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Floating and Sinking: Upthrust and Density

Floating and Sinking: Upthrust and Density

Drop a block into a liquid and compare its weight with the upthrust
v4
Drag the block to push it under or lift it out, then let go. Force arrows to scale. The dashed line is the starting liquid level.
Upthrust grows with the submerged depth until the block is fully under.

Live values

Think first: will this block float or sink in this liquid? If it floats, what fraction will be under the surface?
Weight, W—
Upthrust, U—
Support force, R—
Resultant force—
Fraction submerged—
Average density of the block—
Rise in liquid level—

Results

Worked calculation

Equations

Where upthrust comes from. Pressure in a liquid increases with depth, so the pressure on the bottom face of the block is greater than on the top face. The sideways pressure forces cancel. The difference between the bottom and top forces is the upthrust.

Archimedes' principle. Upthrust equals the weight of the fluid displaced: U = ρf g Vsubmerged. The upthrust grows as more of the block goes under. Once the block is fully submerged, the upthrust is ρf g V, using the whole volume, and pushing the block deeper does not change it.

Floating. A floating block settles where the upthrust equals its weight. Then Vsubmerged / Vtotal = ρbody / ρfluid. If the body is denser than the fluid, even the maximum upthrust is less than the weight, so the body sinks.

Average density. What matters is the body's average density: its total mass divided by its total volume. A hollow steel box can float even though steel is nearly eight times denser than water.

Equal densities. When the densities are equal, the resultant force on a fully submerged body is zero. The body stays wherever it is put, but zero resultant force does not by itself mean the body is stationary.

On the bottom. A sunk block rests on the floor of the tank. The floor pushes up with a support force R, and W = U + R. Upthrust and drag are different: the upthrust still acts on the block at rest.

The model. The tank is 50 cm wide and 20 cm from front to back, so its cross-sectional area is A = 0.100 m². The liquid level rises by Δh = Vsubmerged / A as the block goes in. The bobbing is damped so that the block settles quickly. Force arrows are to scale with each other. Pressure arrows show the pressure above atmospheric pressure, because atmospheric pressure acts on every face and cancels.