Two horizontal cylinders of different diameters are connected and filled with an incompressible fluid. The diameter of the smaller piston is 2 cm and that of the larger piston is 8 cm . The larger piston is attached to a relaxed horizontal spring of spring constant 4000 N/m . A constant force of 50 N is applied to the smaller piston, pushing it inwards. When the system reaches equilibrium, the distance moved by the s

Step-by-step solution

Let the diameters be d₁ = 2 cm and d₂ = 8 cm . The ratio of their areas is: A₂ A₁ = ( d₂ d₁ )^2 = ( 8 2 )^2 = 16 By Pascal's law, the force F₂ exerted by the fluid on the larger piston at equilibrium is: F₂ = F₁ A₂ A₁ = 50 16 = 800 N At equilibrium, this force is balanced by the restoring force of the spring. Using Hooke's law ( F = kx₂ ): 800 = 4000 x₂ x₂ = 0.2 m = 20 cm Since the fluid is incompressible, the volume of fluid displaced by the smaller piston equals the volume entering the larger cylinder: A₁ x₁ = A₂