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8.8 The Parallelogram of Forces

GCSE Physics · AQA · Chapter 8: Forces in Balance

A free GCSE Physics lesson for AQA (8463) from OBA School: (Higher tier) Finding the resultant of two forces that do not act along the same line, using scale drawings. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with every video, practice quizzes and printable worksheets with mark schemes, in our free online course.

What you will learn

  • (Higher tier) explain when and why the parallelogram of forces is used
  • (Higher tier) list what you need to draw an accurate scale drawing of two force vectors
  • (Higher tier) construct a parallelogram of forces and measure the magnitude and direction of the resultant
  • (Higher tier) describe an experiment with pulleys and weights that tests the parallelogram of forces

Key words

  • Resultant: The single force that has the same effect as two or more forces acting together.
  • Parallelogram of forces: A scale drawing in which two forces are drawn as adjacent sides of a parallelogram; the diagonal from their common starting point is the resultant.
  • Scale: The force represented by each unit of length, e.g. 1 cm ≡ 10 N.
  • Adjacent sides: Two sides of a shape that meet at a corner.
  • Protractor: An instrument for measuring and drawing angles.

Forces that do not act along the same line

Forces along one line can be added or subtracted. But when two forces act at an angle to each other – two ropes pulling a boat, wind pushing a flying drone sideways – you cannot simply add them. Because forces are vectors, the direction matters. Instead you combine them with a scale drawing called the parallelogram of forces.

Key fact: The resultant of two forces at an angle is always smaller than their sum and bigger than their difference. For 50 N and 30 N, the resultant lies between 20 N and 80 N depending on the angle.

Drawing the parallelogram of forces

You need a sharp pencil, a ruler, a protractor and a sheet of plain or graph paper.

1. Choose a scale that makes the drawing large, e.g. 1 cm ≡ 10 N. Write it down.

2. From a point O, draw the first force as an arrow of the correct length and direction.

3. From the same point O, use the protractor to draw the second force at the correct angle to the first.

4. Complete the parallelogram: draw a line parallel to each force through the tip of the other.

5. Draw the diagonal from O to the opposite corner. This is the resultant.

6. Measure the diagonal and convert using the scale; measure its angle with the protractor.

Worked example: the forces in the figure

At 1 cm ≡ 10 N: force A is 5.0 cm long; force B is 3.0 cm long at 60° to A.
The diagonal measures 7.0 cm, so the resultant = 7.0 × 10 = 70 N.
The protractor gives the angle between the resultant and A as about 22°.

Answer: 70 N at 22° to the 50 N force

Forces at right angles

When the two forces are perpendicular, the parallelogram is a rectangle. You can still use a scale drawing, but you can check the size of the resultant with Pythagoras’ theorem.

Worked example: the drone

Draw 12 N north and 5 N east from the same point (e.g. 1 cm ≡ 1 N) and complete the rectangle.
Diagonal = 13 cm, so resultant = 13 N. Pythagoras: √(12² + 5²) = √169 = 13 N.
Angle measured from north: about 23° towards the east.

Answer: 13 N at 23° east of north

Two equal forces

If the two forces are equal, the parallelogram is a rhombus and the resultant lies exactly halfway between them. A boat pulled by two ropes, each with a tension of 600 N and 40° apart, feels a forward resultant of about 1130 N – less than 1200 N, because the sideways parts of the two pulls cancel.

Testing the parallelogram of forces

Three strings are knotted together. Two pass over pulleys and carry weights W₁ and W₂; the third hangs straight down with weight W₃. The tension in each string equals the weight it supports. When the knot is at rest, the resultant of the two slanting tensions must be equal and opposite to W₃.

Draw a vertical line, then draw W₁ and W₂ to scale at angles θ₁ and θ₂ on either side of it and complete the parallelogram. The diagonal should be vertical and represent 5.0 N, matching W₃. Friction in the pulleys and errors in reading the angles cause small differences.

Common mistakes

  • Adding the two forces arithmetically when they act at an angle (e.g. saying 50 N + 30 N = 80 N).
  • Drawing the resultant as the other diagonal. The resultant always starts from the point where both forces act.
  • Forgetting to state the scale, or giving the resultant without its direction.
  • Using a small scale. A large drawing reduces the effect of measuring errors.

Revision questions

Try these, then check your answers below each one.

Question 1. (Higher tier) Forces of 6.0 N and 8.0 N act on a point at right angles to each other. What is the magnitude of the resultant?

Answer: 10 N

Working: √(6.0^2 + 8.0^2) = √100 = 10 N.

Question 2. (Higher tier) In a parallelogram of forces, which line represents the resultant? (Choose from: the diagonal that starts where the two forces act, the other diagonal, the longer of the two sides, a line at right angles to both forces.)

Answer: the diagonal that starts where the two forces act

Working: The resultant is the diagonal drawn from the common starting point of the two forces to the opposite corner.

Question 3. (Higher tier) Two ropes pull a post with forces of 500 N each. The angle between the ropes is 60°. Use a scale drawing to find the resultant force.

Answer: 866 N

Working: Scale drawing (e.g. 1 cm ≡ 100 N): diagonal ≈ 8.7 cm, so the resultant ≈ 870 N, along the line halfway between the ropes. (Exact value 866 N.)

Related lessons

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This lesson is section 8.8 (AGCPH0808) of our free GCSE Physics course for AQA. In the full course you get every video for this section, a self-marking practice quiz, a printable worksheet with a mark scheme, chapter tests, and your progress saved. Go to the full GCSE Physics course.

 

 

 

 

 

 

 

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