Question

a ladder is sliding against a vertical wall.its horizontal velocity is v and at a instant it make angle 30 degree with horizontal .then find the velocity of COM

Answer 1

$$\sqrt{x^2+y^2}=L=const\to \frac{d}{dt}\sqrt{x^2+y^2}=0\to \\x\frac{dx}{dt}+y\frac{dy}{dt}=0$$

$$v_y\cos{60}=v\sin{60}\to v_y=\sqrt{3}v$$

$$\vec{v}_{com}=\frac{d}{dt}\vec{r}_{com}=0.5(v_x\vec{i}+v_y\vec{j})=0.5(v\vec{i}-\sqrt{3}v\vec{j})$$

Magnitude of the velocity of cm is

$$v_{com}=v$$