A block of mass M rests on a smooth horizontal table. It is connected by a light inextensible string passing over a smooth, massless pulley at the edge of the table to a hanging block of mass m . When m = M 3 , the acceleration of the system is a₁ . When m = M 2 , the acceleration of the system is a₂ . The ratio a₁ a₂ is

Step-by-step solution

For a system where a block of mass M is on a smooth horizontal table and is pulled by a hanging mass m , the driving force is the weight of the hanging mass, mg . The total mass being accelerated is (M + m) . Therefore, the acceleration of the system is: a = mg M + m Case 1: When m = M 3 a₁ = ( M 3 )g M + M 3 = M 3 4M 3 g = g 4 Case 2: When m = M 2 a₂ = ( M 2 )g M + M 2 = M 2 3M 2 g = g 3 Taking the ratio of the two accelerations: a₁ a₂ = g 4 g 3 = 3 4 Answer: 3 4