Resolving and resultant forces: combining forces into a resultant, using scale vector diagrams, and resolving a single force into perpendicular components.

What this dot point is asking

Edexcel statement 9.3 wants you to use vector diagrams to illustrate the resolution of forces, the addition of forces to find a resultant, and the conditions for equilibrium. You should be able to combine forces along a line and at an angle, and resolve a single force into two perpendicular components.

The resultant force

Finding the resultant is the first step in almost any forces problem, because it is the resultant that goes into Newton's second law ($F = ma$). For forces acting along the same straight line, you simply add or subtract: forces in the same direction add, and forces in opposite directions subtract, with the resultant pointing the way of the larger force.

Forces at an angle: scale vector diagrams

The tip-to-tail (or parallelogram) method turns vector addition into a drawing. Choosing a sensible scale (for example $1\,\text{cm} = 1\,\text{N}$) and drawing accurately are the keys to marks. For two perpendicular forces, you can also calculate the size of the resultant using Pythagoras, $\sqrt{F_1^2 + F_2^2}$, which is quicker and exact.

Resolving a force into components

Resolving is useful when a force acts at an awkward angle, for example a sledge pulled by a rope at an angle to the ground. The pull can be resolved into a horizontal component (which moves the sledge along) and a vertical component (which tends to lift it). At GCSE you usually resolve using a scale diagram by drawing the perpendicular components that add tip-to-tail to give the original force.

How Edexcel examines this

This dot point is examined on both tiers. Forces along a line are a reliable two or three mark calculation, where the mark scheme rewards adding same-direction forces and subtracting opposite ones and stating the direction of the resultant; the classic error is adding forces that oppose each other. Forces at an angle are examined with a scale vector diagram: you are asked to draw the forces tip-to-tail to a stated scale, identify the resultant as the closing arrow, and measure its size and direction. For perpendicular forces, examiners accept (and often expect) Pythagoras for the size and a protractor measurement for the direction. Resolving a single force into perpendicular components appears as a diagram task, rewarding components that are at right angles and add back to the original. Accuracy in drawing, a clearly stated scale, and correct use of subtraction for opposing forces are the main mark-earners, so set diagrams out carefully and label the resultant.

Try this

Q1. Two forces of $9\,\text{N}$ and $4\,\text{N}$ act in opposite directions along a line. Calculate the resultant. [2 marks]

• Cue. $9 - 4 = 5\,\text{N}$ in the direction of the $9\,\text{N}$ force.

Q2. State what is meant by the resultant force. [1 mark]

• Cue. The single force with the same effect as all the forces combined.