To find the RESULTANT of two forces not along the same line:
METHOD 1 — SCALE DRAWING (tip-to-tail):
1. Choose a scale (e.g. 1 cm = 10 N).
2. Draw the first force vector to scale.
3. From the tip of the first vector, draw the second force vector to scale.
4. Draw the resultant from the start of the first to the tip of the second.
5. Measure the resultant length → convert to force. Measure the angle.
METHOD 2 — PARALLELOGRAM OF FORCES:
1. Draw both forces from the same point to scale.
2. Complete the parallelogram.
3. The diagonal = resultant.
METHOD 3 — COMPONENT METHOD:
1. Resolve all forces into x and y components.
2. Sum all x-components: ΣFx.
3. Sum all y-components: ΣFy.
4. Resultant magnitude: R = √(ΣFx² + ΣFy²).
5. Angle: θ = arctan(ΣFy / ΣFx).
Equilibrium and Scale Drawings
EQUILIBRIUM: an object is in equilibrium when the resultant of all forces is zero.
For THREE FORCES in equilibrium:
When drawn tip-to-tail, they form a CLOSED TRIANGLE (the triangle of forces).
If the triangle closes, the object is in equilibrium.
PRACTICAL APPLICATIONS:
Finding the tension in two strings supporting a weight.
Analysing forces on a stationary object on a slope.
Determining the direction of motion when two forces act.
SCALE DRAWING TECHNIQUE:
Accuracy matters — use a ruler and protractor.
Always state the scale used.
Convert measurements back to actual forces using the scale.
LIMITATIONS:
Scale drawings introduce measurement errors — component method is more precise.
For GCSE, scale drawings are acceptable for force problems.
⚠️ Common Mistake
When resolving a force, horizontal component uses COS and vertical uses SIN (for angle measured from horizontal). If angle is measured from the VERTICAL, swap sin and cos. Always check with Pythagoras: Fx² + Fy² should equal F².
📐 Variables
FxHorizontal component (Fx) is measured in newtons (N)
FyVertical component (Fy) is measured in newtons (N)
FResultant force (F) is measured in newtons (N)
θAngle to horizontal (θ) is measured in degrees (°)
📐 Key Equations
Fx = F cos θ
Fy = F sin θ
R = √(Fx² + Fy²)
📌 Key Note
Resolve F at angle θ: Fx = F cosθ, Fy = F sinθ. Find resultant: component method R = √(Fx²+Fy²), or scale drawing tip-to-tail. Equilibrium: three forces form closed triangle. Parallelogram of forces: diagonal = resultant.
🎯 Matching Activity — Vector Components
Match each vector quantity to the correct component formula. — drag the symbols on the right to match the component names on the left.
Horizontal component
Drop here
Vertical component
Drop here
Resultant magnitude
Drop here
Three forces in equilibrium
Drop here
R = √(Fx² + Fy²) — Pythagoras from components
Fx = F cos θ (θ measured from horizontal)
Fy = F sin θ (θ measured from horizontal)
Form a closed triangle when drawn tip-to-tail
⚽ FIFA Worked Examples
Resultant of Two Forces
Forces of 3 N east and 4 N north act on an object. Find the resultant.
F
R = √(Fx² + Fy²); θ = arctan(Fy/Fx)
I
Fx = 3 N (east), Fy = 4 N (north)
F
R = √(3² + 4²) = √(9+16) = √25 = 5 N; θ = arctan(4/3) = 53° north of east
A
Resultant = 5 N at 53° north of east
⭐ Higher Tier Only
HT only — resolve forces into components using trigonometry. Use vector diagrams (scale drawing, tip-to-tail, parallelogram) to determine resultants and solve equilibrium problems.
🔬 Triple Science Only
Resolving forces and vector diagrams (HT only) — part of the physics-only extended forces content.
🎯 Test Yourself
Question 1 of 2
1. A 13 N force acts at 67.4° to the horizontal. What are the horizontal and vertical components? (sin67.4° = 0.923, cos67.4° = 0.385)
2. Three forces acting on an object form a closed triangle when drawn tip-to-tail. What does this tell us?
⭐ How Well Do You Understand This Topic?
Be honest with yourself — this helps you know what to revise!
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