§ 4.1 Turning effects of forces
Key ideas
- The moment of a force is force × perpendicular distance from the pivot to the line of action. Unit: N m.
- A couple is a pair of equal, opposite, parallel forces not in line: it produces rotation with no resultant force.
- The torque of a couple = one force × the perpendicular separation of the two forces.
- The centre of gravity is the single point at which the whole weight appears to act.
Equations
moment = F × dforce × perpendicular distance from the pivotN m
torque = F × sone force of a couple × the separation of the pairN m
Fig. 1 · A couple: the two forces cancel as forces, so nothing accelerates, but together they turn the beam with torque F × s.
Watch out: the distance in a moment is the perpendicular distance from the pivot to the line of action, not the distance to where the force happens to be applied. For slanted forces, resolve first or find the perpendicular arm.
§ 4.2 Equilibrium of forces
Key ideas
- Equilibrium needs both conditions: zero resultant force in every direction and zero resultant torque about any point.
- Principle of moments: in equilibrium, total clockwise moment = total anticlockwise moment about any pivot.
- Three coplanar forces in equilibrium, drawn nose-to-tail, form a closed triangle (a closed vector polygon for more forces).
Fig. 2 · Taken in order, the three forces return to the starting point: a closed triangle means zero resultant, the graphical test for equilibrium.
Watch out: zero resultant force is not enough on its own: a couple has zero resultant force yet still rotates the object. Always check moments as well.
§ 4.3 Density and pressure
Key ideas
- Density ρ = m/V (kg m⁻³); pressure p = F/A (Pa = N m⁻²).
- Pressure in a fluid rises with depth: Δp = ρgΔh, independent of the container's shape.
- Upthrust on a submerged object equals the weight of fluid displaced: F = ρgV with V the submerged volume (Archimedes' principle).
Equations
ρ = m / Vdensity = mass ÷ volumekg m⁻³
Δp = ρ g Δhpressure change with depth in a fluidPa
F = ρ g Vupthrust = weight of the fluid displacedN
Watch out: in the upthrust formula, ρ is the density of the fluid and V the submerged volume of the object, not the object's own density or total volume.