A rod of length $l$ resting on a wall and the floor. Its lower end $A$ is pulled towards right with a constant velocity $v$. Find the velocity of the other end $B$ downward when the rod makes an angle $\theta$ with the vertical. Also, find the angular velocity of the rod.

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Aicharya · Accepted Answer

As rod is a rigid body, the distance between point $A$ and point $B$ should remain unchanged. Hence the relative velocity of ends of rod along its length should be zero.
It means the velocities of ends of rod along its length must be equal.
$\begin{aligned} v_{A} \sin \theta &= v_{B} \cos \theta \Rightarrow v_{B} = v_{A} \frac{\sin \theta}{\cos \theta} = v \tan \theta \\ \text{or } \quad v_{B} &= v \tan \theta \quad \text{...(i)} \end{aligned}$ Then, $\omega_{B A} = \frac{v_{B A_{\perp}}}{l} = \frac{\left|v_{A} \cos \theta - \left(-v_{B} \sin \theta\right)\right|}{l}$ $\Rightarrow \quad \omega_{B A}=\frac{v_{A} \cos \theta+v_{B} \sin \theta}{l}$ ...(ii)
Using Eqs. (i) and (ii),
$\omega_{B A}=\frac{v}{l \cos \theta}$
This is also angular velocity of rigid body about its centre of mass.

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A rod of length \(l\) resting on a wall and the floor. Its lower end \(A\) is pulled towards right with a…

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