### Resolution of Forces

• Resolve forces in horizontal/vertical as well as parallel/perpendicular to a plane.

• Resolution of force means separating a force, into two mutually perpendicular components. These components are normally horizontal and vertical.
• Occasionally, a force is divided into two forces parallel and perpendicular to a plane.
• Resolution of force is done to simplify the process of adding or subtracting multiple forces.
• Typically when you want to add or subtract multiple forces, which are non-collinear, they are broken down into horizontal and vertical forces.
• All horizontal forces are added, similarly all vertical forces are added.
• Finally the resultant is found out from these two components.

### Solved Example:

#### 22-1-01

The method of splitting a single force into two perpendicular components along x-axis and y-axis is called as:

Solution:

Separating a force into two forces for the purpose of addition or subtraction is called resolution.
Resultant of a force is exactly opposite process where two component forces are combined together.
concurrent forces are defined as forces that pass through a common point.

### Solved Example:

#### 22-1-02

What are the X and Y components of the force system shown below?

Solution:
$\theta = \tan^{-1}\left( \dfrac{2}{5}\right) = 21.8~^\circ$
Horizontal Component, $F_x = F \cos \theta = 500 \times \cos 21.8^\circ = 464.23\ N$
Vertical Component, $F_y = F \sin \theta = 500 \times \sin 21.8^\circ = 185.68~N$

### Solved Example:

#### 22-1-03

A force of 10 N is making an angle of 30$^\circ$ with the horizontal. Its horizontal component will be:

Solution:
Horizontal Component, $F_x = F \cos \theta = 10 \times \cos 30^\circ = 5\ N$

### Solved Example:

#### 22-1-04

A man is pulling a trolley with 100 N weight on a horizontal road with a force of 100 N making 45$^\circ$ with the road. The horizontal and vertical components of the net force on trolley will be:

Solution:
Horizontal Component, $F_x = F \cos \theta = 100 \times \cos 45^\circ = 70\ N$
Net Vertical Component, \begin{align*} F_y &= W - F\sin \theta\\ &= 100 - F \sin \theta\\ &= 100 - 100 \times \sin 45^\circ\\ &= 29.23\ N \end{align*}

x component : $v \cos \theta = 110$ $v = \dfrac{110}{\cos 30} = 127 km/hr$