Mechanical – 3B1_GROUP NO. 3

COPLANAR CONCURRENT FORCES

CONTENTS : -

Introduction Equilibrium Conditions Resultant of Coplanar Concurrent Forces System 1. Analytical Method

2. Graphical Method

Lami’s Theorm

INTRODUCTION : -

DEFINITION : - Coplanar Concurrent Forces :- When the acting forces line of action lies in a same plane and meeting at a point to each other, said to be Coplanar Concurrent Forces.

EQUILIBRIUM CONDITIONS : -

If a system of forces acting on a body, keeps the body in a state of rest or in a state of uniform motion, then the system of forces is said to be in equilibrium.

Alternatively, we can say that the resultant force is zero, the system of forces will be in equilibrium.

1) The Coplanar Concurrent Forece is said to be in equilibrium condition if it satisfies following conditions :-

2) Algebraic sum of components of all the forces of the system, along two mutually perpendicular directions, is ZERO.

H = 0 and V = 0

RESULTANT OF COPLANAR CONCURRENT FORCES SYSTEM : -

A resultant force is the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body.

The defining feature of a resultant force is that it has the same effect on the rigid body as the other original system of forces.

The Methods to find out Resultant Force : -

Methods Of Resultant

force

Analytical Methods

Parallelogram law of

Force

Resolution of Forces

Triangle law of Forces

Graphical Methods

Polygon law of Forces

Triangle law of Forces

PARALLELOGRAM LAW OF FORCES : -

If two forces, acting at a point, are represented in magnitude and direction by the two sides of a parallelogram drawn from one of its angular points, their resultant is represented both in magnitude and direction by the diagonal of the parallelogram passing through that angular point.

Assume two forces P and Q on partical at point O, As shown in figure,

= Where,

R = resultant force Angle between P & Q = Angle between P & R

RESOLUTION OF FORCES : -

The process of splitting up the given force in two or more components, in particular direction, without changing effect on the body is called resolution of a force.

Generally a given force is split up in two manually perpendicular force components,

1. Horizontal components ( x-components ) 2. Vertical components ( y-components ).

Here, For force R,

Horizontal component = Rcos

Vertical component = Rsin

TRIANGLE LAW OF FORCES : -

If two forces acting on a body, it can represented by two adjacent sides of triangle with magnitude and directions and resultant can be given by other side of triangle; which is the statement of law of triangle.

Assume that, we have a pair of forces acting on a point than Triangle law of forces can be applied as,

For image, Resultant can be given by, R = .

The direction of the resultant can be given by,

= ( ).

Graphically, Triangle Law of Forces can be represented by taking two forces at two adjacent sides of triangle, which are acting on a body or particle and the resultant force can be represented by the other side of the triangle.

The angle can be also obtained from figure by using specific instrument.

POLYGON LAW OF FORCES : -

If a number of forces acting at a point be represented in magnitude and direction by the sides of polygon taken in order, then the resultant of all these forces may be represented in magnitude and direction by the closing side of polygon taken in opposite order.

Graphically , It can be represented by making a polygon as shown as shown in figure.

GRAPHICAL METHOD : -

Resultant of Coplanar Concurrent Forces can be graphically represented by using Space diagram and Vector diagram.

Space Diagram : - If a force acting in a system are represented by direction in the form of diagram by denoting every force with specific character , the diagram is known as space diagram.

Vector Diagram : - A space diagram of coplanar concurrent forces with magnitude and direction of all forces, the diagram is known as vector diagram.

Space Diagram Vector Diagram

LAMI’S THEORM : -

If three coplanar forces acting at a point be in equilibrium , then each force is proportional to the sine of angle between other two sides.

Applying Lami’s theorm on given figure, we have;

= =