## Addition of Forces

Since force is a vector quantity, forces can be represented by an arrow diagram.

- The magnitude of the force is represented by the length of the arrow
- The direction of the force is represented by the direction in which the arrow is pointed.

### Resultant Force

**Resultant force** is the combination of 2 or more forces.

- The effect on a body produced by 2 or more forces acting on it will be the
**same**as that produced by their resultant force. - Hence, resultant force is used to simplify force diagrams – it is easier to deal with one resultant force than multiple forces.

### Adding Forces in Same Direction

The above figure shows two forces – 10 N and 20 N acting on a car. The resultant force will be 30 N to the right, which is obtained by adding the two forces numerically.

In the general case, where there is $F_{1}$ to $F_{n}$ acting on an object in the same direction,

$$F_{\text{resultant}} = F_{1} + F_{2} + … + F_{n}$$

### Adding Forces in Opposite Direction

When the applied forces are in the opposite direction, the resultant force is dependent on the magnitude of the forces. Using the first car in the above figure, there are two forces 20 N and 40 N acting on it. The resultant force will be 20 N as $F_{\text{resultant}} = 40 \, – 20 = 20 \, \text{N}$ to the right.

When the two forces are same in magnitude but different in direction, the resultant force will be 0 (as seen above)

### Slightly more advanced trick:

Taking rightward as positive, the forces acting on the car will be -20 N (Negative because the force is to the left) and + 40 N.

Using the addition of forces, $F_{\text{resultant}} = (\, – 20) + 40 = + 20 \, \text{N}$, which is 20 N to the right.

## Worked Example

### Three forces of 3 N, 1.5 N and 2 N are acting on an object, as shown in the picture below.

### What is the resultant due to the three forces?

**Show/Hide answer**

From the diagram above and taking rightward as positive, the resultant force is given by:

$$\begin{aligned} F_{\text{resultant}} &=3-1.5-2 \\ &=-0.5 \text{ N} \end{aligned}$$

## Forces at an angle to each other (Drawing force diagrams)

This will be more complicated than the previous two cases. We can use the parallelogram law of vector addition to find the resultant force.

Consider the above diagram, we are given two forces – A and B. We will shift B to match up with A as seen below. Ensure that force A and B are drawn to scale (e.g. 1 cm to 5 N)

After shifting B, we draw two more lines to make it a parallelogram. The resultant force will just be the red line in the diagram below. You can measure the diagonal (A + B line) to find the resultant force.

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