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Question: Consider a body, shown in figure, consisting of two identical balls, each of mass M connected by a l...

Consider a body, shown in figure, consisting of two identical balls, each of mass M connected by a light rigid rod. If an impulse J = Mv is imparted to the body at one of its ends, what would be its angular velocity

A

v/L

B

2v/L

C

v/3L

D

v/4L

Answer

v/L

Explanation

Solution

Initial angular momentum of the system about point O

= Linear momentum × Perpendicular distance of linear momentum from the axis of rotation =Mv(L2)= Mv\left( \frac{L}{2} \right) ....(i)

Final angular momentum of the system about point O =I1ω+I2ω=(I1+I2)ω=[M(L2)2+M(L2)2]ω= I_{1}\omega + I_{2}\omega = (I_{1} + I_{2})\omega = \left\lbrack M\left( \frac{L}{2} \right)^{2} + M\left( \frac{L}{2} \right)^{2} \right\rbrack\omega ....(ii)

Applying the law of conservation of angular momentum

⇒ Mv(L2)=2M(L2)2ωMv\left( \frac{L}{2} \right) = 2M\left( \frac{L}{2} \right)^{2}\omega

⇒ ω=vL\omega = \frac{v}{L}