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# A rope thrown over a pulley has a ladder with a man of mass m on one of its ends and a counter balancing mass M on its other end. The man climbs with velocity $V_r$ relative to the ladder. Ignoring the masses of the pulley as well as the friction on the pulley axis, the velocity of the centre of mass of this system is

$\begin {array} {1 1} (a)\;\frac{m}{M}Vr & \quad (b)\;\frac{m}{2M}Vr \\ (c)\;\frac{M}{m}V_r & \quad (d)\;\frac{2M}{m}Vr \end {array}$

Mass of Man $=m$
Man of ladder $=M-m$
If man moves up, ladder moves down and counter balancing man moves up.
Velocity of man relative to ladder $=V_r$
Velocity of ladder $= V$
Velocity of man relative to ground $=(V_r-V)$
$V_{cm}=\large\frac{MV+m(V_r-v)-(M-m)V}{2M}$
$V_{cm} =\large\frac{mv_r}{2M}$