Unbalanced Force - Newton's Second Law of Motion

1. Definition

A. Unbalanced forces

Forces on a body are unbalanced when the resultant force is not zero:

Fᵣₑₛᵤₗₜₐₙₜ ≠ 0

B. Newton’s second law (O Level form)

The resultant force on a body causes it to accelerate in the direction of the resultant force.

For constant mass:

Fᵣₑₛᵤₗₜₐₙₜ = ma

• Fᵣₑₛᵤₗₜₐₙₜ = resultant force (N)
• m = mass (kg)
• a = acceleration (m s⁻²)

2. Key Ideas

• Always use the resultant force, not just one force.
• If Fᵣₑₛᵤₗₜₐₙₜ = 0, forces are balanced and a = 0 (see Balanced Forces and Newton’s First Law).
• If Fᵣₑₛᵤₗₜₐₙₜ ≠ 0, the object accelerates:
  • same direction as velocity → speeds up
  • opposite direction to velocity → slows down (decelerates)
• Units check: N = kg × m s⁻².
• 1 N is the force needed to give a mass of 1 kg an acceleration of 1 m s⁻².

Newton’s 2nd law (constant mass): force vs acceleration

Force–acceleration graph for a constant mass, showing direct proportionality (straight line through origin).

Scroll across the graph to read all labels.

Open full-size graph

View figure data

Values for Newton’s 2nd law (constant mass): force vs acceleration

Resultant force (N)	m = 2 kg
0	0
2	1
4	2
6	3
8	4
10	5

Values for Newton’s 2nd law (constant mass): force vs acceleration

Resultant force (N)	m = 2 kg
0	0
2	1
4	2
6	3
8	4
10	5

3. Detailed Explanations

A. Resultant force decides acceleration

Resultant force is the vector sum of all forces on the object:

• if forces cancel → Fᵣₑₛᵤₗₜₐₙₜ = 0 → no acceleration
• if one direction “wins” → Fᵣₑₛᵤₗₜₐₙₜ ≠ 0 → acceleration

B. Exam workflow (how to use F = ma)

1. Draw a free body diagram (FBD) and label all forces.
2. Choose a positive direction (often the direction of motion).
3. Add the forces using signs; with forward positive, subtract the backward-force magnitudes.
4. Use Fᵣₑₛᵤₗₜₐₙₜ = ma.
5. Check the unit and use the sign to state the direction.

See: Free Body Diagrams (FBD) and How To Add Forces.

C. Including friction and air resistance

In many questions, the applied force is not the same as the resultant force because resistive forces act in the opposite direction.

• friction: Friction
• air resistance and terminal velocity: Terminal Velocity

4. Common Mistakes

• Using the applied force instead of the resultant force.
• Forgetting friction/air resistance when the question does not say “neglect”.
• Mixing up mass and weight (weight is a force; mass is not).
• Unit errors (g instead of kg, km/h instead of m/s, minutes instead of seconds).
• Sign mistakes (not stating a positive direction, or giving an acceleration direction that contradicts the forces).

5. Exam Tips

• Start with an FBD, even for a 1-line calculation.
• Write “resultant force = (forward forces) − (backward forces)” before substituting.
• If the object slows down, its acceleration is opposite to its velocity.
• Always show units and give your final answer in N or m s⁻² as needed.

6. Worked Examples

7. Mind Stretchers