Forces, density and pressure

A-Level Physics 9702 · Topic 4 revision notes · AS

Moments, couples and centre of gravity, the two conditions for equilibrium, and density, fluid pressure and upthrust.

10 definitions to learn word for word, 9 equations in their own column, 6 diagrams and graphs. Read it, cover it, test yourself.

Definitions (mark-scheme wording)EquationsExam traps

4.1Turning effects of forces

Moment of a force: the product of the force and the perpendicular distance from the pivot to the line of action of the force.
Couple: a pair of forces that are equal in magnitude, opposite in direction and act along parallel but different lines of action.
Torque of a couple: the product of one of the forces and the perpendicular distance between the lines of action of the two forces.
Centre of gravity: the point at which the whole weight of an object may be considered to act.

Moment unit: N m (not J, even though the base units match).
A slanted force: moment = F sin θ × d, using only the perpendicular component.
A couple gives zero resultant force but a non-zero torque: pure rotation, no translation.
Torque of a couple is the same about any point.
Uniform object: centre of gravity at its geometric centre.
Freely suspended object hangs with its centre of gravity directly below the pivot.

Moment of a slanted force: use the perpendicular distance d_⊥ = d sin θ to the line of action.

A couple: forces cancel (no acceleration) but give torque F × s.

Using the distance to the point of application instead of the perpendicular distance to the line of action.
Torque of a couple is F × s, not 2F × s.

Equilibrium: the state in which there is no resultant force and no resultant moment (torque) on an object.
Principle of moments: for an object in equilibrium, the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about the same point.

Two conditions: resultant force = 0 (no linear acceleration) and resultant torque = 0 (no angular acceleration).
Take moments about the point where an unknown force acts: it drops out.
Resolve forces into two perpendicular directions; each sum must be zero.
Three coplanar forces in equilibrium form a closed vector triangle when drawn tip-to-tail.
Three non-parallel forces in equilibrium have lines of action that pass through one point.
Weight of a uniform beam acts at its midpoint.

Beam balanced on a pivot: take moments about the pivot. Here 30 N × 0.80 m = 60 N × 0.40 m.

Three forces in equilibrium drawn tip-to-tail close the triangle: zero resultant.

Zero resultant force alone is not equilibrium: a couple has zero resultant force but still rotates.
Forgetting the beam's own weight, acting at its centre of gravity.

Density: mass per unit volume.
Pressure: force per unit area, where the force acts normal (perpendicular) to the area.
Upthrust: the upward force exerted by a fluid on a submerged or floating object, due to the pressure difference between its lower and upper surfaces.
Archimedes' principle: the upthrust on an object in a fluid equals the weight of fluid it displaces.

Pa = N m^-2; density of water = 1000 kg m^-3.
Fluid pressure acts equally in all directions at a given depth.
Pressure increases linearly with depth; it does not depend on the shape of the container.
Total pressure at depth h = atmospheric pressure + ρgh.
Upthrust arises because pressure on the bottom face exceeds that on the top: F = ρgV.
Floating object: upthrust = weight.

Deriving p = ρgh: the column above area A has mass ρAh and weight ρAhg, so p = W/A = ρgh.

Upthrust: the bottom face is deeper, so the upward force exceeds the downward force by ρgV.

In F = ρgV, ρ is the density of the fluid and V the submerged volume, not the object's.
Δp = ρgΔh gives the pressure difference; add atmospheric pressure for the total pressure.