Upthrust, floating and sinking

When something is put in a liquid it pushes some liquid aside, and the liquid pushes back up with a force called upthrust. If the upthrust can balance the weight, the object floats. Whether something floats depends on its density compared with the liquid, not just on how heavy it is.

Key facts

  • Upthrust is an upward force from a fluid on any object in it.
  • Archimedes’ principle: upthrust = weight of fluid displaced = ρ(fluid) × V(displaced) × g.
  • A floating object: upthrust = weight.
  • Objects less dense than the liquid float; denser ones sink.
  • Apparent weight in water = real weight − upthrust.
  • Fraction of a floating object under the surface = ρ(object) ÷ ρ(liquid).

Worked examples

Example 1

A beach ball is pushed under water and let go. Which forces act on it, and why does it shoot up?

  1. Gravity pulls the ball down: its weight. The ball is light, so this arrow is short.
  2. The water pushes up on it: upthrust. A big ball pushes aside lots of water, so this is big.
  3. Upthrust is bigger than weight, so the ball shoots up until it floats with the forces balanced.

Answer: Weight (down) and upthrust (up). Upthrust is bigger, so it rises.

Example 2

A stone weighs 8.0 N in air and 5.0 N in water. Find the upthrust and the stone’s volume. (ρ water = 1000 kg/m³, g = 9.8 N/kg)

  1. The water holds up part of the weight: that is the upthrust. upthrust = 8.0 − 5.0 = 3.0 N
  2. Upthrust = weight of water pushed aside = ρVg, so V = 3.0 ÷ (1000 × 9.8) = 3.06 × 10^−4 m³ ≈ 306 cm³

Answer: 3.0 N; 306 cm³

Example 3

A block of mass 120 g and volume 150 cm³ is put in water. Does it float? What fraction is under water?

  1. Density of the block: ρ = 120 ÷ 150 = 0.80 g/cm³
  2. 0.80 is less than 1.0, so the block floats.
  3. Fraction under water = its density ÷ water’s density. 0.80 ÷ 1.0 = 0.80

Answer: It floats, with 0.80 (80%) under water.

Example 4

A raft has volume 0.60 m³ and mass 180 kg. What extra mass can it carry before it is fully submerged in fresh water?

  1. Fully under, it pushes aside 0.60 m³ of water: 1000 × 0.60 = 600 kg
  2. That is the most it can weigh in total (raft + load). load = 600 − 180 = 420 kg

Answer: 420 kg