A block of mass 4 kg is placed on a horizontal rough surface and connected to a hanging block of mass 1 kg by a light inextensible string passing over a smooth pulley. The coefficient of kinetic friction between the 4 kg block and the surface varies with the distance x moved by the block as = x 8 . If the system is released from rest at x = 0 , the maximum speed attained by the blocks is (Take g = 10 m s ⁻² )

Step-by-step solution

The net driving force on the system is the weight of the hanging block minus the frictional force on the block on the horizontal surface. F_ net = m₂ g - m₁ g = 1(10) - ( x 8 )(4)(10) = 10 - 5x The acceleration of the system is: a = F_ net m₁ + m₂ = 10 - 5x 4 + 1 = 2 - x The speed of the system is maximum when the acceleration becomes zero. a = 0 2 - x = 0 x = 2 m Using the work-energy theorem for the system from x = 0 to x = 2 m : W_ net = K ₀² F_ net , dx = 1 2 (m₁ + m₂) v_ max ^2 - 0 ₀² (10 - 5x) , dx = 1 2 (5)