COMPOSITION AND RESOLUTION OF FORCES FOR RESULTANT FORCE

Every object is said to be in a state. Basically there are two states, as Rest and Motion. In both the states, forces have the prime importance in keeping the object in a state, or changing the state of the object from rest to motion or vice-versa. So let’s study about force and methods of composition and resolution of forces for evaluating resultant force of number of forces acting simultaneously on a body



 

Force
“Force is something which changes or tends to change the state of a body from rest to motion or vice-versa.”
Unit of force is Newton, Denoted by capital letter ‘N’.


ILLUSTRATION
Suppose a body is resting on the floor and if we are pushing it horizontally with a force with specific magnitude. This horizontal force will tend to move the body and if the force is sufficient enough to move it, then it will cause a change in state of the body from rest to motion. The body will start moving in the direction of applied force. This is the change of state from rest to motion. Alternatively, if a body is moving and a pushing effort is applied on the body against the motion. The pushing force will retard the motion of the body and can stop it if strong enough. This is the change of state from motion to rest.



TERMINOLOGIES RELATED TO FORCE

Magnitude
Magnitude is the value i.e. 100N, 200N, 5KN. In terms of Unit.

Line of Action
Direction in which force is acting. Say horizontal, Vertical or at an angle with horizontal or vertical.

Nature of Force
Nature of force whether it is pull or push. Pull or push is denoted by an arrow head on end of line of action.

Point of Concentration
Point of Concentration is the point at which or through which force is acting.



PRINCIPLE FOR NUMBER OF FORCES ACTING ON A BODY OR POINT

If a number of forces acting simultaneously on a body then the effect produced by all the forces will be same as produced by a single resultant of all the forces.

Composition of forces

FR = F1 + F2
FR = 30 + (-20)         F2 is –ve, is opposite or in –ve direction
FR = 10 KN

Two forces F1 and F2 are acting simultaneously opposite to each other on a body (a). The resultant of both, FR will produce the same effect on the body as produced by F1 and F2 (b).


SYSTEM OF FORCES

If two or more than two forces, with different magnitude and direction, act simultaneously on a body, then they form a system of forces. Some common systems of forces are
Coplanar Forces     –    Forces acting on the same plane or parallel to the plane.
Collinear Forces     –    Forces having same line of action.
Concurrent Forces   –    Forces acting on a same point irrespective of the plane and line of action.


COMPOSITION OF FORCES

Composition of Forces or compounding is the method to find  resultant force of a number of forces acting simultaneously on a body.

Composition of forces

Methods for finding resultant force(FR) are

Parallelogram Law

Triangle Law

Polygon law

Resolution of forces or Method of Resolution


 


GRAPHICAL METHOD

In Graphical method, forces are drawn on graph with some suitable scale and their resultant is measured by Parallelogram law, Triangle Law or Polygon Law. While Parallelogram law is also used as analytical method for resolution of forces. Let’s study the laws used for graphical method for composition of forces.


PARALLELOGRAM LAW OF FORCES

“If two consecutive sides of a parallelogram represents the magnitude and direction of two coplaner forces acting on a particle, then its diagonal represents the Resultant Force(FR) in magnitude and direction.”

Illustration 1
Two forces F1 and F2 of magnitudes 80KN and 50KN respectively acting simultaneously on a particle such that the angle between the two forces is 60o. Considering bigger force as F1.

Composition of forces

 

Taking some suitable scale and drawing a parallelogram ABCD with Sides AB and AD representing the forces F1 and F2 and having an angle (θ) between them.

F1 as AB = 80 mm = 8 cm,
F2 as AD = 50 mm = 5 cm
θ = 60o

 

Composition of forces

According to law of parallelogram diagonal AC is representing the resultant force (FR). Length of the diagonal AC is the magnitude of the resultant force and the angle (α), which diagonal AC is making with the side AB, represents the direction of resultant force (FR).

By Measurement

Composition and resolution of forces

Thus the Resultant FR is equal to 113.5 KN acting at an angle (α) 22.5o with F1.


TRIANGLE LAW OF FORCES

“If two sides of a triangle, taken in order, represents the magnitude and direction of two coplaner forces acting on a partical, then its third side, taken in opposite order represents the Resultant Force(FR) in magnitude and direction.”

Solving previous illustration with triangle law.

Taking some suitable scale (1KN = 1 mm) drawing a triangle ABC with sides AB and BC representing the forces F1 and F2 and having an angle (θ) between BC and extended side AB

F1 as AB = 80 mm = 8 cm,
F2 as BC = 50 mm = 5 cm
θ = 60o

Composition of forces

According to triangle law of forces side AC is representing the resultant force (FR). Length of side AC is the magnitude of the resultant force and the angle (α), which side AC is making with side AB represents the direction of resultant force (FR).

By Measurement

Composition and resolution of forces

Thus the Resultant FR is equal to 113.5 KN acting at an angle (α) 22.5o with F1.

Triangle and parallelogram law are best suited for system of forces having two forces acting simultaneously on a body. For a system of forces having more than two forces, Polygon law of forces is applied.


POLYGON LAW OF FORCE

“If sides of a polygon, taken in order, represents the magnitude and direction of more than two coplaner forces, acting on a partical, then its closing side, taken in opposite order, represents the Resultant Force(FR) in magnitude and direction.”

Illustration 2
Let us consider a system of forces F1, F2, F3 & F4.  Having magnitude as 25 KN, 35 KN, 30 KN & 20 KN, making an angle of 30o, 45o, 40o and 80o with horizontal as shown in figure below.

Composition of forces

Taking some suitable scale (1KN = 1 mm) drawing the sides of polygon ABCDE with Sides AB, BC, CD & DE  representing the forces  F1, F2, F3 & F4, having respective angle between them.

Drawing
F1 as AB = 25 mm = 2.5 cm,
F2 as BC = 35 mm = 3.5 cm
F3 as CD = 30 mm = 3 cm,
F4 as DE = 20 mm = 2 cm

Angle between F1 and F2 is 105o
Angle between F2 and F3 is 95o
Angle between F3 and F4 is 60o

Joining point A and E to draw the closing side AE.

Composition and resolution of forces

According to polygon law of forces, side AE is representing the resultant force (FR). Length of side AE is the magnitude of the resultant force and the angle (α) which side AE is making with side AB represents the direction of resultant force (FR).

By Measurement

Composition and resolution of forces

Thus the Resultant FR is equal to 22.6 KN acting at an angle (α) 154.4o with F1.


ANALYTICAL METHOD

Analytical method is different from graphical method. Let us study analytical method for finding the resultant force for number of forces acting simultaneously on a particle. There are two analytical methods for composition of forces, Parallelogram law and Method of Resolution.


PARALLELOGRAM LAW OF FORCES

“If two consecutive sides of a parallelogram represents the magnitude and direction of two coplaner forces acting on a partical,

Composition and resolution of forces

Then the Resultant Force (FR) and angle α is obtained by

Composition and resolution of forces

Solving Illustration 1 by parallelogram law

Composition and resolution of forces

Resultant Force FR will be

Composition and resolution of forces

The angle (α) which resultant FR is making with force F1   

Composition and resolution of forces


RESOLUTION OF FORCES FOR RESULTANT FORCE

Before understanding the method of finding resultant force by method of Resolution let us understand what resolution of forces is.


Resolution of  Forces

Resolution of forces is the process of breaking a force into two components, basically horizontal and vertical components.

Suppose force (F) acting on a particle such that, it is making an angle (θ) with horizontal axis (AB).

Composition and resolution of forces

Composition and resolution of forces

Horizontal Component
It is the product of magnitude of force and the cosine of the angle, it is making with horizontal axis. Direction of the component is considered same as the direction of horizontal axis, with which it is making the angle θ.

Vertical Component
It is the product of magnitude of force and the sine of the angle, it is making with horizontal axis. Direction of the component is considered same as the direction of vertical axis, normal to the horizontal axis with which it is making the angle θ.


PRINCIPLE OF FINDING RESULTANT FORCE BY THE METHOD OF RESOLUTION OF FORCES

“The resolved parts of the forces in a particular direction will be equal to the resolved parts of resultant force in same direction.”

Method of resolution of forces starts with summation of Horizontal Components in a system of forces is represented By ∑H while summation of Vertical Components is represented by ∑V.

∑H = Sum of Horizontal Components in a system of forces.

      = F1cos θ1 + F2cos θ2 + F3cos θ3 +_ _ _+ Fn cos θn

∑V = Sum of Vertical Components in a system of forces.

      = F1sin θ1 + F2sin θ2 + F3sin θ3 +_ _ _+ Fn sin θn

Magnitude of resultant force is obtained by

Composition and resolution of forces

Direction of resultant in terms of angle(α) is obtained by

 

Composition and resolution of forces


Solving Illustration 2 with method of resolution

Composition and resolution of forces

 

 Evaluating ∑H and ∑V (Using sign convention)

Composition and resolution of forces

∑H  =  25 cos 30o + 20 cos 80o + (-30 cos 40o) + (-35 cos 45o)

       =  25 cos 30o + 20 cos 80o – 30 cos 40o – 35 cos 45o

       =  – 22.60

∑H  =  – 22.60 (-ve)

Similarly

∑V  =  25 sin 30o + 35 sin 45o + (- 30 sin 40o) + (- 20 sin 80o)

       =  25 sin 30o + 35 sin 45o – 30 sin 40o – 20 sin 80o

       =  – 1.73

∑V  =  – 1.73 (-ve)

Resultant Force FR

Composition and resolution of forces

Composition and resolution of forces

The angle (α), which resultant (FR) making with horizontal axis.

 

Composition and resolution of forces

Composition and resolution of forces

 

In Solution of illustration 2 by polygon law, α is the angle which resultant FR is making with force F1. Here α is the angle, which resultant FR is making with horizontal axis in negative direction.

Thus angle between FR and F1 will be

 

Composition of forces

 

Thus sum of both horizontal and vertical Component of  forces is negative, the resultant will lie in 3rd quadrant.

Composition of forces

 

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