# Resultant Force

## Key Stage 3

### Meaning

The Resultant Force is the overall force on an object.

A resultant force can be calculated by taking forces that act along the same line and add them together.
If the forces are in the same direction as each other, then they have a positive value.
If one force is acting in the opposite direction to another one of the forces is given a negative value.
 If the 'up' direction is chosen to be positive then any vector pointing 'up' has a positive value and any vector pointing 'down' has a negative value. In this diagram the green vector has a value of +2 while the red vector has a value of -4. If the number lines represent the number of Newtons force then the Resultant Force is the two values added together. FR = (+2N) + (-4N) FR = -2N upwards
 If the 'right' direction is chosen to be positive then any vector pointing 'right' has a positive value and any vector pointing 'left' has a negative value. In this diagram the blue vector has a value of +3 while the yellow vector has a value of -2. If the number lines represent the number of Newtons force then the Resultant Force is the two values added together. FR = (+3N) + (-2N) FR = 1N to the right

### Examples

 Choose which directions are positive. Up is positive. Calculate the Resultant Force. There is 4N up and 7N down. FR = (+4N) + (-7N) FR = -3N up Choose which directions are positive. Right is positive. Calculate the Resultant Force. There is 4N left and 3N right. FR = (+3N) + (-4N) FR = -1N right Choose which directions are positive. Up is positive. Right is positive. Calculate the Resultant Force. There is 6N up and 6N down. FVertical = (+6N) + (-6N) FVertical = 0N up There is 6N left and 4N right. FHorizontal = (+4N) + (-6N) FHorizontal = -2N right

## Key Stage 4

### Meaning

The resultant force is the sum of all forces acting on an object.

When forces act in the same direction their magnitudes are added together.
When forces act along the same line but in opposite directions; one is subtracted from the other.
When forces are at right angles to one another they can be added using Pythagoras' Theorem to find the magnitude of the resultant force and then trigonometry can be used to find the direction of the resultant force.

### Examples

 Choose which directions are positive. Up is positive. Calculate the Resultant Force There is 29N up and 37N down. FR = (+29N) + (-37N) FR = -8N upwards

 Choose which directions are positive. Right is positive. Calculate the Resultant Force There is 114N right and 105N left. FR = (+114N) + (-105N) FR = 9N right
 Choose which directions are positive. Right is positive. Up is positive. Calculate the Resultant Force There is 38N up and 38N down. FR = (+38N) + (-38N) FR = 0N upwards Calculate the Resultant Force There is 50N right and 40N left. FR = (+50N) + (-40N) FR = 10N right
 Choose which directions are positive. Right is positive. Up is positive. Calculate the Resultant Force There is 22N up and 16N down. FVertical = (+22N) + (-16N) FVertical = 6N upwards Calculate the Resultant Force There is 26N right and 18N left. FHorizontal = (+26N) + (-18N) FHorizontal = 8N right Since there are two resultant forces that are perpendicular they form a right angle triangle and their hypotenuse is the magnitude of the resultant force. This can be found using Pythagoras' Theorem. $${F_R}^2 = {F_V}^2 + {F_H}^2$$ $$F_R = \sqrt{6^2 + 8^2}$$ $$F_R = \sqrt{100}$$ $$F_R = 10N$$ The angle from the horizontal can be found using Trigonometry. $$\tan \theta = \frac{Opposite}{Adjacent}$$ $$\tan \theta = \frac{6}{8}$$ $$\theta = 36.9^\circ$$ up from horizontal

### References

#### AQA

Resultant force, pages 155-8, GCSE Physics; Student Book, Collins, AQA
Resultant force, pages 205, 212, GCSE Combined Science; The Revision Guide, CGP, AQA
Resultant force, pages 53, 54, 64, 71, GCSE Physics; The Revision Guide, CGP, AQA
Resultant forces, page 131, 132, 162-165, GCSE Combined Science Trilogy; Physics, CGP, AQA
Resultant forces, pages 118-119, 144-145, 148-149, 168-169, GCSE Physics; Third Edition, Oxford University Press, AQA
Resultant forces, pages 120-1, GCSE Physics, Hodder, AQA
Resultant forces, pages 151, 152, 194-197, GCSE Physics; The Complete 9-1 Course for AQA, CGP, AQA
Resultant forces, pages 210-11, GCSE Combined Science Trilogy 2, Hodder, AQA

#### Edexcel

Resultant forces, page 209, GCSE Physics, CGP, Edexcel
Resultant forces, pages 12-13, 134, GCSE Physics, Pearson Edexcel
Resultant forces, pages 181, 182, GCSE Combined Science; The Revision Guide, CGP, Edexcel
Resultant forces, pages 67, 68, GCSE Physics; The Revision Guide, CGP, Edexcel

#### OCR

Resultant forces, pages 163, 164, Gateway GCSE Combined Science; The Revision Guide, CGP, OCR
Resultant forces, pages 26-28, Gateway GCSE Physics; The Revision Guide, CGP, OCR
Resultant forces, pages 65-68, Gateway GCSE Physics, Oxford, OCR