## What is Resolution of Forces?

It is basically the procedure in which the given amount of force is split into various numbers of components; it is done in the way that there are no changes in the effect on the body. It is generally done in the two mutually perpendicular directions.

It can also be classified as the breaking a single vector into multiple ones. This can be seen with the help of example:

In the given figure the gardener is using the grass cutting machine for cutting the grass and is applying the force from the inclined angle so that the machine will move in the further direction and then the force is applied by the machine blades on the grass as well.

One more example is of the cart which is moved from one location to the other.A cart is been moved when the force is applied to drag it from one place to another. During this the force is applied in the inclined direction and the forces are distributed in two forms or components known as horizontal or the vertical components i.e.,${F}_{x}$and ${F}_{y}$respectively. In this figure, the force component which is responsible for moving the cart from one place to another is horizontal component of applied force i.e., in ${F}_{x}$ direction. And if the angle of inclination increases the size of the vertical component also increases and the horizontal component decreases.

## Resolution of a Force into Components

Any force F, which is applied on a body can be replaced into two or more forces. Both these forces which have been replaced in place of the singe force will cause the same effect on the system as that is produced by the force F. This is the resolution of force into the components, where a single force is divided into two or more components without causing any change in the effect of the system. The characteristic of the resolving force is that it can be resolved into two or more components which can be either perpendicular with each other or they can be inclined to each other. When the two components are in perpendicular direction to each other, then such type of forces are said to have the rectangular components. And when the two components are in inclined direction with each other, then such type of forces are said to have inclined components.

## Principle of Resolution

It tells how the resolution of forces takes place for a certain body. For any kind of body, the total sum of all the resolving parts of the number of forces, in the given direction is equal to the resultant of their resolved part in the same direction as the applied force.

## Method of Resolution for the Resultant Force

First of all, all the forces are to be resolved in the horizontal form and the total sum for the horizontal components is calculated. Then the forces are resolved in the vertical form and the total sum for the vertical components is calculated. Then the resultant of the resolution of horizontal as well as the vertical component is given by the equation i.e.

$$R=\sqrt{{\left({\displaystyle \sum H}\right)}^{2}}+\sqrt{{\left({\displaystyle \sum V}\right)}^{2}}$$The resultant force will be given by some angle with respect to the horizontal as,

$$\mathrm{tan}\theta =\frac{{\displaystyle \sum V}}{{\displaystyle \sum H}}$$The values of$\sum V$ and $\sum H$ will be responsible for the change in value of the angle.When the value of $\sum V$is positive, the resultant will make an angle in between 0 to 180 degrees. But in case of negative value of$\sum V$, the resultant will make an angle in between 180 to 360 degrees.

On the other hand, when the value of $\sum H$ is positive, the resultant will make an angle in between 0 to 90 degrees or it can be 270 to 360 degrees. But in case of negative value of$\sum H$, the resultant will make an angle in between 90 to 270 degrees.

## Resolution of Forces into the Rectangular Components

A force on a body or a system is considered to be resolved into an infinite number of components which can be done by Parallelogram method, but when the forces are resolved into two components that are parallel to both the X and Y axes, then such type of components are known as rectangular components.

This can be considered as any force F which is acting on any particle O at any angle $\theta $, two axes namely X and Y are passing through the origin and are perpendicular to each other, then these two axes are called as rectangular axes, which can be either horizontal or vertical and can be inclined as well.

The force R can be resolved from the given figure into two different components ${R}_{x}$and Ry along both the axes i.e., X and Y respectively. These ${R}_{x}$ and${R}_{y}$are known as the rectangular components. With the help of these components, we will be constructing the adjacent sides so that it will work as a polygon, the constructed polygon will form the rectangle OABC.

Now taking the right-angled triangle i.e., OAB and applying the function of trigonometry we get;

$$\mathrm{cos}\theta =\frac{OA}{OC}$$Therefore,

$$\begin{array}{l}OA=OC\left(\mathrm{cos}\theta \right)\\ {R}_{x}=OA=R\mathrm{cos}\theta \\ \mathrm{sin}\theta =\frac{AC}{OC}\end{array}$$Therefore,

$$\begin{array}{l}AC=OC\left(\mathrm{sin}\theta \right)\\ {R}_{y}=AC=OB=R\mathrm{sin}\theta \end{array}$$Therefore, we got the two rectangular components of the force R, i.e.

${R}_{x}=R\mathrm{cos}\theta $ and ${R}_{y}=R\mathrm{sin}\theta $

Sign conventions also play a major role in resolution of forces. While resolving the forces, the sign conventions are also recommended to be checked. As the conventional coordinates’ directions are used for knowing the sign convention of the components of force, when the direction of component is similar to the direction of coordinates i.e., forward or in upward direction, the signs are to be taken as positive, while when the direction of the component is opposite to the direction of coordinates i.e., backward or in downward direction, then in such cases the signs are taken as negative.

## Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

- Bachelors in Science (Physics)
- Masters in Science (Physics)
- Bachelors in Science (Mathematics)
- Masters in Science (Mathematics)

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