Question

A stationary pulley carries a rope one end of which supports a ladder with a man and the other a counterweight of mass $M$ . The man of mass $m$ climbs up a distance $l$ with respect to the ladder and then stops. The displacement of the centre of mass of this system is

• A. $\frac{m l}{M + m}$
• B. $\frac{m l}{2 M}$
• C. $\frac{m l}{M + 2 m}$
• D. $\frac{m l}{2 M + m}$

Answer 1

Answer: (B)

From the question mass of the ladder is $\left(M - m\right)$ .
Let's assume that counterweight goes up by distance $x$ then ladder also goes down by $x$ . Therefore from the ground frame, man goes up by $\left(l - x\right)$ .

Then distance moved by the center of mass is given by,
$x_{c m}=\frac{M \left(x\right) + m \left(l - x\right) + \left(M - m\right) \left(- x\right)}{2 M}$
$x_{c m}=\frac{m l}{2 M}$