Vectors, Forces and Moments

Scalars and vectors

• A scalar has only magnitude (size): mass, speed, distance, energy, temperature.
• A vector has magnitude and direction: displacement, velocity, acceleration, force, momentum.

Adding vectors

Vectors are added tip-to-tail. For two perpendicular vectors, the resultant is found using Pythagoras and trigonometry:

resultant = √(x² + y²)      direction: tan θ = y ÷ x

For non-perpendicular vectors, use a scale drawing or components.

Resolving vectors

Any vector can be split into horizontal and vertical components:

horizontal = F cos θ        vertical = F sin θ

This is essential for inclined planes and projectile problems — treat the two directions independently.

Forces and equilibrium

An object is in equilibrium when the resultant force and resultant moment are zero — it stays still or moves at constant velocity.

• For forces in equilibrium, the components in each direction balance (the vectors form a closed triangle).

Moments

A moment is the turning effect of a force:

moment = force × perpendicular distance from the pivot   (N m)

• The principle of moments: for a body in equilibrium, the total clockwise moments = total anticlockwise moments about any point.
• A couple is two equal, opposite, parallel forces; its moment (torque) = one force × the perpendicular distance between them.

Centre of mass

The centre of mass is the single point where the whole weight of an object appears to act. An object topples when the line of action of its weight falls outside its base.

Worked example

A 3 N force acts east and a 4 N force acts north on a point. Find the resultant.

• Magnitude = √(3² + 4²) = √25 = 5 N. Direction: tan θ = 4/3 → θ = 53° north of east. ✓

Common mistakes

• Confusing scalars and vectors (e.g. speed vs velocity, distance vs displacement).
• Swapping sin and cos when resolving — the component adjacent to the angle uses cos.
• Forgetting the moment needs the perpendicular distance from the pivot.

Exam tips

• Resolve forces into perpendicular components and treat each direction separately.
• Use the principle of moments for equilibrium of beams/ladders.
• State both magnitude and direction for a resultant vector.

Key facts to remember

• Scalar = magnitude only; vector = magnitude + direction; resultant of perpendicular vectors via Pythagoras + trig.
• Resolve: horizontal = F cos θ, vertical = F sin θ.
• Moment = force × perpendicular distance; equilibrium needs zero resultant force and balanced moments (principle of moments).