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

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**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.

F_{R} = F_{1} + F_{2}

F_{R} = 30 + (-20) F_{2} is –ve, is opposite or in –ve direction

F_{R} = 10 KN

Two forces F_{1} and F_{2} are acting simultaneously opposite to each other on a body (a). The resultant of both, F_{R} will produce the same effect on the body as produced by F_{1} and F_{2} (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.

Methods for finding resultant force(F_{R}) are

Parallelogram Law

Triangle Law

Polygon law

Resolution of forces or Method of Resolution

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**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(F _{R}) in magnitude and direction.”*

*Illustration 1*

Two forces F_{1} and F_{2} of magnitudes 80KN and 50KN respectively acting simultaneously on a particle such that the angle between the two forces is 60^{o}. Considering bigger force as F_{1}.

Taking some suitable scale and drawing a parallelogram ABCD with Sides AB and AD representing the forces F_{1} and F_{2} and having an angle (θ) between them.

F_{1} as AB = 80 mm = 8 cm,

F_{2} as AD = 50 mm = 5 cm

θ = 60^{o}

According to law of parallelogram diagonal AC is representing the resultant force (F_{R}). 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 (F_{R}).

By Measurement

Thus the Resultant F_{R} is equal to 113.5 KN acting at an angle (α) 22.5^{o} with F_{1}.

**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(F _{R}) 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 F_{1} and F_{2} and having an angle (θ) between BC and extended side AB

F_{1} as AB = 80 mm = 8 cm,

F_{2} as BC = 50 mm = 5 cm

θ = 60^{o}

According to triangle law of forces side AC is representing the resultant force (F_{R}). 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 (F_{R}).

By Measurement

Thus the Resultant F_{R} is equal to 113.5 KN acting at an angle (α) 22.5^{o} with F_{1}.

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(F _{R}) in magnitude and direction.”*

*Illustration 2*

Let us consider a system of forces F_{1}, F_{2}, F_{3} & F_{4}. Having magnitude as 25 KN, 35 KN, 30 KN & 20 KN, making an angle of 30^{o}, 45^{o}, 40^{o} and 80^{o} with horizontal as shown in figure below.

Taking some suitable scale (1KN = 1 mm) drawing the sides of polygon ABCDE with Sides AB, BC, CD & DE representing the forces F_{1}, F_{2}, F_{3} & F_{4}, having respective angle between them.

Drawing

F_{1} as AB = 25 mm = 2.5 cm,

F_{2} as BC = 35 mm = 3.5 cm

F_{3} as CD = 30 mm = 3 cm,

F_{4} as DE = 20 mm = 2 cm

Angle between F_{1} and F_{2} is 105^{o}

Angle between F_{2} and F_{3} is 95^{o}

Angle between F_{3} and F_{4} is 60^{o}

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

According to polygon law of forces, side AE is representing the resultant force (F_{R}). 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 (F_{R}).

By Measurement

Thus the Resultant F_{R} is equal to 22.6 KN acting at an angle (α) 154.4^{o} with F_{1}.

**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,

Then the Resultant Force (F_{R}) and angle α is obtained by

Solving *Illustration 1* by parallelogram law

Resultant Force F_{R} will be

The angle (α) which resultant F_{R} is making with force F_{1}

**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).

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
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*“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.

= F_{1}cos θ_{1} + F_{2}cos θ_{2} + F_{3}cos θ_{3} +_ _ _+ F_{n} cos θ_{n}

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

= F_{1}sin θ_{1} + F_{2}sin θ_{2} + F_{3}sin θ_{3} +_ _ _+ F_{n} sin θ_{n}

Magnitude of resultant force is obtained by

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

Solving Illustration 2 with method of resolution

Evaluating ∑H and ∑V (Using sign convention)

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

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

= – 22.60

∑H = – 22.60 (-ve)

Similarly

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

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

= – 1.73

∑V = – 1.73 (-ve)

Resultant Force F_{R}

The angle (α), which resultant (F_{R}) making with horizontal axis.

In Solution of illustration 2 by polygon law, α is the angle which resultant F_{R} is making with force F_{1}. Here α is the angle, which resultant F_{R} is making with horizontal axis in negative direction.

Thus angle between F_{R} and F_{1} will be

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