## Parallelogram Rule Of Forces (Engineering Science)

The parallelogram rule of forces is used to a method used to find the resultant of 2 forces acting coplanar forces forces acting at an angle to each other. This law takes takes into account both the magnitude and direction of a force. The most common methods for finding the resultant of forces include:

- parallelogram rule of forces
- triangle rule of forces
- polygon rule of forces
- resolution of forces

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## Parallelogram rule of forces

The parallelogram rule of forces states that, "If two forces, acting simultaneously on a particle, be represented in magnitude and direction by the two adjacent sides of a parallelogram then their resultant may be represented in magnitude and direction by the diagonal of the parallelogram which passes through their point of intersection."

For example, let us say we have 2 forces OA and OB acting at a point O but in different directions as shown in the diagram below:

To find the resultant, we complete the force diagram into a parallelogram in which the two given forces are adjacent sides. Then the diagonal that passes through the forces is the resultant force, as shown in the diagram below:

### Worked Example

##### The angle between the two forces of magnitude 20 N and 15 N is 60°. If the 20 N force is horizontal, determine the magnitude and direction of the resultant force.

Here are the stage you should follow to find the resultant:

- Sketch the diagram as described by the question.

- Close the diagram into a parallelogram to show the resultant, R.

- Since opposite sides of a parallelogram are parallel and equal, they are equal vectors. This means that it you extend the base of the parallelogram, you get 2 corresponding angles that are also equal. In this case, the corresponding angles are both 60°, as shown in the diagram below:

- Angles on a straight line add up to 180, therefore the inside angle of the 60° angle is 120°.

- To unclutter the diagram, you can then focus your attention on the triangle that has two sides and an included angle to find the resultant by cosine rule. So, let us remove the unnecessary parts from the diagram.

From that diagram we can now find the magnitude of the resultant force, R, and its direction to the horizontal, θ.

**Using cosine rule to find the magnitude of the resultant:**

R^{2} = 15^{2} + 20^{2} – 2(15)(20)cos 120°

R = 30.4 N

**Using sine rule to find the direction of the resultant:**

$latex \frac{\sin \theta}{15} = \frac{\sin 120}{30.4}$

$latex \sin \theta = \frac{15\sin 120}{30.4}$

θ = 25.3 °

**Therefore, the resultant is 30.4 N at 25.3 ° to the horizontal.**

## Final word

When finding the resultant of 2 forces using parallelogram rule, you:

- create a parallelogram with the 2 forces as the adjacent sides.
- use cosine rule to calculate the resultant, which is the diagonal through the two forces.
- use sine rule to calculate the direction of the resultant, which is the angle between the resultant and the horizontal force.