Relative Equilibrium of Liquids — Worked Examples

An open rectangular tank is $3.00\ \text{m}$ long and $2.00\ \text{m}$ high and initially contains water to depth $1.50\ \text{m}$. It accelerates horizontally at $2.50\ \text{m/s}^2$. Determine the two end depths, check for spilling, and find the maximum bottom gage pressure.

A rectangular tank is $4.00\ \text{m}$ long, $2.00\ \text{m}$ wide, and $2.00\ \text{m}$ high. It initially contains water to depth $1.80\ \text{m}$ and accelerates horizontally at $3.00\ \text{m/s}^2$. Determine the volume spilled after relative equilibrium is reached.

A water tank accelerates vertically upward at $3.00\ \text{m/s}^2$. Determine the gage pressure $1.20\ \text{m}$ below its open free surface.

A water tank accelerates vertically downward at $6.00\ \text{m/s}^2$. Determine the gage pressure $1.50\ \text{m}$ below its open free surface and identify the free-fall limit.

During an idealized free-fall interval, a connected pressure boundary fixes a liquid at $95.0\ \text{kPa}$ absolute. The container and liquid accelerate downward at $g$, and other effects are neglected. What absolute pressure exists at a point $2.00\ \text{m}$ vertically away within the connected liquid?

A $5.00\ \text{m}$ long tank accelerates horizontally at $4.00\ \text{m/s}^2$ while accelerating upward at $2.00\ \text{m/s}^2$. Determine the free-surface angle magnitude and end-to-end elevation difference.

A cylindrical tank is $1.20\ \text{m}$ in diameter and $1.80\ \text{m}$ high and initially contains water to depth $1.20\ \text{m}$. It rotates at $60.0\ \text{rpm}$. Determine center and wall depths and check for spilling or bottom uncovering.

An open cylindrical tank has radius $0.750\ \text{m}$, height $1.50\ \text{m}$, and initial water depth $1.00\ \text{m}$. Assuming the surface vertex remains above the tank bottom, determine the speed at which the wall first reaches the rim.

A cylindrical tank has radius $0.500\ \text{m}$, height $1.00\ \text{m}$, and initial liquid depth $0.400\ \text{m}$. Determine the angular speed at which the rotating free-surface vertex just reaches the tank bottom. Verify that the wall is still below the rim at that instant.

A completely filled cylindrical tank of water rotates at $8.00\ \text{rad/s}$. Determine the pressure increase from the rotation axis to radius $0.500\ \text{m}$ at the same elevation.

Take the free-surface vertex of a rotating open water tank as $r=0$, $z=0$, with $z$ positive upward. At a point $r=0.400\ \text{m}$ and $z=-0.300\ \text{m}$, the liquid rotates at $\omega=5.00\ \text{rad/s}$. Determine gage pressure.

A cylindrical tank has radius $0.500\ \text{m}$, height $1.20\ \text{m}$, and initial water depth $1.00\ \text{m}$. After rotation and spillage reach relative equilibrium, the free surface is at the rim at the wall and the center-to-wall elevation difference is $0.800\ \text{m}$. The center remains wetted. Determine the spilled volume.