Forces, Density and Pressure

This guide covers Topic 4, Forces, density and pressure, in full — subtopics 4.1 Turning effects of forces, 4.2 Equilibrium of forces and 4.3 Density and pressure — from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is AS Level content.

Before studying this

This resource assumes vector resolution from Physical Quantities, Units and Measurement and Newton’s laws from Dynamics: Newton’s Laws and Momentum.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — AS Level, Topic 4

4.1 Turning effects of forces — weight acting at the centre of gravity; defining and applying the moment of a force; a couple as a pair of forces producing rotation only; defining and applying torque.

4.2 Equilibrium of forces — stating and applying the principle of moments; understanding equilibrium as no resultant force and no resultant torque; using a vector triangle to represent coplanar forces in equilibrium.

4.3 Density and pressure — defining and using density and pressure; deriving and using ∆p = ρg∆h for hydrostatic pressure; understanding upthrust as arising from a difference in hydrostatic pressure; calculating upthrust using F = ρgV (Archimedes’ principle).

Centre of gravity, moments and couples

An object’s weight can be treated as acting at a single point, its centre of gravity. The moment of a force about a point is force × perpendicular distance from the line of action of the force to that point, and has a turning effect (measured in N m).

A couple is a pair of equal, opposite, parallel forces whose lines of action don’t coincide — it has zero resultant force (so no translational acceleration) but a nonzero resultant moment, giving an angular acceleration. This doesn’t mean the body has no translation at all: a body already moving uniformly stays in that same uniform translation while a couple acts on it, gaining rotation on top of it. The torque of a couple is one force × the perpendicular distance between the two forces.

Equilibrium

An object is in equilibrium when there is no resultant force and no resultant torque acting on it — both conditions are needed; zero resultant force alone permits rotation, and zero resultant torque alone permits acceleration.

The principle of moments states that, for a system in equilibrium, the sum of clockwise moments about any point equals the sum of anticlockwise moments about that same point.

For three coplanar forces in equilibrium, a closed vector triangle (each force represented as an arrow, tip-to-tail, returning to the start) is a standard way to represent and solve for an unknown force or angle.

Worked example. A uniform beam of weight 40 N is pivoted at its centre (so the beam’s own weight produces no moment about the pivot), with a 25 N weight hung 0.6 m from the pivot on one side. How far from the pivot must a 15 N weight be hung on the other side to balance it? Using the principle of moments (the beam’s own 40 N weight is excluded because it acts through the pivot itself and so has zero perpendicular distance to it):

25 × 0.6 = 15 × d
15 = 15d
d = 1.0 m

Density and pressure

Density ρ = mass / volume. Pressure p = force / area (acting perpendicular to the surface).

Hydrostatic pressure. The pressure difference between two points in a fluid at different depths is:

∆p = ρg∆h

where ∆h is the depth difference. This can be derived from the definitions of pressure and density by considering the weight of a column of fluid acting over its base area.

Upthrust. Because pressure increases with depth in a fluid, the upward force on the bottom of a submerged object is greater than the downward force on its top — the resulting net upward force is upthrust, calculated using Archimedes’ principle:

F = ρgV

where ρ is the fluid’s density and V is the volume of fluid displaced (not necessarily the object’s total volume, if only partly submerged).

Floating and sinking. An object floats when the upthrust on it equals its weight, so it displaces exactly its own weight of fluid; it sinks when its weight exceeds the maximum upthrust available even when fully submerged. Equivalently, an object floats if its average density is less than the fluid’s density. This is why a steel ship floats even though steel itself is denser than water: the ship’s overall density, including the large volume of enclosed air within its hull, is below the density of water. If the hull is breached and fills with water, the average density rises above that of water and the ship sinks.

Common mistakes

• Forgetting the second equilibrium condition. Zero resultant force is not sufficient on its own — zero resultant torque is also required for true equilibrium.
• Confusing a couple with a single force or with two forces that aren’t parallel and opposite. A couple specifically produces rotation with no net linear force.
• Using the object’s total volume instead of the volume of fluid displaced in Archimedes’ principle — this only coincide when the object is fully submerged.
• Taking moments about the wrong point, or forgetting that the principle of moments can be applied about any point in a system in equilibrium, not just an obvious pivot.
• Judging whether an object floats by the density of its material alone, rather than its overall average density — a hollow or air-filled object made of a dense material can still have an average density below that of the fluid it sits in.

Quick revision checklist

• Moment = force × perpendicular distance; couple and torque definitions
• The two conditions for equilibrium (no resultant force, no resultant torque)
• The principle of moments, and using a vector triangle for three coplanar forces
• Density and pressure definitions
• Deriving and using ∆p = ρg∆h
• Archimedes’ principle, F = ρgV, using displaced fluid volume

Related resources

• Dynamics: Newton’s Laws and Momentum — the force concepts this topic extends
• Work, Energy and Power — the next AS topic
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.