Consider a system of blocks X. A and B as shown in the figure. The blocks A and B have equal mass and are connected by a massless string through a massless pully. The coefficient of friction between block A & X and B & X is 0.5. If block ‘X’ moves on horizontal frictionless surface, what should be its minimum acceleration such that blocks A and B remain stationary. (g= acceleration due to gravity)

Block X moves with acceleration a,  such that block A and B remains stationary. Block B is stationary only when, Tension force on block B is equal to the friction force on B.

i.e., T = μR

T = μma ...(i) [∵R=ma]

Block A is stationary only when the friction force on block A is equal to the sum of the tension force on block A and the applied force on block A.

μR = T+ma

μmg = μma+ma [From Eq. (i)]

μg = μa+a

μg = a(μ+1)

a = μg/(μ+1) = 0.5g(0.5+1) = g/3

∴a = g/3

Thus, g/3 should be its minimum acceleration such that block A and B remains stationary.