Philosophy of Mechanics Unravelled

Philosophy of Mechanics Unravelled

First of all happy New year to all iitbandas . Its hardly 4 months left for IIT JEE 2010 . So now is the time to take stock of your preparation . So i will be writing now more articles to help you in this preparation . This article is on Mechanics .  You must have read numerous articles  and books on mechanics , but here i will try to explain things such that a layman will understand  Newton laws . Physics is subject which is closely associated with nature and i bet if your concepts are clear , you will solve any numericals of Mechanics . So put a full stop on endless numericals that you have been solving ..this is what you require before delving deep into numericals ….Enjoy and post any doubts on the forum .

What is physics all about ? Physics is study of different facets of systems which nature has created.

System can be of two kinds : – Living & Non Living

Till now, we have been applying physics to the system of non-living things.

The system can be

• Mechanical System
  
• Electromagnetic system
  
• Thermodynamic system
  
• Quantum system
  
• Optical system
  
• Electronic system
  

In this article we will confine ourselves to mechanical system. we will discuss what mechanics is all about? What do we read in mechanics?

Mechanics is all about studying statics and dynamics of a system.

In statics we determine the true state of the system. For a mechanical system, determining true state or statics of a system means obtaining position and velocities of all constituents of the mechanical system at a particular time t .

In Dynamics, we study how the state of system changes when a system is subjected to an external disturbances.

For example if we take a system of one mass kept on a horizontal floor

Figure :

Statics of system means knowing exactly the position and velocity of block at any particular time t.

Suppose the mechanical system of single block is subjected to an external disturbance in the form of force F. The block will respond to this cause (Force F) by getting accelerated. The acceleration is an effect produced by the cause (Force F). It characterizes the rate with which the state of the system changes.

This is true for any physical system which nature has created.

If any natural system is subjected to any external disturbance or cause, then as an effect, the state of the system changes. Also there is inherent tendency in the system to oppose any change in the state of system. The measure of strength of this tendency is called inertia of the system. Inertia of the system determines the rate at which the state of the system changes with time as an effect of the case.

If effect produced depends linearly on the cause then the system is said to be linear. Almost all the system created by God is linear.

The cause & effect produced are related by equation

… (1)

So, if inertia of the system is high, then effect produced will be small and vice-versa.

For mechanical system

Force = mass × acceleration

Here force is cause, acceleration is an effect and mass measures the translational inertia.

Mass is the measure of tendency in the mechanical system to oppose any change in the translational state of the system brought upon the application of force.

Similarly, Torque = Moment of Inertia × angular acceleration.

Here, Torque is the cause, angular acceleration is the effect and moment of inertia is the measure of rotational inertia. Moment of inertia measures the tendency of any mechanical system to oppose any change in rotational state of the system produced as a result of application of torque.

Similarly, for an electrostatic system

This is not an algebraic equation of the form (1)

But, still it is a linear differential equation of first order. Hence, electrostatic system is a linear system where an electrostatic system is a effect and change density is a cause.

For electrodynamics system

Potential difference = Resistance × current

Here, Potential difference is a cause and current is an effect and resistance is the measure of inertia

For magnetic system

MMF = Reluctance × Flux.

Here, Magneto motive force is the cause and flux produced is the effect and Reluctance measures the inertia.

For Thermal System

Temperature difference = Thermal resistance × rate of flow of heat

Here, Temperature difference is the cause and is the effect.

For wind system, pressure difference is the cause and wind the effect.

Similarly, this philosophy of cause and effect applies well to all the system living or non-living created by nature. The system may be linear or non-linear which will determine the effect produced.

The philosophy applies to the living system as well. When any living system is subjected to any external disturbance (cause) then also an effect is produced and magnitude of the effect produced depends on inertia of the system.

For example, if you talk to a young student he will immediately get influenced. The effect produced is high because the inertia is low.

On the other hand if you talk to an adult person. The effect produced is low, because with the age he/she has accumulated lot of inertia. If you talk to an old people, there is almost no effect or little effect because their inertia is too high!

The equation which enable us to determine the effect produced by the cause is called dynamical equation of system. Dynamical equation tells you about how the state of the system will change with time and hence, we can know about future state of the system.

So, the basic objective of mechanics is given as

The most important aspect of mechanics is to know how the state of the system changes when system is subjected to an external disturbance(cause)

For a mechanical System , the state of the system is defined by two variables i.e. position & Velocity of constituent bodies . Based on methods to determine the position & velocity of Constituents bodies of mechanical System , we have different formulations of Mechanics :

1. Newtonian Formulation of mechanics
   
2. Lagrangian Formulation of Mechanics
   
3. Hamiltonian Formulation of Mechanics
   

We will confine ourselves to the Newtonian Formulation of Mechanics .

In Newtonian formulation of mechanics , WE make of Newton Dynamical Equation

————————(2)

—————————(3)

Equation(2) correlates the force with acceleration produced & equation(3) correlates torque with angular acceleration produced . So , algorithm for solving a dynamical problem of classical mechanics using Newtonian formulation is

1. Show all the forces acting on the body by drawing a free body Diagram
   
2. Choose a co-ordinate System
   

   The Considerations while choosing a co-ordinate system are :
   

   1. Maximum number of forces should lie along axes.
      
   2. Acceleration should always lie along axes if there is any contradiction in these two, then second should be given the preference.
      
   3. Apply Newton laws of Motion and obtain the equation.
      
   4. Solve the Equation of motion and calculate the acceleration.
      

      Before winding up this article, let us discuss about Newton First & third law .
      

      All the Newton Laws are very basic laws of Nature. We have been discussing that force produces a change in velocity i.e. the state of the system. So , thinking on this line , we can also say that if the net force acting on the system is Zero , then the state of the system should remain same . This means if body is in rest or moving with a constant velocity then it will continue to be in rest or moving with constant velocity unless an unbalanced external force is applied on it.
      

      In other Words , system will remain in the state of equilibrium unless an unbalanced external force is applied on it .
      

      This is called Newton First Law of Motion .
      

      Newton Third law deals with action and reaction . It states that every action has equal and opposite reaction such that action and reaction act on different bodies .