Resultant Forces (Oxford AQA IGCSE Combined Science Double Award)

Revision Note

Resultant Forces

What is a resultant force?

  • When a number of forces are acting on a single object, the forces can be replaced by a single force that has the same effect on the motion of the object as all the original forces acting together

  • This single force is called the resultant force

  • The resultant force determines:

    • The direction in which the object will move as a result of all of the forces

    • The magnitude of the final force experienced by the object

What is a Free-Body force diagram?

  • Free-body force diagrams are vector diagrams showing the multiple forces acting on a single object

  • The vectors can be drawn onto a picture of the object, or the object can be represented by a dot

Free-Body force diagram of a car

Free-body force diagram of a car showing the forward acting thrust force of the engine on the car, the backwards acting frictional force of the road on the car (tyres), the upward acting normal contact force of the  road on the car, and the downward acting force of weight (gravitational pull of the Earth) on the car
Free-body force diagrams show multiple forces acting on a single object
  • The vertical forces add together

    • In the example of the car, the normal contact force and the weight are equal in magnitude and opposite in direction

    • Therefore, the vertical forces add up to zero

    • We say the forces are balanced

  • The horizontal forces add together

    • In the car example, the thrust force has a greater magnitude than the frictional force and acts in the opposite direction

    • Therefore, the net horizontal force is in the forward direction

  • Taking all the forces into account, the resultant force is in the forward direction

  • A non-zero resultant force will cause a change in the object's motion

  • A change in the object's motion is an acceleration

    • In the example of the forward travelling car, the car will accelerate or speed up

Examiner Tips and Tricks

Force diagrams are any diagrams which show forces acting, free-body force diagrams specifically show the forces acting on a single object. You don't need to know this terminology for your exam, but you do need to recognise if the forces shown in a diagram are acting on a single object or on multiple objects.

Calculating resultant forces

  • Force is a vector quantity, it has both magnitude and direction

  • When adding forces together it is important to assign positive and negative values to show the direction in which the forces are acting

  • If a 5 N force acts to the right and a 5 N force acts to the left on an object, then we assign one of the values as positive and one as negative

  • So the resultant force acting on the object is

Resultant space force space equals space 5 space plus space open parentheses negative 5 close parentheses

Resultant space force space equals space 0 space straight N

  • The forces acting on the object are equal in magnitude and opposite in direction therefore they cancel one another out

  • This is like two people pushing a box with equal force from opposite sides, the box doesn't move

  • If two people push the box from the same side in the same direction, one with a 3 N force and one with a 7 N force, then the forces will add together and the box will move in the direction of the resultant force

Resultant space force space equals space 3 space plus space 7

Resultant space force space equals space 10 space straight N

  • If two people push the box in opposite directions, one with a 7 N force to the left (negative) and one with a 3 N force to the right (positive), then the forces will add together and the box will move in the direction of the resultant force

Resultant space force space equals space open parentheses negative 7 close parentheses space plus space 3

Resultant space force space equals space minus 4 space straight N

Zero and Non-Zero resultant forces

Three rectangles representing objects with two forces acting on each. Object 1 has a 5 N force to the right and a 5 N force to the left. Object 2 has a 3 N force and a 7 N force both acting to the right. Object 3 has a 3 N force acting to the right and a 7 N force acting to the left.
Zero and non-zero resultant forces acting on three objects

Worked Example

Calculate the magnitude and direction of the resultant force in the diagram below.

A rectangular block with a 14 N force acting to the left, a 4 N force acting to the right and an 8 N force acting to the right

Answer:

Step 1: Assign a direction to the forces

  • Forces acting to the right are positive

  • Forces acting to the left are negative

Step 2: Add together all the forces acting on the object

Resultant space force space equals space open parentheses negative 14 close parentheses space plus space 4 space plus space 8

Resultant space force space equals space open parentheses negative 14 close parentheses space plus space 12

Resultant space force space equals space minus 2 space straight N

Step 4: State the magnitude and direction of the resultant force

  • The resultant force is 2 N to the left

Examiner Tips and Tricks

Mathematically, it doesn't matter which direction you assign to be positive or negative, as long as you are consistent throughout your calculation.

Using scale diagrams to determine resultant force

  • There are two methods that can be used to combine vectors using a scale diagram: the triangle method and the parallelogram method

  • To combine vectors using the triangle method:

    • Step 1: link the vectors head-to-tail

    • Step 2: the resultant vector is formed by connecting the tail of the first vector to the head of the second vector

    • Step 3: The magnitude of the resultant vector can be found by measuring the length of the arrow

  • To combine vectors using the parallelogram method:

    • Step 1: link the vectors tail-to-tail

    • Step 2: complete the resulting parallelogram

    • Step 3: the resultant vector is the diagonal of the parallelogram

    • Step 4: The magnitude of the resultant vector can be found by measuring the length of the arrow

Triangle and Parallelogram method of combining vectors

In the triangle method, place the vectors head to tail and draw the missing side of the triangle as the resultant vector. For the parallelogram method, link the vectors tail to tail, complete the parallelogram, then draw the resultant vector as the diagonal of the parallelogram
Force vectors can be combined to determine the resultant force using the triangle method or the parallelogram method

Worked Example

Two ropes are used to pull a shopping trolley out of a ditch. The angle between the forces is 40°

A free-body force diagram with a dot representing the shopping trolley. A horizontal vector points to the right labelled 200 N. A second vector, which is half the length, points diagonally upwards to the right labelled 100 N. The angle between the vectors is 40 degrees.

Draw a vector diagram to determine the magnitude of the resultant force from the ropes on the shopping trolley.

Answer:

Step 1: Draw the known vectors on the scale diagram

The vectors have been drawn onto graph paper with a scale of 2 cm = 100 N
  • Select a suitable scale that allows room for the parallelogram to be constructed

  • The scale used here is 2 cm = 100 N

  • Draw the vectors tail to tail at an angle of 40°

Step 2: Construct the parallelogram

The parallelogram lines have been constructed at the same angles as the vectors
  • The lines should be the same length as the vectors and at the same angles

Step 3: Draw the resultant vector

The diagonal to the parallelogram has been drawn in. This is the resultant force.
  • Draw in the resultant vector as the diagonal of the parallelogram

Step 4: Determine the magnitude of the resultant force

  • Measure the length of the vector

The length of the resultant vector is measured to be 5.7 cm
  • Use the scale to work out the magnitude

    • The scale is 2 cm = 100 N

    • Therefore, 1 cm = 50 N

F space equals space 5.7 space cross times space 50

F space equals space 285 space straight N

  • An answer between 280 N and 290 N would gain the mark

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Leander Oates

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