In the given figure, a long platform of mass m is placed on a frictionless surface. Two blocks of masses 4m and m (where m=10 kg) are placed on the platform. For both blocks, the coefficient of static friction with the platform is equal to 0.16, and the coefficient of kinetic friction is equal to 0.10. The blocks are connected by a light ideal string through a light pulley (mounted at a movable massless stand), which is acted upon by an unknown horizontal force F. If the acceleration of the platform is $2 m / s^{2},$ find the value of the unknown force $F and acceleration$ of blocks 4m and m as $a_{1}$ and $a_{2}$ respectively.
$F (N), a_{1} (m / s^{2}), a_{2} (m / s^{2})$

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$180 N, 2 m / s^{2}, 8 m / s^{2}$

$180 N, 8 m / s^{2}, 2 m / s^{2}$

$90 N, 2 m / s^{2}, 8 m / s^{2}$

$90 N, 8 m / s^{2}, 2 m / s^{2}$

answer is A.

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Detailed Solution

Both blocks 4 m and m have a tendency of motion towards rightwards so friction will act on both blocks leftward, hence on the platform rightward. So

$\begin{array}{l} a = \frac{F_{1} + F_{2}}{m} = 2 m / s \\ F_{1} + F_{2} = 20 N \\ F_{1} \leq 64 N_{1} F_{2} \leq 16 N \end{array}$

Case I: Suppose block of mass 4 m has a relative motion with respect to platform So,

$\begin{array}{l} F_{1 k} = 40 N \\ F_{2} = - 20 N (not possible) \end{array}$

Case II: Suppose block of mass m has a relative motion with respect to platform, so

$\begin{array}{l} F_{2 k} = 10 N \Rightarrow F_{1 s} = 10 N (possible) \\ \frac{F}{2} - F_{1 s} = 4 m \times 2 \Rightarrow \frac{F}{2} = 1 + 4 \times 10 \times 2 = 90 \\ \Rightarrow F = 180 N \Rightarrow \frac{F}{2} - F_{2 k} = m a \Rightarrow 90 - 10 = 10 a \Rightarrow a = 8 m / s^{2} \end{array}$