Understanding the parallelogram law of forces through problems : Part-I

It is impossible to make a concept gullible without applying it. In other words no concept has ever evolved; in someone's brain, without he or she encountering a problem at some point in their life and squeezing the brain to find a way out of the problem. The famous story of Sir Issac Newton challenging an apple's falling, which eventually turns into the concept of Gravitational force is the best example related to it..

In this piece of blog (Part-I and Part-II), we will see practical problems from some of the best books available for Engineering Mechanics and will solve them applying our learning from previous blogs.         

Before, going into the problems there are few concepts needed to be understood. And they are in fact handy and very useful while solving the problems. And these are :

• The Triangular Law of Forces
• The Polygon Law of Forces
• The Resolution of Forces  

Laws of The Resultant Force

• The Triangular Law of Forces
• The Polygon Law of Forces

The Triangular Law of Forces :

"If two forces acting simultaneously in a particle, be represented in magnitude and direction by the two sides of a triangle, taken in order; their resultant may be represented in magnitude and direction by the third side of the triangle, taken in opposite order".  

Figure-1 | Triangular Law of Forces

Basically, when two forces are arranged in such a way that they are any two sides of a triangle, and their directions are same. Then their resultant will be the third side of the same triangle in magnitude, and the direction will be opposite to the other forces. The concept of Triangular forces are directly imported from Vector Algebra.

This law along with Parallelogram law of forces can be useful tools to solve the problems graphically. That means a geometry box is sufficient to determine resultant from a force system. And also no equation, calculation and remembrance of formula is required. Though the answer may not be accurate all the time rather they will be close to the answers of analytical solutions due to many factors like accuracy of drawing and precision of the instruments of geometry box. The process is also time consuming for solving any problem. In spite of that, such learning enhances our understanding of the fact that, how these laws works!     

To understand the practical aspect of the above discussion, we can solve a problem graphically to show the readers, how the concept works!

(Q1): A disable automobile is pulled by means of ropes subjected to the two forces as shown. Determine graphically, the magnitude and direction of their resulting using (a) the parallelogram law (b) the triangle rule. 

Figure-2 

Solution, 
To approach the problem for a graphical solution, we need a sharpened pencil, a scale, a protractor and a compass from geometry box.

Let,
1 mm length in the geometry scale is representing 1 KN of force shown in Fig-2

(a) Solution with Parallelogram Law   

Steps

• We draw a horizontal straight line of 4 mm which represents the 4 KN force
• Then we draw an another co-planer and concurrent force line of 2 mm, and inclined to 55⁰ (i.e. 30⁰+25⁰) with the horizontal force. Which represents the 2 KN force from Fig-2.
• Their directions are shown as given in the Fig-2.
• Now we make a parallelogram by connecting the parallel lines of those we have drawn, with the help of compass.  
• Now we have to join the two points of the distant corners of the parallelogram thus drawing its major diagonal.
• The length of the Diagonal comes out to be 5.4 mm in geometry scale. From which we can say the magnitude of the resultant is 5.4 KN 

Figure-2A

(b) Solution with Triangular Law
 Steps

• First we draw a horizontal line of 4 mm representing the 4 KN force.
• Then we draw an another co-planer concurrent force line of 2 mm length, inclined to the horizontal at 125⁰ (i.e. 180⁰- 55⁰). Which represents the 2 KN force.
• Their directions are shown as per Fig-2    
• Now we join the open parts of the figure we have drawn, thus making it a triangle. As this line is representing the resultant force, so as per Triangular law, the direction should be opposite to the other forces.
• The length of this line represents the magnitude of resultant force. By measuring the length we can say the magnitude of resultant force is 5.3 KN     

Figure-2B

Now let us compare the results with the analytical (trigonometric) solution of the same problem

Figure-2C | Trigonometric solution of problem (Q1)

In analytical solution the magnitude of resultant force comes out to be s 5.276 KN, which is satisfactory and all the values of the answer are near to each other     

Answers

(a) 5.4 KN

(b) 5.3 KN

(c) Trigonometric (Analytical) solution = 5.276 KN

The Polygon Law of Forces :

This law is an extension of Triangle Law of Forces for more than two forces, and it states that :

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

Let us take six forces and arrange them as they represents the sides of a polygon. Their directions are taken as similar to each other. 

Now, the Fig-3 shows, how using the Triangular law, we can find the ultimate resultant and also validate the statement of Polygon Law of forces,

Figure-3|Polygon Law of Forces

Where  

Resolution of Forces : 

The process of splitting up the given force into a number of components, without changing its effect is known as Resolution of force.

The principle of resolution may be precisely stated as "The algebraic sum of the resolved parts of a number of forces, in a given direction, is equal to the resolved part of their resultant in the same direction"

Generally forces are resolved in vertical and horizontal directions.

For better understanding of the resolution of force, let's solve a problem

(Q2): A machine component of 1.5 m long and weight 1000 N is suspended by two ropes AB and CD as shown in Fig. 4 given below.

Figure-4

Calculate the tensions T1 and T2 in the ropes AB and CD.

Solution, 

In this problem, first we can check visually. There are 3 forces acting in the system. The tension force T1 and T2 with their respective angle and the weight of the machine part, which is given as 1000N.

Now, let us resolve the forces horizontally; that means we are equating the horizontal forces of opposite directions :

T1 cos 60⁰ = T2 cos 45⁰

    

T1 = 1.414 X T2............(i) 

Now, let us resolve the forces horizontally,

T2 sin 45⁰ + T1 sin 60⁰ = 1000

T2 sin 45⁰ + (1.414 X T2) X sin 60⁰ = 1000

0.707 X T2 + (1.414 X T2) X 0.707  = 1000

               

1.93 X T2 = 1000..........(ii)

So,   

T2 = 518.1 N

      

T1 = 1.141 X 518.1 = 732.6 N

Hence, our required answers are calculated above.

We will continue our journey of solving problems in the next piece of blog. We will also enjoy learning the method of resolution for the resultant force. The readers must be assured that their author will be working hard to make the coming blogs exciting and satisfying too. 

Readers are also encouraged to comment on the content and suggest ways of making the presentation better and also requested to share the blog.  

The Force systems and the Principles of Mechanics : Parallelogram Law of Forces

Understanding the Principles of Mechanics 

To understand the principles of the elementary mechanics, it is necessary  to know about the various kinds of orientations of forces or the system of forces. 

"When two or more forces act on a body, they are called the system of forces".  

Following systems of force are defined below to address the need of their references and understanding in future topics. 

1. Coplanner Forces :  The forces, whose lines of action lies on the same plane, are known as co planner forces.
2. Co-linear Forces : The forces, whose lines of action lies on the same line, are known as co linear forces. Co-linear forces may not be on the same plane but they will be linear in acting all the time 
3. Concurrent Forces : The forces, Which meet at one point, are known as concurrent forces. (Irrespective to their lines and planes.)  
4. Coplanner concurrent Forces : The forces, which meet at one point and their lines of action also lie on the same plane, are known as co-planner concurrent forces.
5. Coplanner non-concurrent Forces : The forces, which don not meet at one point, but their lines of action lie on the same plane, are known as coplanner non-concurrent forces. 
6. Non-coplanner concurrent Forces : The forces, which meet at one point but their lines of action do not lie on the same plane, are known as Non-coplanner concurrent forces.
7. Non-coplanner non-concurrent Forces : The forces, which don not meet at one point, and their lines of action do not lie on the same plane, are known as Non-coplanner non-concurrent forces. 

Resultant of A Force : 

The idea of resultant is very important in mechanics as it accommodates the effect of all the acting forces and turn them together into a single representative force. A resultant force properly can be defined as, "If a number of forces acting simultaneously on a particle, then it is possible to find out a single force which would produce the same effect as produced by all the given forces. This single force is called the resultant force" 

POINTS

• The number of given forces are known as component forces.
• The process of finding out the resultant force, of a number of given forces, is called composition of forces.

As mentioned in my earlier blog, we are now ready to learn the the first foundational Principle of elementary mechanics, which is the "Parallelogram Law of Forces". The above written ideas and their understanding is important for moving further into the coming topic.

Parallelogram Law of Forces : 

This law essentially gives the clear idea of a resultant and it shows the way of calculating in the force systems. The statement of the law can be written as :

"If two forces acting simultaneously on a particle, be represented in magnitude and direction by two adjacent sides of a parallelogram; their resultant may be represented in magnitude and direction by the diagonal of the parallelogram, which passes through their intersection".   

        Figure - 1

Now, referring to figure-1, 

Let, F1, F2 be the forces whose resultant R is required to be found out and q is the angle between the forces F1 & F2.

Let, a be the angle which the resultant force makes with one of the forces (Say F1)  

Then, as per Vector Algebra we can write 

 

Now, If the angle (a) which the resultant force makes with the other force F2.

Some important corollaries from Eqn(1)  

• If q=0⁰, when the forces act along the same line and direction, then                  R = F1 + F2                                   (Since cos 0⁰= 1)
• If q=90⁰, when the forces act at right angles                                                       
  
                       (Since cos 90⁰= 0)                                                                                                                                                                                                                                                              
• If q=180⁰, when the forces act along the same line and opposite direction, then,                                                                                                      R = F1 - F2                                          (Since cos 180⁰= -1)                        In this case resultant forces will act in the direction of the greater force       
• If the two forces are equal i.e., when F1 = F2= F then                               
  
  In the next piece of blog we will continue to travel through the learning of this blog and also explore its application in some exciting problems.                                                                                

Concept of Force and its Features in Engineering Mechanics

A force can simply be defined as

"Its an agent which produces or tends to produce, destroys or tends to destroy motion"

To enhance the understanding, some helpful examples are as; when we see a horse applying force to pull a cart and to set it in motion or even a couple of magnets  attracting or repelling each other. Apart from those examples one can think of many examples by observing our day to day life.  

The feature and effects of Force can be listed as :  

• A force is a vector quantity, that means it has a magnitude and a direction as well.
• A force represents the action of one body on another.
• Force can be exerted by actual contact or at a distance (Gravitational, Magnetic force etc.) 
• It may change the motion of a body (i.e if a body is at rest, the force may set in motion. And if the body is already in motion it may accelerate it.)
• It may retard the motion of a body.
• It may retard the forces, already acting on a body, thus bringing it to rest or in equilibrium.
• It may give rise to internal stresses in the body, on which it acts.

Characterization of Force : 

A force is characterized in order to determine its affects. There are mainly Four such characteristics of a force, and they are :

• Point of application of the force
• Magnitude of the force 
• Direction of the force 
• Nature of the Force(Push or pull). This is denoted by placing an arrow head on the line of action of the Force.  

LEARNING POINTS :

1. In statics we consider a force which tends produce a motion. Sometimes the applied force may not be sufficient to move a body. Like if we try to lift a body of a weight;  beyond our capacity, and we fail to do so. In that case we have exerted a force undoubtedly, but the weight appears to be motionless. This kind of act teaches us that a force necessarily may not produce a motion in the body; but it may, simply tend to do so.
2. In Newtonian mechanics, Space, Time and Mass are absolute concepts and they are independent of each other (Not applicable in relativistic mechanics). On the other hand the concept of force is independent of the other three.

Unit and Dimension of Force : 

The SI Unit of Force is NEWTON represented as N. That also can be defined as 

"The amount of force when applied to a 1 KG weight body, then the body will attain an acceleration of 1 m/s^2".

The  CGS Unit of Force is DYNE. "A unit of force that, acting on a mass of 1 Gram, increases its velocity by 1 Centimeter per Second every second along the direction that it acts". 

To learn the dimension of a Force, its unit comes handy. Though unlike the unit, the dimension of force (or any other property) is fixed, but its easy to derive the dimension from the definition of the unit.

The unit of force can be unfolded as, 

Hence, the Dimension of force is derived from its SI unit as above.   

Continuing in the journey of learning statics; one need to understand two important ideas. They are the ''Particle" and the "Rigid body".

  

Concept of Particle : 

In statics, its important to underscore the significance of particle and 

"It may be defined as a body of infinetly small volume and is considered to be a concentrated point". 

Here the term infinetly small volume indicates that the physical presence of such a particle is not possible, rather the concept is ideal and valid for mathematical analysis only.  

Concept of Rigid Body :

"A Rigid Body may be defined as a body which can retain its shape and size, even if subjected to some external forces"

Like the particle concept, there is no body which is perfectly rigid in actual practice. But for the sake of simplicity, we take all the bodies as rigid bodies in mathematical analysis and modelling works. 

 A Rigid Body can be imagined as a combination of large number of particles occupying fixed positions with respect to each other. The study of mechanics of particles  is obviously prerequisite to that of Rigid Bodies.

The Physical Independence Principle of Force :

The principle of physical independence gives the hint to the idea of a "Resultant", as it states that

"If a number of force is simultaneously acting on a particle, then the resultant of these forces will have the same effect as produced by all the forces"

It clearly indicates and teaches us that; the combined effects of multiple forces can be seen in one separate force as a resultant. And its affect can be seen in practice along with its magnitude and direction.     

The study of elementary mechanics rests on six fundamental principles based on experimental evidence. And they are listed as :

1. The Parallelogram Law for the addition of forces.
2. The Principle of Transmissiblity.
3. Newtons First Law
4. Newtons Second Law
5. Newtons Third Law
6. The Newtons Law of Gravitation.       

We will continue enjoy learning the principles in detail and in more elaborate manner in the coming blog. Please do share and comment in the comment box below about the article and suggest your ideas to make the presentations better.  

References :

1. "Vector Mechanics for Engineers, Statics" by Ferdinand P Beer, E. Russell Johnson, David F. Mazurek, Phillip J Cornwell, Elliot R Eisenberg & Sanjeev Sanghi.
2. "Materials Science and Engineering and Introduction" by William D Callister Jr, David G Rethwisch
3. "A Textbook of Engineering Mechanics" by R.S Khurmi