2. Moments and Their Applications
Chapter 3
Moments and
Their Applications

3. Learning Objectives Introduction Moment of a Force
Graphical Representation of Moment
Units of Moment
Types of Moments
Clockwise Moment
Anticlockwise Moment
Varignon’s Principle of Moments (or Law of Moments)
Applications of Moments
Position of the Resultant 
Force by Moments
Levers
Types of Levers
Simple Levers
Compound Levers

4. Introduction Moment of a Force
In this chapter, we shall discuss the effects of these forces, at some other point, away from the point of intersection or their lines of action.
Moment of a Force
It is the turning effect produced by a force, on the body, on which it acts. The moment of a force is equal to the product of the force and the perpendicular distance of the point, about which the moment is required and the line of action of the force.
Mathematically, moment,
  M = P X l
 where P = Force acting on the body, and
l = Perpendicular distance between the point, about which the moment is
required and the line of action of the force.

5. Graphical Representation of Moment
Consider a force P represented, in magnitude and direction, by the line AB. Let O be a point, about which the moment of this force is required to be found out, as shown in Fig From O, draw OC perpendicular to AB. Join OA and OB. Now moment of the force P about O = P X OC = AB X OC Fig Representation of moment But AB X OC is equal to twice the area of triangle ABO. Thus the moment of a force, about any point, is equal to twice the area of the triangle, whose base is the line to some scale representing the force and whose vertex is the point about which the moment is taken.

6. Units of Moment Types of Moment
Since the moment of a force is the product of force and distance, therefore the units of the moment will depend upon the units of force and distance. Thus, if the force is in Newton and the distance is in meters, then the units of moment will be Newton-meter (briefly written as N-m). Similarly, the units of moment may be kN-m (i.e. kN X m), N-mm (i.e. N X mm) etc.
Types of Moment
Broadly speaking, the moments are of the following two types:
Clockwise moments.
Anticlockwise moments.

7. Varignon’s Principle of Moments (or Law of Moments)
Types of Moment
Clockwise moments It is the moment of a force, whose effect is to turn or rotate the body, about the point in the same direction in which hands of a clock move as shown in Fig. 3.2 (a).
Anticlockwise moments It is the moment of a force, whose effect is to turn or rotate the body, about the point in the opposite direction in which the hands of a clock move as shown in Fig. 3.2 (b).
Varignon’s Principle of Moments (or Law of Moments)
It states, “If a number of coplanar forces are acting simultaneously on a particle, the algebraic sum of the moments of all the forces about any point is equal to the moment of their resultant force about the same point.”

8. Example
A uniform wheel of 600 mm diameter, weighing 5 kN rests against a rigid
rectangular block of 150 mm height as shown in Fig. 3.8.
Fig.3.8.
Find the least pull, through the centre of the wheel, required just to turn the
wheel over the corner A of the block. Also find the reaction on the block. Take
all the surfaces to be smooth.

9. Solution

10. Applications of Moments
Though the moments have a number of applications, in the field of
Engineering science, yet the following are important from the subject point
of view : 
Position of the resultant force
Levers
Position of The Resultant Force
It is also known as analytical method for the resultant force. The position of a resultant force may be found out by moments as discussed below : 
First of all, find out the magnitude and direction of the resultant force by the method of resolution as discussed earlier in chapter ‘Composition and Resolution of Forces’.
Now equate the moment of the resultant force with the algebraic sum of moments of the given system of forces about any point. This may also be found out by equating the sum of clockwise moments and that of the anticlockwise moments about the point, through which the resultant force will pass.
An engineer designing a suspension bridge like one above, takes account of forces acting at points within the structure and the turning moment of forces.

11. Example
ABCD is a square. Forces of 10, 8 and 4 units act at A in the directions AD,
AC and AB respectively. Using the analytical method, determine
(i) resultant force in magnitude and direction ;
(ii) magnitude and sense of two forces along the directions AJ and AH, where J
and H are the mid-points of CD and BC respectively, which together will
balance the above resultant.

12. Solutions

13. Levers Type of Levers Simple Levers Compound Levers
A lever is a rigid bar (straight, curved or bent) and is hinged at one point. It is free to rotate about the hinged end called fulcrum. The common examples of the use of lever are crow bar, pair of scissors, fire tongs, etc.
It may be noted that there is a point for effort (called effort arm) and another point for overcoming resistance or lifting load (called load arm).
Type of Levers
Simple Levers
Compound Levers

14. Simple Levers
Fig Simple levers A lever, which consists of one bar having one fulcrum is known as *simple lever as shown in Fig (a) and (b). Let P = Effort applied W = Weight lifted a = Length between fulcrum and effort, and b = Length between fulcrum and weight. Now taking moments of the effort and load about the fulcrum (F) and equating the same, consideration will show that in order to increase the mechanical advantage, either length of the lever arm (a) is to be increased or length of the load arm (b) is to be reduced. Note. A simple lever may be straight, curved or even bent.

15. Example
The lever ABC of a component of a machine is hinged at B, and is subjected to a system of coplaner forces as shown in Fig. 3.19
Neglecting †friction, find the magnitude of the force (P) to keep the lever in
equilibrium. Also determine the magnitude and direction of the reaction at B.
Solution. Given : Vertical force at C = 200 N and horizontal force at C = 300 N.
 Magnitude of the force (P)
Taking moments about the hinge B and equating the same,

16. Solution

17. Compound Levers
A lever, which consists of a number of simple levers is known as a compound lever, as shown in Fig (a) and (b).
Fig Compound levers
A little consideration will show, that in a compound lever, the mechanical advantage (or leverage) is greater than that in a simple lever. Mathematically, leverage in a compound lever
 = Leverage of 1st lever × Leverage of 2nd lever × ...
The platform weighing machine is an important example of a compound lever. This machine is used for weighing heavy loads such as trucks, wagons along with their contents. On smaller scales, these machines are used in
godowns and parcel offices of transport companies for weighing consignment goods.

18. Example A compound lever shown in Fig. 3
Example A compound lever shown in Fig is required to lift a heavy load W
Find the value of W, if an effort (P) of 100 N is applied at A.
Solution. Given : Effort, (P) = 100 N 
From the geometry of the lever, we find that the leverage of the upper lever AB