Density and pressure

AS Level Physics · Forces, density and pressure. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

Density is mass per unit volume. Pressure is force per unit area, where the force acts at right angles to the area.

In a fluid at rest, pressure increases with depth, because deeper layers support the weight of the fluid above them. For a column of height Δh and cross-sectional area A, the weight is ρAΔhg. Dividing by A gives Δp = ρgΔh. The pressure depends on depth and density only, not on the shape of the container.

Because pressure is greater at greater depth, the upward force on the bottom of a submerged object is larger than the downward force on its top. This difference is the upthrust. It equals the weight of the fluid displaced.

Rules to remember

Common mistake

Using the density of the object in the upthrust equation. The ρ in F = ρgV is the density of the fluid, and V is the volume of the object that is below the surface.

Worked example

A metal block of weight 39 N and volume 5.0 × 10−4 m3 hangs from a newton meter and is fully submerged in oil of density 900 kg m−3. What does the newton meter read? (g = 9.81 m s−2)

  1. Upthrust = ρgV = 900 × 9.81 × 5.0 × 10−4 = 4.41 N upwards.
  2. The block is in equilibrium: reading + upthrust = weight.
  3. Reading = 39 − 4.41 = 34.6 N.

Answer: About 35 N (34.6 N).

Practice questions

Try each one, then open the answer.

1. What is the pressure due to the water at a depth of 12 m in a lake? (density of water 1000 kg/m3, g = 9.81 m/s2)

  1. A
    1.2 × 103 Pa
  2. B
    1.2 × 104 Pa
  3. C
    1.2 × 105 Pa
  4. D
    1.2 × 106 Pa
Show answer

Answer: C. Δp = ρgΔh = 1000 × 9.81 × 12 = 1.18 × 105 Pa.

2. A box of weight 600 N rests on the floor on a face measuring 0.50 m by 0.30 m. What pressure does it exert on the floor?

  1. A
    90 Pa
  2. B
    1200 Pa
  3. C
    2000 Pa
  4. D
    4000 Pa
Show answer

Answer: D. Area = 0.50 × 0.30 = 0.15 m2; p = F/A = 600 ÷ 0.15 = 4000 Pa.

3. Why does an object fully submerged in a liquid experience an upthrust?

  1. A
    liquids are less dense than solids
  2. B
    the pressure on its lower surface is greater than on its upper surface, giving a net upward force
  3. C
    the liquid pushes equally on all sides
  4. D
    gravity is weaker in liquids
Show answer

Answer: B. Pressure increases with depth, so the upward push on the bottom exceeds the downward push on the top. The difference is the upthrust.

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