(c) p = ρgh; upthrust on an object in a fluid; Archimedes’ principle.

The idea

In a fluid at rest, pressure increases with depth according to p = ρgh, where ρ is the fluid density in kg m⁻³, g is the gravitational field strength in N kg⁻¹ and h is the depth below the surface in m. This pressure acts equally in all directions. A submerged or partly submerged object experiences upthrust because the pressure on its lower surface exceeds the pressure on its upper surface. Archimedes’ principle states that the upthrust equals the weight of fluid displaced. For a 0.20 m³ object fully submerged in water of density 1.0 × 10³ kg m⁻³ with g = 9.81 N kg⁻¹, upthrust = ρVg = 1.0 × 10³ × 0.20 × 9.81 = 1.96 × 10³ N. An object floats when upthrust balances its weight.

What earns the marks

• States p = ρgh and uses it to find pressure at a depth in a fluid of uniform density.
• Identifies upthrust as the resultant upward force on an object in a fluid caused by the pressure difference between its lower and upper surfaces.
• States Archimedes’ principle: the upthrust equals the weight of fluid displaced by the object.
• Calculates upthrust using weight of displaced fluid, ρVg, where V is the displaced volume.
• Explains floating and sinking in terms of the balance between upthrust and the object’s weight.

Common mistakes and how to put them right

• Using the total depth to the bottom of the container instead of the depth below the fluid surface.

  Put it right: The error is measuring from the wrong reference, and the correction is to measure h from the free surface of the fluid.

• Confusing the object’s volume with the displaced volume for a floating object.

  Put it right: The error is using the whole volume, and the correction is to use only the submerged volume, which equals the volume of fluid displaced.

• Treating upthrust as a fixed property of the object.

  Put it right: The error is ignoring the fluid, and the correction is to recognise that upthrust depends on the density of the fluid and the displaced volume.

Exam tips

• Sketch the object in the fluid and mark the depths of its top and bottom surfaces to see why a pressure difference produces upthrust.
• For floating objects, equate upthrust to weight and use the submerged volume, not the total volume, in ρVg.
• Keep g as 9.81 N kg⁻¹ unless the question states otherwise, and check that pressure answers are in pascals.