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

We can consider this motion as rotation with respect to instant centre of rotation (ICR).

In the picture, point D is ICR. Point B is the middle of AC and centre of the mass as well. The angular frequency is the same for all points of the ladder. Since we know velocity of point C, we can express angular frequency via v

$\omega = \frac{v}{R} = \frac{v}{DC} = \frac{v}{l \sin{30^\circ}}$

where l is length of ladder.

Then to find we $v_{COM}$ multiply $\omega$ by the distance to the ICR (which is equal to BD).

$v_{COM}= \omega \cdot BD = \omega \cdot \frac{l}{2}= \frac{v}{l \cdot 0.5} \cdot l \cdot 0.5 =v$.

Answer: $v_{COM} = v$