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P8.5 The Parallelogram of Forces

GCSE Combined Science · AQA · Chapter P8: Forces in Balance

A free GCSE Combined Science: Trilogy lesson for AQA (8464) 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 a scale drawing is needed to find a resultant force
  • (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) check a scale drawing for two forces at right angles using Pythagoras’ theorem

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 60 N and 40 N, the resultant lies between 20 N and 100 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 6.0 cm long; force B is 4.0 cm long at 50° to A.
The diagonal measures 9.1 cm, so the resultant = 9.1 × 10 = 91 N.
The protractor gives the angle between the resultant and A as about 20°.

Answer: 91 N at 20° to the 60 N force

Forces at right angles

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

Worked example: the drone

Draw 8 N north and 15 N east from the same point (e.g. 1 cm ≡ 1 N) and complete the rectangle.
Diagonal = 17 cm, so resultant = 17 N. Pythagoras: √(8² + 15²) = √289 = 17 N.
Angle measured from north: about 62° towards the east.

Answer: 17 N at 62° 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 canal boat pulled by two ropes, each with a tension of 750 N and 50° apart, feels a forward resultant of about 1360 N – less than 1500 N, because the sideways parts of the two pulls cancel.

As the angle between two equal forces increases from 0° to 180°, the resultant falls from twice one force to zero. At 120°, the resultant of two equal forces is equal to each of them.

Worked example: two equal forces at 120°

Two forces of 30 N act on a hook with an angle of 120° between them. Scale 1 cm ≡ 5 N: each arrow is 6.0 cm long.
Completing the rhombus and measuring the diagonal gives 6.0 cm, so the resultant = 6.0 × 5 = 30 N.
The resultant points along the line halfway between the two forces (60° from each).

Answer: 30 N, bisecting the angle between the forces

Common mistakes

  • Adding the two forces arithmetically when they act at an angle (e.g. saying 60 N + 40 N = 100 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 9.0 N and 12 N act on a point at right angles to each other. What is the magnitude of the resultant?

Answer: 15 N

Working: Resultant = √(9.0² + 12²) = √225 = 15 N.

Question 2. (Higher tier) The diagram shows a completed parallelogram of forces for forces P and Q acting at point O. Which line represents the resultant? (Choose from: P, Q, R, S.)

Answer: R

Working: R is the diagonal that starts at O, where both forces act, so R is the resultant. S is the other diagonal and does not start at O.

Question 3. (Higher tier) Two ropes pull a post, each with a force of 400 N. The angle between the ropes is 60°. Use a scale drawing to find the resultant force.

Answer: 693 N

Working: Scale 1 cm ≡ 50 N: two 8.0 cm sides at 60°. The diagonal measures about 13.9 cm, so the resultant ≈ 690 N, halfway between the ropes.

Related lessons

Continue in the full course

This lesson is section P8.5 (AGCSC3705) of our free GCSE Combined Science 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 Combined Science course.

 

 

 

 

 

 

 

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