Question #81242

A U-tube of uniform cross-sectional area $A$ containing an incompressible liquid of density $\rho$ is placed inside an elevator. The left arm of the U-tube is open to the atmosphere (pressure $P_0$). The right arm is sealed with a frictionless, leak-proof piston of mass $m$. A light inextensible string is attached to the piston, passes over a fixed ideal pulley, and then goes around a massless movable pulley before being attached to the ceiling of the elevator. All string segments are vertical. The movable pulley is connected to the floor of the elevator via an ideal vertical spring of stiffness $k$. Initially, the elevator is at rest, the system is in equilibrium, and the fluid levels in both arms are equal. The elevator then begins to accelerate upwards with a constant acceleration $a$. Assuming the liquid does not spill, what is the downward displacement $y$ of the piston relative to the U-tube once the new equilibrium is established?