Laws of forces

There are mainly three laws of forces to solve a force sytem.

1. Parallelogram law of forces
2. Triangle law of forces
3. Polygon law of forces

Parallelogram law of forces:-

It states that "If two concurrent forces are represented in magnitude and direction on two adjacent sides of a parallelogram then the diagonal of the parallelogram represents the resultant, in magnitude and direction, of the two forces."

Consider the diagram shown below

Let P and Q be two forces concurrent at point O represented by the two sides of a parallelogram OB and OA respectively.

Then the diagonal of the parallelogram OADB i.e OD represents the resultant of the two forces P and Q.

Now extend the line OB and join with the line DM to form a right angle triangle. Let θ be the angle between two forces P and Q. Now ∠AOB = ∠DBM = θ

Direction of the resultant can be calculated as follows 

tanβ=QSinθ/(P+QCosθ)

Let us analyse the resultant a bit more.

1.Assume θ=0°

Cos0°=1

Then resultant

R=√P²+Q²+2PQ

2.Assume θ=90°

Cos90°=0

Hence,

R=√P²+Q²

3.Assume θ=180°

Cos180°=-1

Hence,

R=√P²+Q²-2PQ

Triangle law of forces :-

It states that "If two concurrent forces acting on a body can be represented, in direction and magnitude, by the two sides of a triangle then the third side of the triangle represents the resultant in direction and magnitude, arranged in reverse order."

In the figure above forves P and Q are represented by the two sides of a triangle and the third side represents the resultant of those forces placed in reverse order.

The result so obtained above for paralellogram law of forces to find the resultant can also be used to find the resultant in triangle law of forces.

i.e R=√P²+Q²+2PQcosθ.

Where R= resultant of forces.

Polygon law of forces:-

Polygon law of forces states that "if there are more than two forces that can be represented in magnitude and direction by the sides of a polygon than the closing side represents the resultant of the forces."

In the figure below:-

Four forces namely F1,F2,F3 and F4 are placed on the side of a polygon according to magnitude and direction. The side that closes the polygon,if measured, gives the resultant of the all the forces represented in reverse order.

Now to obtain magnitude of resultant R one need to find summation of all the forces in x direction i.e. Σx and summation of all the forces in y direction i.e. Σy.

Then resultant 

R=√(Σx)²+(Σy)²