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Figure shows block A a motor C mounted on a with combined mass m and another block B of mass 2m. Block B is attached to shaft of motor with a light inextensible string. Motor is switched on at t =0 and string starts winding on its shaft and block B starts moving at acceleration a. If initial separation between blocks is l then find the time after which the two bodies will collide. The friction coefficient between blocks and ground is $μ$ .

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$\sqrt{\frac{2 l}{a + μ g}}$

$\sqrt{\frac{2 l}{3 a + μ g}}$

$\sqrt{\frac{l}{3 a + μ g}}$

$\sqrt{\frac{2 l}{2 a + μ g}}$

answer is A.

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Detailed Solution

Equation of motions of (A + C) and B as

$T - f = {ma}_{1}$

$T - μmg = {ma}_{1}$ ………(i)

$T - 2 μmg = 2 ma$ ………(ii)

$T = 2 μmg + 2 ma$

from (i) $2 μmg + 2 ma - μmg = {ma}_{1}$

$a_{1} = μg + 2 a$

relative acceleration of B w.r.t (A+C) a_BA=a+a₁ $= 3 a + μg$

time taken to cover a length l is $t = \sqrt{\frac{2 l}{a_{BA}}} = \sqrt{\frac{2 l}{3 a + μg}}$

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