1. Law of Polygon

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LAW OF POLYGON OF FORCES

AIM: To verify the law of polygon of forces for a given coplanar concurrent system of forces.

APPARATUS: Force table, Strings, Weights.

THEORY:

Law of polygon of forces is used to determine the resultant of coplanar concurrent force system consisting of more than two forces.

The law states “When more than two coplanar forces acting at a point are represented in magnitude and direction by the sides of a polygon taken in order, the closing side from the first to the last point of the polygon represents the resultant of the force system in magnitude and direction”.

Graphical Method

Let forces P1, P2, P3 and P4 acting at O are represented in magnitude and direction by the

sides AB, BC, CD and DE respectively of a polygon ABCDE taken in order. We are to prove that the resultant of these forces is represented in magnitude and direction by the side AE. According to the triangle law of forces, AC is the resultant of AB and BC, AD represents the resultant of AC and CD, and AE represents the resultant of AD and DE, that is, AE represents the resultant of P1, P2, P3 and P4.

2 Graphical conditions for equilibrium

Since the forces are in equilibrium, their resultant is zero. This means that the closing line of polygon (i.e. force diagram) drawn to represent the given forces will have zero length. In other words, the polygon drawn with the given forces will be closed figure. Hence, we can state: If a system of coplanar concurrent forces be in equilibrium, their vector diagram will be a closed figure.

Space diagram, Force diagram and Bow’s notation

Space diagram is that diagram which shows the lines of action of forces in space. In a space diagram, the actual directions of forces are marked by straight lines with arrows to indicate the sense in which the forces act. In the space diagram the space between any two forces is designated by Bow’s notation.

Force diagram is a diagram that is drawn according to some suitable scale to represent a given system of forces in magnitude, direction and sense. The closing line of the Force diagram represents the resultant of the given system of forces and its direction is from starting point towards the end point.

Bow’s notation is the method of designating forces in a space diagram. According to this

notation, each force in space diagram is denoted by two capital letters, each being placed P3 P1 P1 P2 P4 O a b c d e Space Diagram A C D E E P1

B

3 on two sides of the line of action of a force. Bow’s notation is particularly helpful in graphical solutions.

Analytical Method

The P1, P2, P3, P4 and P5 are the five non-concurrent forces acting on a plate in

equilibrium at an angle of 1, 2, 3, 4 and 5 with respect to positive X-axis measured in

anticlockwise direction then the magnitude of resultant is given by,

√

Where,

∑Fx = P1cos1 + P2cos2 + P3cos3 + P4cos4

= summation of all forces along positive X-axis

∑Fy = P1sin1 + P2sin2 + P3sin3 + P4sin4

=summation of all forces along positive Y-axis

The direction of Resultant force is given by

If the forces are in equilibrium the value of resultant (R) is zero. This method of finding the resultant is called Resolution of forces.

PROCEDURE:

 First adjust five strings to the desired angle. Four of them represent four forces of

which the resultant is desired and the fifth one is the equilibrant force.

 In the experimental set-up the angle between any two strings can be read directly

from the circular graduated disc. Now suspend any arbitrary masses on four strings and then try to adjust the fifth mass such that the inner ring appears concentric with the graduated disc. In this state five forces are in equilibrium.

 Note down the masses suspended on each string along with one representing an

equilibrant force.

 All we need is to see that all the forces including equilibrium force are proper and

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 Then mark the exact positions of the strings with the angle between them on a

plain paper and also note the masses suspended on them.

 Further sketch the space and force diagrams and get the graphical solution. Do the

necessary calculations and get the analytical solution.

 For getting variations, each time change the positions of the strings and the values

of the masses suspended on them. Repeat the procedure for getting 4 or 5 readings.

ASSUMPTIONS:

It is assumed that the pulleys are frictionless.

CALCULATIONS: ANALYTICAL: ∑Fx = ∑Fy = RA = √ Experimental, RE = Graphical, RG = Graphical Error = (RG- RA)/ RA x100 Experimental Error = (RE- RA)/ RA x100

5 RESULT: Resultant Force: RE = RA = RG = Percentage Error: Graphical Error = Experimental Error = CONCLUSION:

7 OBSERVATION TABLE Sr. No Masses suspended (kg) Force exerted

(N) Angle with x-axis

Resultant force (N) Percentage error m1 m2 m3 m4 m5 P1 P2 P3 P4 P5 θ1 θ2 θ3 θ4 θ5 Exp. RE Anal. RA Grap. RG Experimental (RE-RA/RA) x100 Graphical (RG-RA/RA) x100