How To Add Forces

1. Definition

A. Resultant force

The resultant force is the single force that has the same effect as two or more forces acting together on an object.

Since force is a vector, you must combine forces using magnitude and direction.

2. Key Ideas

• Force is a vector, so you must state the direction (e.g. “20 N to the right”).
• For forces in a straight line (1D), choose a positive direction and add using signs:
  • Fᵣₑₛᵤₗₜₐₙₜ = ∑ F
• If forces act in the same direction, the resultant is the sum of magnitudes.
• If forces act in opposite directions, the resultant is the difference, in the direction of the larger force.
• For forces at an angle (2D), use a scale drawing (triangle or parallelogram method) and measure the resultant.
• Resultant force = 0 means forces are balanced (object at rest or moving with constant velocity).

Resultant depends on the angle between forces

Resultant force magnitude versus the angle between two equal forces (10 N each).

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Values for Resultant depends on the angle between forces

Angle between two equal forces (°)	F1 = F2 = 10 N
0	20
30	19.32
60	17.32
90	14.14
120	10
150	5.18
180	0

Values for Resultant depends on the angle between forces

Angle between two equal forces (°)	F1 = F2 = 10 N
0	20
30	19.32
60	17.32
90	14.14
120	10
150	5.18
180	0

3. Detailed Explanations

A. Why we use resultant force

When several forces act on an object, we often replace them with one force: the resultant.

This matters because motion depends on the resultant force:

• if forces are balanced ( Fᵣₑₛᵤₗₜₐₙₜ = 0), there is no acceleration
• if forces are unbalanced, the object accelerates (see Unbalanced Force)

B. Forces in the same line (1D): same direction

If forces are along the same straight line and point the same way, add the magnitudes.

Example: 10 N right and 20 N right gives 30 N right.

C. Forces in the same line (1D): opposite directions

If forces oppose each other, subtract the magnitudes. The resultant is in the direction of the larger force.

Example: 40 N right and 20 N left gives 20 N right.

D. Using signs (quick method for 1D)

Choose a positive direction (e.g. right is positive). Then:

• rightward force: + F
• leftward force: -F

Example: + 40 + (-20) = +20 N (so 20 N to the right).

E. Forces at an angle (2D): scale drawing

When forces are at an angle, you cannot add or subtract magnitudes directly. Use a scale drawing (graphical vector addition).

Workflow:

1. Choose a scale (e.g. 1 cm represents 5 N).
2. Draw the first force as an arrow to scale.
3. Draw the second force as an arrow to scale in the correct direction.
4. Use either method:
   • triangle method (head-to-tail), or
   • parallelogram method (complete the parallelogram).
5. Draw the resultant from the start point to the end point, and measure its length (then convert using the scale).

F. Special case: forces at 90°

If two forces are perpendicular, the vector diagram forms a right-angled triangle, so you can use Pythagoras’ theorem for the magnitude:

Fᵣₑₛᵤₗₜₐₙₜ = square root of (F₁² + F₂²)

4. Common Mistakes

• Adding magnitudes even when forces are in opposite directions.
• Giving a force value without a direction (e.g. writing “20 N” when “20 N to the right” is needed).
• Sign mistakes in 1D (choosing right as positive but treating a leftward force as positive).
• Drawing an inaccurate scale diagram (no scale stated, ruler not used, wrong angles).
• Treating “resultant force” as an extra force on the diagram (it replaces the forces).

5. Exam Tips

• Start with a quick sketch showing force directions.
• For 1D questions, state the sign convention (e.g. “right is positive”) and show the signed sum.
• For 2D graphical questions:
  • state your scale
  • use a ruler/protractor
  • give the magnitude and direction of the resultant (if asked)
• If the object is at rest or moves with constant velocity, write: Fᵣₑₛᵤₗₜₐₙₜ = 0 (balanced forces).

6. Worked Examples

7. Mind Stretchers