A rod of length l $$=$$ $$1$$ m resting on a wall and the floor. Its lower end A is pulled towards right with a constant velocity v $$=$$ $$2$$ $$m/s$$. Find the angular velocity of the rod when it makes an angle θ$$=60$$∘ with the vertical.

Question

A rod of length l $$=$$ $$1$$ m resting on a wall and the floor. Its lower end A is pulled towards right with a constant velocity v $$=$$ $$2$$ $$m/s$$. Find the angular velocity of the rod  when it makes an angle θ$$=60$$∘ with the vertical.

Infinity Learn · 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.vAsin⁡θ$$=vBcos$$⁡θ⇒$$vB=vAsin$$⁡θ$$cos$$⁡θ$$=vtan$$⁡θor $$vB=vAtan$$⁡θ$$=vtan$$⁡θ             …(i)Then$$, ω$$BA=vBAl=vBA$$⊥$$l=vAcos$$⁡θ−−vBsin⁡θl⇒ ω$$BA=vAcos$$⁡θ$$+vBsin$$⁡θl            …$$(ii)Using$$ Eqs. $$(i) and (ii), ω$$BA=vlcos$$⁡θ$$=21$$×$$cos$$⁡$$60$$∘$$=4m/sThis$$ is also angular velocity of rigid body about centre of mass.

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