Classical Mechanics

Classical mechanics deals with the study of the motion of an object. Classical mechanics acts as a base for all the physics concepts. It gives the answers to, why if we throw an object straight above us in a moving platform fall right back in our hands but not behind us. Mathematical methods of classical mechanics explain the motion of a body, which our brain can’t think of. This article will help you get a basic idea of what classical mechanics is and what it has to offer to the world of physics. 

Classical Mechanics

Classical mechanics is a physical theory that determines the motion of an object in the future if the present values are known to us. In classical mechanics, all the objects are considered to be point masses and the speed is not anywhere near to the speed of light. Let us look at all the quantities that can be studied on a moving body and mathematical methods of classical mechanics.

Speed and Velocity

Similar to distance and displacement which have different meanings, even though their meaning is almost similar. Speed and velocity are exactly the same. Speed is defined as the rate at which an object covers a distance. The total distance covered by the object, regardless of its direction, is divided by the time taken for it to cover the distance. Speed is a scalar quantity. 

    Speed = Total Distance / Time taken

Velocity is defined as the rate of change of position, it can also be said as a derivative of position with time. As displacement is a directional measurement. Velocity is a vector quantity. Unlike speed where total distance is considered, in velocity, change in position is taken. So if the object starts and ends at the same point, no significant change is recorded so it can be said velocity is zero. 

     v = drdt.

Acceleration

Acceleration is defined as the rate of change of velocity, it can also be said as a derivative of velocity with time or the second derivative of position with respect to time. 

     a = dvdt = d2r/dt2.

Velocity can change in both magnitude and direction. If velocity decreases over time it is referred to as deceleration. 

Frames of reference 

The frame of reference is a set of coordinates from which the distance, velocity of an object is measured. An inertial frame is usually taken as a reference frame to measure the quantities of an object. An inertial frame is either at rest or moving at a constant velocity. Whereas non-inertial frames accelerate with respect to the inertial frame. Objects in the non-inertial frame appear to move in ways not explained by the forces existing in the inertial frame. It happens only because of the relative motion between these two frames. Those forces are named fictitious force, pseudo force, or inertia force. 

Consider two frames, P and P’. Observers from each reference have their own coordinates, P has (x,y,z,t) and P’ has (x’,y’,z’,t’). The relation between their coordinates while observing an activity from their respective reference frame P’ and P, which is moving at a relative velocity of u in the x-direction is given by:

     x’ = x – ut;

     y’ = y;

     z’ = z;

     t’ = t.

These are transformation equations called Galilean transformations. These transformations affect the other set of equations as follows:

• v’ = v – u ( from the perspective of s’ the velocity v’ of the particle is slower by u compared to its velocity v in the perspective of s.
• a’ = a (acceleration of a particle remains the same in any inertial frame of reference.)
• f’ = f ( force of a particle remains the same in any inertial frame of reference. )

Force 

Force is defined as an action that causes the body to change its velocity. It can also be said as the rate of change of momentum of the particle with respect to time. 

F = dpdt = d(mv)dt

Since mass is constant and a = dvdt the force equation can be simplified to,

F = ma.

Force is a vector quantity. Whenever force F is applied to a body it exerts equal and opposite force – F. 

Work and Energy

If a constant force F is applied and it causes displacement of the object, it can be said that work is done on the object. Work can also be defined as the scalar product of force and displacement. 

W = F.Δr 

As only, the displacement of an object is considered, no matter the path taken by an object, only the initial and final position is considered to calculate the work. 

The kinetic energy of the particle is given as,

Ek = ½ mv2. 

If different energies are applied to a body, the sum of kinetic energies of the particles is considered. 

Work-Energy Theorem states that for a particle of mass m, the work done on the particle as it moves from r1 to r2 is equal to the change in its kinetic energy. 

W= ΔEk = Ek2 – Ek1= ½ m ( v22 – v12).

If the forces are conservative it can be said that with decrease in the potential energy of a body its kinetic energy increases. The energy of the body is conserved.

(summation) ∑ E = Ek + Ep.

Conclusion

Classical mechanics is the basement of physics. Its application is wide, and it forms the base for all the calculations that happen in whole physics. This article covered the Introduction to classical mechanics and the Mathematical methods of classical mechanics. It is a vast topic for further deep understanding, Classical Mechanics: The Theoretical Minimum can be referred for further knowledge on the topic.