Question

A rod of length $l$ is moving in a vertical plane $x-y)$ when the lowest point $A$ of the rod is moved with a velocity $v$. Find the (a) angular velocity of the rod and (b) velocity of the end $B$

Answer 1

Since the rod is rigid, the components of $v_A$ and $v_B$ along the rod are equal. Thus,
$v_B\cos \theta =v_A\sin \theta =v\sin \theta$
$\Rightarrow v_B=v\tan \theta ....\left(i\right)$
Then, ${\omega }_{BA}=\frac{v_{BA}}{l}=\frac{v_{{BA}_{\perp }}}{l}=\frac{|v_A\cos \theta -(-v_B\sin \theta )|}{l}=\frac{v_A\cos \theta +v_B\sin \theta }{l}....\left(ii\right)$
Using Eqs. (i) and (ii), ${\omega }_{BA}=\frac{v}{l\cos \theta }$