In the figure shown, the instantaneous speed of end \(A\) of the rod is \(v\) to the left. The angular…

In the figure shown, the instantaneous speed of end \(A\) of the rod is \(v\) to the left. The angular velocity of the rod of length \(L\) must be

1. \(\frac{\nu}{2 L}\)
2. \(\frac{v}{L}\)
3. \(\frac{v \sqrt{3}}{2 L}\)
4. none of these

Solution

The velocities of the ends \(A\) and \(B\) along the length of rod should be the same.
Hence,
\(\mathrm{v}_{\mathrm{A}} \cos 30^{\circ}=\mathrm{v}_{\mathrm{B}} \cos 30^{\circ} \Rightarrow \mathrm{v}_{\mathrm{A}}=\mathrm{v}_{\mathrm{B}}=\mathrm{v}\)
Hence, the angular velocity of the rod is
\(\omega=\frac{\left(\mathrm{v}_{\mathrm{AB}}\right) \perp}{\mathrm{L}}=\frac{2 \mathrm{v} \sin 30^{\circ}}{\mathrm{l}} \Rightarrow \omega=\frac{\mathrm{v}}{\mathrm{L}}\)

Asked in: JEE Mains - Rotational Motion - Chapter Test

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