# Forces

Force is a vector quantity. This means it has a magnitude (size) and a direction. Force is measured in Newtons (N). Figure 1 shows a 7 N force acting at 45° above the horizontal.

Figure 1

To solve questions in M1 that involve forces you will usually have to find components of forces or find resultant forces. Let’s start with finding components.

### Components of forces

Finding components is a case of using simple trigonometry to find the vertical and horizontal distances covered by a diagonal line. This will also tell you how much of a force is acting vertically and how much horizontally.

In Figure 1 part of the force is acting vertically, so start by resolving upwards:

$\begin{array}{rl} R(\uparrow) & 7 \sin(45\textdegree)\\= & \dfrac{7 \sqrt{2}}{2}\\= & 4.95\end{array}$

There is also a component of the force acting horizontally, so resolving to the right:

$\begin{array}{rl} R(\rightarrow) & 7 \cos(45\textdegree)\\= & \dfrac{7 \sqrt{2}}{2}\\= & 4.95\end{array}$

In this example, because of the 45° angle, the horizontal component and the vertical component are equal. Notice that when resolving through the angle you should use the cosine, and resolving away from the angle use the sine of the angle.

The same approach can be used to resolve parallel and perpendicular to a surface. Do this by using the angle between the surface and the force you need to split into components.

### Resultants of forces

The resultant of two or more forces is the single force that is equivalent to the combined forces. This can be most easily represented by a force triangle.

Figure 2

Figure 2 shows two forces, one of 7N acting at 45° above the horizontal and one of 3N acting horizontally, and the resultant force which is the single force equivalent to the two combined forces. Putting the forces end to end gives two sides of the triangle and the resultant is the third side of the triangle.

You will be required to find the magnitude of the force (the length of the side if drawn to scale) and the angle at which it acts. This can be done using the cosine rule but it is more normal to find the horizontal and vertical components of each force, add them, and then use Pythagoras to find the magnitude and the arc tangent to find the angle.

Resolve horizontally for 7N force:
$7 \cos(45\textdegree)\\= \dfrac{7}{\sqrt{2}}\\= \dfrac{7\sqrt{2}}{2}$
Resolve vertically for 7N force:
$7 \sin(45\textdegree)\\= \dfrac{7}{\sqrt{2}}\\= \dfrac{7\sqrt{2}}{2}$
The 3N force has only a horizontal component and no vertical component. Use Pythagoras to find the magnitude of the resultant:
$\begin{array}{lcl}R^2 & = & \left(\dfrac{7\sqrt{2}}{2} + 3\right)^2 + \left(\dfrac{7\sqrt{2}}{2}\right)^2\\R^2 & = & 87.69848481\textellipsis\\R & = & \sqrt{87.69848481\textellipsis}\\R & = & 9.36\end{array}$
So the resultant force has a magnitude of 9.36 N. Now work out the arc tangent of the components to find the direction in which the resultant acts:
$\begin{array}{lcl}\theta & = & \tan^{-1}\left({\dfrac{\frac{7\sqrt{2}}{2}}{\frac{7\sqrt{2}}{2}+3}}\right)\\\theta & = & \tan^{-1}(0.622629522\textellipsis)\\\theta & = & 31.9\textdegree \end{array}$
The direction of the resultant is 31.9° above the horizontal.

The single force that is equivalent to the combined forces above therefore has magnitude 9.36 N acting at 31.9° above the horizontal