# Equilibrium of Forces Study Notes for GATE ME Exam

By BYJU'S Exam Prep

Updated on: September 25th, 2023

Equilibrium of Forces Study Notes for GATE ME Exam for Mechanical Engineering: In this article, know the important pointers for Equilibrium of Forces like subdivisions and other terminologies, which can help you to score well in GATE ME, SSC JE ME and other competitive exams.

## Equilibrium of Forces

The branch of science that considers the motion of bodies and the effects of forces on that motion is known as Engineering Mechanics. It can be subdivided as:

• Statics: The special case of a body at rest or a body that moves with a constant velocity is dealt with under statics.
• Dynamics: Dynamics is subdivided into Kinematics and Kinetics. Kinematics deals with rates of change of geometrical quantities in a moving system; it does not involve the concept of force. Kinetics deals with the causes and the nature of motion resulting from specified forces.

FORCE- An agent which destroys or tends to destroy, produces or tends to produce motion.

• SI Unit: Newton (N)
• A system of forces is obtained when two or more forces of different magnitude and direction act upon a body.
• Concurrent forces are those forces (two or more) that act at the same point. Concurrent forces do not need to have the same direction. Their point of action must be the same. In case they have the same direction, they are collinear forces.
• If two or more forces whose directed arrows lie in the same plane, then they are called coplanar forces. Two concurrent forces always lie in a common plane, hence are always coplanar. But it is not necessary to have three or more concurrent forces to be coplanar.

RESULTANT OF FORCE- A single force that will have the same effect as that of several forces acting on a body is known as the resultant force. The single force is called resultant force and the process of finding out the resultant force is called composition of forces. The reverse of the composition of forces is called the resolution of force.

## COPLANAR FORCE SYSTEM

• PARALLELOGRAM LAW OF FORCES- If two adjacent sides of a parallelogram represent two forces both in magnitude and direction which act on a particle, then the diagonal of the parallelogram obtained by these two sides represent the resultant of these force both in direction as well as magnitude. Suppose two forces P and Q are represented by OP and OQ respectively, two sides of the parallelogram. Now the parallelogram OPRQ is completed and the resultant is represented by diagonal OR.

In triangle ORS

In triangle PRS, PS = Q cosΦ , RS = Q sinΦ

In triangle ORS

• ANALYTICAL METHOD- Resolve all forces horizontally, then determine SH = algebraic sum of all horizontal components. Resolve all forces vertically, then determine SV = algebraic sum of all vertical components.

• TRIANGLE 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 will represent the results of the two forces in the direction and magnitude took in the opposite order.
• POLYGON LAW OF FORCES- When the forces acting on a body are more than two, the triangle law can be extended to polygon law. Polygon Law states that if several coplanar concurrent forces acting simultaneously on a body are represented in magnitude and direction by the sides of a polygon, taken in order, then their results can be represented by the closing side of the polygon in the opposite direction and of the same magnitude as that depicted by the length.

MOMENTS- The turning effect produced by a force, on the body, on which it acts.

Moment M = F × L, SI Unit: Nm

Where

F = Force acting on the body, and

L = Perpendicular distance of the point, about which the moment is determined and the line of action of the force.

• DIRECTIONS OF MOMENTS-  Anticlockwise Moment whose effect is to turn or rotate the body, in the opposite direction in which the hands of the clock move. If the effect of the moment is such that its effect is to turn the hands of a clock in the opposite direction, then that moment is the anticlockwise moment.
• VARIGNON’S THEOREM- The moment of a force about any point is equal to the algebraic sum of the moments of the components of that force about the same point.

• Couples- Couple is the moment produced by two opposite, non-collinear and equal forces. It does not produce any translation but produces only rotation. The resultant force of a couple is zero. The product of the distance between the lines of action of forces and the magnitude of one of the forces is the moment of a couple.

## CONDITIONS OF EQUILIBRIUM

The term equilibrium means either the body moves with constant velocity or is at rest. A body is said to be in static equilibrium when the body does not tend to rotate and the resultant force acting on it is zero.

The resultant force causing motion should be zero.

i.e. ΣF = 0. It means that ΣH = 0 And ΣV = 0

The resultant moment causing rotation should be zero.

i.e. ΣM = 0 about any point

• FREE BODY DIAGRAM- One of the most useful aids for solving a statics problem is the free body diagram (FBD). A free-body diagram is a sketch of the body that shows the body (by itself, free of the other part of the system) and all the forces applied to it, that is, every force acting on it. Fig shows free body diagrams.

• LAMI’S THEOREM- When three coplanar forces act at a point and the equilibrium is maintained, then each force is proportional to the sine of the smaller angle between the other two forces.

ACCORDING TO LAMI’S THEOREM, where P, Q & R are the three coplanar forces and a, b, g are three angles.