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.