What is Resolution of Forces?

What is the Resolution of Forces?

Resolution of forces is essentially the technique by which a given quantity of force is divided into a variable number of components; it is done in such a way that the effect on the body remains constant. In general, it is done in two mutually perpendicular directions.

Resolution of Forces into Components

The process of splitting up a given force into different components without changing its effect on the body is called the resolution of a force. If a force 'F' is resolved or replaced by two forces using the principle of resolution of forces, it will together produce the same effect as that of force 'F'. These forces are known as the components of the force 'F'.

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Resolution of Forces into Rectangular Components

The rectangular component of any force is two components such that they are perpendicular to each other, and the resultant of the two components will give the resultant force. Let us consider a force 'F' inclined at an angle 'θ' to the horizontal as shown in the figure below. X and Y are two axes passing through 'O' and perpendicular to each other.

Applying the principle of resolution of forces, resolve the force 'F' into Fx and Fy. The polygon constructed with these two components as adjacent sides will form a rectangle OABC, therefore, the components are known as rectangular components. For the sign convention of components of forces, conventional co-ordinate directions are considered.

X-component (F_x)→Positive

←Negative

Y-component (F_y) ↑ Positive ↓ Negative

Example:

Determine the X and Y components of the force shown in the figure.

Solution:

Consider the triangle OAB,

cos θ = OA/OB

OA = OB cosθ

OA = F_x

OB = F

F_x = F cosθ

Sin θ = AB/OB

AB = OB sinθ

AB = F_y

OB = F

F_y = F sinθ

The two rectangular components of the force 'F' after resolution of force are:

Fx = F cosθ

Fy = F sinθ

Applying the principle of the resolution,

1. Component along X-direction

F_x = 20 cos30°

F_x = 17.32N

2. Component along Y-direction

F_y = 20 sin30º

F_y = 10N

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Triangular Parallelogram Law of Forces

In the Triangular Law of Forces, If two forces acting simultaneously on a body are represented in magnitude and direction by two sides of a triangle in order, then the third side represents the resultant of the two forces in the opposite direction and magnitude.

The resultant of two concurrent and coplanar forces, i.e, forces lying in the same plane whose line of action passes through a single point may be obtained using the theorem of the parallelogram of forces which states that "If two forces acting at a point are represented by the sides of a parallelogram drawn from the point in magnitude and direction, their resultant force is represented by the diagonal of the parallelogram drawn through that point in magnitude and direction."

Resolution of Forces MCQs

1. A force of 50N is acting at a point making an angle 35^o with the horizontal. Determine the components of this force in X and Y directions.

Component along X-direction

F_x = 50 cos 95^o = 40.95N

Component along Y-direction

F_y = 50 sin 35^o = 28.68N

2. Find the resultant for the given force system using the principle of superposition of forces. 

∑F_x = 0

30 + 15 cos 60^o – 25 cos 30^o = 15.85N

∑F_y = 0

50 + 15 sin 60^o + 25 sin 30^o = 75.5N

Resultant Force (R) = √(F_x²+ F_y²)

= 15.852+ 75.52

= 77.15N

3. Resolve the 400 N force acting on a block as shown into horizontal and vertical components.

Angle of inclination of 400 kN force with respect to horizontal x-axis = 40^o-20^o = 20^o

Component along Horizontal axis = 400 cos 20^o = 375.35N

Component along Vertical axis = -400 sin 20^o = -136.8N