The Floating Block and the Leaking Tank
A wooden cube (side 0.10 m, density 600 kg/m³) floats in a large tank of water (density 1000 kg/m³). A small hole is opened 0.50 m below the water surface on the side of the tank, and water streams out horizontally. A student predicts the cube will sink lower as water drains, and separately predicts the efflux speed using v = √(2gh) with h measured from the tank bottom. Use g = 10 m/s².
1. Determine the cube's submerged depth while floating, and state whether it changes as the water level drops (tank cross-section is large).
Read the worked answer
Floating → buoyant force = weight: ρ_w g V_sub = ρ_cube g V_total. Fraction submerged = ρ_cube/ρ_w = 600/1000 = 0.60. Submerged depth = 0.60(0.10) = 0.060 m.
Because the cube floats by displacing its own weight, the submerged fraction depends only on the density ratio - not on the tank's water depth. As water drains (slowly, quasi-statically), the cube rides down with the surface but its submerged depth stays 0.060 m until the water is shallower than 0.060 m, at which point it rests on the bottom. The student's 'sinks lower' is wrong while it floats.
2. Evaluate the efflux-speed prediction. What is the correct speed at the instant shown?
Read the worked answer
Torricelli: v = √(2gh) where h is the depth below the free surface, not above the bottom. Here h = 0.50 m:
- v = √(2·10·0.50) = √10 ≈ 3.2 m/s.
The student's use of height-from-bottom is a category error in the Bernoulli derivation (the relevant pressure difference is ρg × depth below surface). As the surface falls, the instantaneous efflux speed decreases because h decreases - a time-dependent problem, not a single constant.
3. As the tank drains, does the cube's submerged depth change at the moment the water depth equals the cube's submerged depth? Explain.
Read the worked answer
While floating, submerged depth is fixed at 0.060 m regardless of surface height (density ratio argument). The transition occurs when the water depth equals 0.060 m: below that, the cube cannot displace enough water to support its weight, so the bottom of the tank supplies a normal force and buoyancy is less than weight. At exactly 0.060 m the cube is on the verge - normal force → 0. So the submerged depth is constant at 0.060 m for all depths ≥ 0.060 m and then decreases (limited by available water) below that. The judgment is separating the floating regime (Archimedes sets geometry) from the grounded regime (contact force appears).