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Consider a body shown in the figure, consisting of two identical balls, each of mass $\mathrm{M}$ connected by a light right rod. If an impulse $J=M V$ ( $V=$ linear velocity of ball)is imparted to the body at one of its ends, what would be its angular velocity?
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The correct answer is:
$\frac{V}{L}$
Impulse $J=\Delta P$
$\begin{aligned} & \Rightarrow M V=2 M v^{\prime}-0 \\ & \Rightarrow v^{\prime}=\frac{V}{2}\end{aligned}$
The rod will rotate about its COM.
Thus, $v^{\prime}=\omega R=\omega\left(\frac{L}{2}\right)$
$\Rightarrow \omega=\frac{2 v^{\prime}}{L}=\frac{V}{L}$
$\begin{aligned} & \Rightarrow M V=2 M v^{\prime}-0 \\ & \Rightarrow v^{\prime}=\frac{V}{2}\end{aligned}$
The rod will rotate about its COM.
Thus, $v^{\prime}=\omega R=\omega\left(\frac{L}{2}\right)$
$\Rightarrow \omega=\frac{2 v^{\prime}}{L}=\frac{V}{L}$
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