Question

Point $A$ of rod $AB(l=2m)$ is moved upwards against a wall with velocity $v=2m/s.$ Find angular speed (in rad/s) of the rod at an instant when $\theta ={60}^{\circ }$.

Answer 1

As the end A of the rod moves up along the wall with a velocity, 2m/s directed along +ve Y axis as shown in the figure. The other end of the rod, B, meanwhile moves with velocity, v , along -ve X direction as shown in the figure. So, resolving velocity at end A in two components, v1 and v2

and resolving velocity at B in two components in v3 and v4

$v_3=v_2$

$\therefore v\cos {60}^{\circ }=2\cos {30}^{\circ }=\sqrt{3}$

$\therefore v=2\sqrt{3}m/s$

Now looking at the components v1 and v4, both tending to move the rod in clockwise direction,

we can say that the net $\omega$

$\omega =\frac{v_1}{l}+\frac{v_4}{l}$

$=\frac{\left(2\sin {30}^{\circ }\right)+\left(v\sin {60}^{\circ }\right)}{2}$$=\frac{(2\times 1/2)+(2\sqrt{3}\times \sqrt{3}/2)}{2}$

$=2rad/s$