In this tutorial, you will study about the Parallelogram of Forces. This is an important law in Statics. This includes the definition of Parallelogram Law of Forces, Composition of Forces, Resultant and all the methods to draw and solve Parallelogram of two different forces acting on a body.

**Contents:**

- Parallelogram of Forces
- Composition of Forces
- Triangle Law of Forces
- Parallelogram Law of Forces
- Derivation of Resultant of Two Forces
- Analytical Method of Parallelogram Law
- Special Cases in Analytical Method
- Special Cases in Graphical Method

## Parallelogram of Forces

Usually, we find that objects in the real life are under several loads. This means that there is more than one force acting on the bodies. Such complications are solved by Parallelogram of Forces.

- When there are several forces with different magnitudes acting on a body, they are said to be a system of forces.
- The general problem of Statics is to find out conditions that help the System of Forces to remain or hold in equilibrium.
- The constraint conditions that are found as solutions should keep the body in equilibrium in both horizontal and vertical directions as well as in rotation also.
- The various methods of solution to this problem are based on several axioms, called the Principles of Statics, out of which Parallelogram of Forces is studied at the first place.
- The principle of Parallelogram of Forces was first employed indirectly by Stevinus in 1586 and finally formulated by Varignon and Newton in 1687.

## Composition of Forces

The composition of forces means reducing a given system of forces to the simplest system that will be its equivalent.

- The equivalent system consists of only one force that can contribute all the effects of the original forces on the body.
- The Parallelogram of Forces is drawn for a number of forces but only two at a time. Hence as per the number of forces the parallelograms drawn will be high.
- Mostly we deal with the composition of two forces initially. The Parallelogram Law and the Triangle Law defines the ways and methods to calculate resultant of two given forces.
- The principle of Parallelogram of Forces was first employed indirectly by Stevinus in 1586 and finally formulated by Varignon and Newton in 1687.

## Triangle Law of Forces

The Triangle Law of Forces provides basic definition of Parallelogram Law. The law is used when two forces are acting in different directions starting from a same point.

**Definition:**

If two forces are acting in different directions having their starting point in common, then the force which forms the third side of the triangle by closing it will be the resultant of the two forces.- Closing of a polygon means the resultant force must be taken in the reverse order i.e. in opposite direction of the two forces.
- The Triangle Law is also useful for other vectors. Triangle Law of Vector Addition also follows the same principle.
- When a body is in equilibrium under the action of three forces acting at a point, then those forces can be completely represented by three sides of a triangle taken in order.
- If forces are represented in vectors, then the resultant vector is the sum of the two initial vectors. For example, if A and B are two vectors acting on the body, then the resultant of the two vectors is given by A+B, according to Triangle Law of Vector Addition.

The figure given below is a representation of Triangle Law of Forces

The above figure clearly explains how the Triangle Law of Forces acts. It shows two forces X and Y and their resultant is given by another force R which is in the opposite direction to either X or Y.

## Parallelogram Law of Forces

The Parallelogram Law of Forces states that if two forces are acting under some angle at a same point, then they can be represented by two adjacent sides of the parallelogram. And the resultant of the two forces is given by any one of the diagonals of the parallelogram.

The steps to be followed while drawing a Parallelogram are:

- At first, draw a vector that exactly represents one of the forces to be resolved in the given problem.
- Now draw the second force through the tail of the first force.
- Similarly, draw parallel forces to both the forces with exactly same length as of original length of the vector.
- The parallelogram is complete now. Now draw a diagonal from one vertex joining the opposite vertex of the parallelogram.
- By calculating the length of the diagonal either theoretically or practically, we get the resultant of the two forces.

## Derivation of Resultant of Two Forces

The purpose of Parallelogram Law is to determine the resultant of two forces acting on a particle. Hence, the following method is to be followed to derive the resultant of two forces.

- Consider two vector forces acting on a particle. Let them be AB and AC. And the force AD will be the resultant of the two forces. The forces AB and AC are called the components of the force AD.
- Instead of constructing the complete parallelogram, the resultant can also be obtained by constructing the triangle ACD. Here, we take the vector AC from its end C draw the vector CD, equal and parallel to AB.
- Thus, after completing the triangle, the longest side AD will be the resultant of the two forces. It is obtained as the geometric sum of the two vectors representing the forces.
**Free Vectors:**

The vectors which do not show the point of application of forces that they represent are called Free Vectors.

The diagram given below shows the resultant of two forces.

As explained, the two forces AB and AC constitute a parallelogram. Now they are drawn separately and the triangle is closed by AD, which is the resultant of the two forces drawn graphically.

## Analytical Method of Parallelogram Law

If two forces P and Q are acting on a body, with an angle of α between them, we will now analytically find out the magnitude of the resultant R.

- Angle between two forces is α.
- Angles made by the line of action of resultant with the given forces = β and γ.
- It is sometimes convenient to use these formulae for determining the resultant instead of making accurate construction, to scale, of the triangle of forces.
- The magnitude of resultant of two forces is obtained by using mathematical operations such as Pythagoras rule, Sine rule etc.

The figure given below shows the analytical method of calculation.

The above figure gives out the magnitude of the resultant of two forces. Here R is the resultant, α is the angle between the two forces, β and γ are the angles made by the vectors with the resultant respectively.

R = \(\sqrt{P^2+Q^2+2PQcosα}\)

sin β = \(\frac{Q}{R}\)sin α

sin γ = \(\frac{P}{R}\)sin α

These results are also obtained from the above figure.

## Special Cases in Analytical Method

Here are some of the special cases that are observed in Parallelogram Law.

**Angle α=0**^{o}

Then the two forces are said to be acting in same direction. Then the magnitude of the resultant is the addition of two forces.**Angle α = 180**^{o}

If the angle between two forces is 180^{o}, then the two forces are said to be acting in exact opposite directions. In this case, the magnitude of resultant will be the difference of the magnitude of two forces.**Angle α = 90**^{o}

When the angle between two forces is 90^{o}, the two forces will become the horizontal and vertical components of the resultant force. The magnitude of resultant is given by sum of the squares of the two forces under root.- The angle between one of the forces and the resultant is given by the inverse tangent of the ratio of the two forces. Usually, they are denoted by β and γ.

## Special Cases in Graphical Method

Graphically, if two forces are drawn and a parallelogram is constructed, the measurements of the each of the forces must be taken according to some defined scale. However, for smaller angles, it is impossible to draw a parallelogram. Hence the following method is to be followed.

- The triangle of forces become very narrow and we conclude that, in the limiting case, where the two forces act along the same line and in the same direction, the resultant is equal to the sum of forces and act in the same direction.
- Again, in the same manner, if two forces are acting in the opposite directions almost along the same line, then the resultant will be their difference in magnitudes.
- By taking these simple conditions, we may refer to the special cases in analytical methods also.
- The forces which act along the same line are called Collinear Forces, to which a parallelogram cannot be constructed.

## Key Points to Remember

Here are the key points that we need to remember about “Parallelogram of Forces”.

- When there are several forces with different magnitudes acting on a body, they are said to be a system of forces.
- The Parallelogram of Forces is drawn for a number of forces but only two at a time. Hence as per the number of forces the parallelograms drawn will be high.
- When a body is in equilibrium under the action of three forces acting at a point, then those forces can be completely represented by three sides of a triangle taken in order.
- The angle between one of the forces and the resultant is given by the inverse tangent of the ratio of the two forces. Usually, they are denoted by β and γ.
- The forces which act along the same line are called Collinear Forces, to which a parallelogram cannot be constructed.
- The angle between the two forces plays a major role in determining the resultant of the two forces.
- The Parallelogram Law and Triangle Law are the easiest ways to reduce a System of Forces into simplified form.
- The magnitude of the resultant of two forces will be their algebraic sum.

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