Scalars and vectors - Revision Guide

Three forces in equilibrium forming a closed triangle

In exam questions, you can prove three forces are in equilibrium (or find a missing force) in two ways:

By resolving: Resolve all forces horizontally and vertically. The sum of the upward forces must equal the downward forces, and the sum of leftward forces must equal rightward forces.
By drawing a closed triangle: Draw a scale diagram of the forces tip-to-tail and verify that they form a closed loop.

Example

Finding an unknown force in equilibrium

An object is hanging from a ceiling, suspended by two strings. The tension in string A is $\text{ N}$ pulling at an angle of $40∘40^\circ$ to the vertical. The string B pulls completely horizontally. The object is stationary. Calculate the weight of the object.

Recognise the state: The object is stationary, meaning it is in equilibrium. The sum of all vertical forces is zero, and the sum of all horizontal forces is zero.
Identify vertical forces: The only force pulling downwards is the weight, $W$ . The only force with an upward component is the tension in string A. String B is perfectly horizontal, so it has no vertical component.
Resolve the upward force: The vertical component of tension A is adjacent to the $40∘40^\circ$ angle (from the vertical).

\text{Upward component} = 15 \times \cos(40^\circ)

Equate upward and downward forces:

\begin{aligned} W &= 15 \times \cos(40^\circ) \\ W &= 15 \times 0.766 \\ W &= 11.5 \text{ N} \end{aligned}

Exam technique

In the exam

Watch out for "constant velocity" / "terminal velocity". Examiners love stating a car is moving at constant velocity, and then asking for the resultant force. Don't waste time calculating – if velocity is constant, the resultant force is $\text{ N}$ .
Identify the angle before you resolve. Never guess whether to use sine or cosine. Quickly draw the right-angled triangle. Ask yourself: "Is the component I want opposite or adjacent to the angle I have?"
Check your scale diagram arrows. When drawing closed triangles for equilibrium, make sure the arrows actually follow each other around the loop tip-to-tail. If two arrows meet tip-to-tip, your triangle is wrong!

Self review

Check yourself

Can you name three scalar quantities and three vector quantities?
If an object is moving at $\text{ m s}^{-1}$ in a straight line with no acceleration, what can you say about the coplanar forces acting on it?
When resolving the weight of an object on a ramp, which trigonometric function gives the component parallel to the slope: sine or cosine?
How do you graphically arrange two vectors to find their resultant?