figure. As a result of the collision, the centre of mass of the rod attains a speed of $v_{0} / 8$ and the particle rebounds back with a speed of $v_{0} / 4 .$ Find the following:

(a) The ratio $M / m .$
(b) The angular velocity of the rod about $O .$
(c) The coefficient of restitution $' e'$ for the collision.
(d) The velocities of the ends $' A'$ and $' B'$ of the rod, namely, $v_{A}$ and $v_{B},$ respectively.

Accepted Answer

Mass of rod $A B$ is $= M$

Length of rod $' m'$

Mass of particle $= m$

Velocity of particle $= v_{0}$

Speed of centre of mass of the rod after collision $= v_{0} / 8$

Particle rebounds back with a speed of $v_{0} / 4$ after collision.

We assume particle plus rod as a system.

There is no external force and torque on system.

So, In the process of collision, the linear momentum remains constant.

$(a)$ By Conservation of linear momentum

$' A'$
$\Rightarrow m v_{0} = M \frac{v_{0}}{8} - m \frac{v_{0}}{4}$
$\Rightarrow 8 m = M - 2 m \Rightarrow 10 m = M$
$\Rightarrow M / m = 10$
There is no external torque on system. So, Angular momentum of the system also remains constant.

$(b)$ Therefore, Conservation of angular momentum about the point of collision

$L_{i} = L_{f}$
$\Rightarrow 0 = \frac{M {(8 l)}^{2}}{12} ω - M \frac{v_{0}}{8} (l)$
$\Rightarrow ω = \frac{3 v_{0}}{128 l}$

$(c)$ Using coefficient of restitution equation between points of collision

$e = \frac{v_{2} - v_{1}}{u_{1} - u_{2}}$

$\Rightarrow e = \frac{[(\frac{v_{0}}{8} + ω l) - (- \frac{v_{0}}{4})]}{v_{0} - 0}$
$\Rightarrow [(\frac{v_{0}}{8} + ω l) - (- \frac{v_{0}}{4})] = e (v_{0} - 0) \Rightarrow e = \frac{51}{128}$
$(d)$ So, Velocity at end $A$
$v_{A} = \frac{v_{0}}{8} + ω . 4 l = \frac{28}{128} v_{0}$
$\Rightarrow v_{A} = \frac{7}{32} v_{0}$
Velocity at end $B$
$v_{B} = \frac{v_{0}}{8} + ω . 4 l = \frac{4}{128} v_{0}$
$\Rightarrow v_{B} = \frac{1}{32} v_{0}$

Important Questions on Rigid Body Dynamics: Part 2

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