Question

A small bloc of mass m kept at the left end of a larger block of mass M & length l. The system can slide on horizontal road. The system is started towards right with an initial velocity v. The friction coefficient between road & bigger block is u & that between block is u/2. Find the time elapsed before the smaller block separates from the bigger block.

Answer 1

Draw the free body diagrams for both blocks separately.  See diagram enclosed.
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Let a1 = deceleration of mass M.   initial velocity = v.
N1 = (M+m) g,       m a1 = - μ1 N1 - μ2 N2 = - μ1 (M+m) g - μ2 m g
    => a1 = - g [ μ1 M/m + (μ1+μ2) ]
    => v1 wrt ground = v + a1* t = v - g [μ1*M + (μ1+μ2) m] t

Let  a2 = acceleration of mass m.    u2 = 0 initial velocity wrt Mass M.  We add a pseudo force acting on mass m as it is on a non-inertial frame attached to mass M.

N2 = mg,    m*a2 = μ2 N2 - m a1

m * a2 = μ2 mg + g [ μ1 M + (μ1+μ2) m ]          in the forward direction.
     => a2 = g [ μ1 M/m + μ1 + 2 μ2 ]
     => v2 = 0 + a2 t =  g [ μ1 M/m + μ1+ 2 μ2 ] t

The distance to travel before smaller block m separates from the bigger block M = L = width of block M.
 
$L = 0*t + \frac{1}{2} a_2\ t^2\\t = \sqrt{\frac{2 L}{a_2}} = \sqrt{\frac{2L}{g\ [ \mu_1\frac{M}{m} +\mu_1+2 \mu_2 ]}}\\Given\ \mu_2=\frac{1}{2}\mu_1\\\\t=\sqrt{\frac{2L}{g\ \mu\ (\frac{M}{m} +2)}}$

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