Physics · Lesson 2

Scalars & vectors, equilibrium of forces & moments

JAMBWAECNECO

What the syllabus asks

JAMB: Scalars and vectors: addition, resolution, resultant; equilibrium of forces: moments, principle of moments, couples, centre of gravity, stable, unstable and neutral equilibrium

WAEC: Scalars and vectors, addition and resolution of vectors, resultant and equilibrant, triangle and parallelogram of forces, equilibrium of forces, moments, principle of moments, couples, conditions of equilibrium, centre of gravity and stability

NECO: Scalars and vectors, addition and resolution of vectors, resultant and equilibrant, triangle and parallelogram of forces, equilibrium of forces, moments, principle of moments, couples, conditions of equilibrium, centre of gravity and stability

What you will learn

Distinguish scalars from vectors with examples.
Find the resultant of two vectors in the same line, at right angles and at any angle.
Resolve a vector into perpendicular components and use components to add several forces.
State and apply the conditions for equilibrium of concurrent forces, including the triangle of forces and Lami’s theorem.
Define the moment of a force and apply the principle of moments.
Solve problems on beams, metre rules and hinged rods using both conditions of equilibrium.
Explain couples, centre of gravity and the three states of equilibrium.

Key points

Scalars: magnitude only; vectors: magnitude and direction.
Same-line vectors add with signs; perpendicular: R = √(a² + b²), tan θ = b/a.
Any angle: R² = P² + Q² + 2PQ cos θ (parallelogram law).
Components: F cos θ along, F sin θ across; on a slope mg sin θ down the plane.
Equilibrium of concurrent forces: ΣFx = 0, ΣFy = 0; equilibrant opposes the resultant.
Moment = F × perpendicular distance (N m).
Principle of moments: clockwise = anticlockwise in equilibrium.
Rigid body equilibrium: forces balance and moments balance.
Couple: equal opposite parallel forces; torque = F × separation.
C.G.: low and over a wide base means stable; three states: stable, unstable, neutral.

Start here: Scalars and vectors

If a friend says “I walked 2 km”, you know how far she went but not where she ended up. If she says “I walked 2 km due north from the school gate”, you know exactly where she is. The second statement includes a direction. Physics separates quantities into these two types.

Scalar: a quantity that has magnitude (size) only. Vector: a quantity that has both magnitude and direction.

Scalars	Vectors
Distance, speed	Displacement, velocity
Mass, time, temperature	Force, weight, acceleration
Energy, work, power	Momentum, impulse
Volume, density, pressure	Electric field intensity, magnetic flux density

A vector is drawn as an arrow: its length shows the size (to a chosen scale) and its arrowhead shows the direction. In print, vectors are often written in bold (F) or with an arrow above.

Pressure is a common trap: although it comes from a force, pressure itself acts equally in all directions at a point in a fluid, so it is treated as a scalar. Work and energy are also scalars, even though they come from force and displacement.

⚠️ Common mistake: listing speed as a vector. Speed is a scalar; velocity (speed in a stated direction) is the vector.

💡 Exam tip: JAMB often gives four quantities and asks “which is a vector?” or “which pair are both scalars?”. Memorise the table above.

Try it yourself. Classify: kinetic energy, displacement, power, impulse.

Show solution

Scalars: kinetic energy, power. Vectors: displacement, impulse.

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Practice questions

1. Find the resultant of 5 N and 12 N acting at right angles.

Show worked answer

13 N at tan⁻¹(12/5) = 67.4° to the 5 N force.

2. Find the resultant of 4 N and 6 N at 60° to each other.

Show worked answer

R² = 16 + 36 + 24 = 76, R = 8.72 N.

3. Resolve 80 N acting at 40° above the horizontal.

Show worked answer

61.3 N horizontal, 51.4 N vertical.

4. A 2 kg mass rests on a smooth plane inclined at 37°. Find the force along the plane to hold it (sin 37° ≈ 0.6).

Show worked answer

20 × 0.6 = 12 N.

5. A 250 N child sits 1.6 m from a seesaw pivot. Where must a 400 N adult sit?

Show worked answer

1.0 m from the pivot.

Key words

Scalar: A quantity with magnitude only.
Vector: A quantity with magnitude and direction.
Resultant: The single vector with the same effect as two or more vectors together.
Equilibrant: The single force that balances the others; equal and opposite to the resultant.
Resolution: Splitting a vector into two perpendicular components.
Component: The effective part of a vector in a given direction.
Concurrent forces: Forces whose lines of action pass through one point.
Moment of a force: Force × perpendicular distance of its line of action from the pivot.
Fulcrum (pivot): The point about which a body turns.
Couple: Two equal, opposite, parallel forces not in the same line; they cause rotation only.
Torque: The turning effect of a couple or force; unit N m.
Centre of gravity: The point where the whole weight of a body appears to act.

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