# General Mechanics/Fundamental Principles of Dynamics

## History of Dynamics

### Aristotle

Aristotle expounded a view of dynamics which agrees closely with our everyday experience of the world. Objects only move when a force is exerted upon them. As soon as the force goes away, the object stops moving. The act of pushing a box across the floor illustrates this principle -- the box certainly doesn't move by itself!

However, if we try using Aristotle's dynamics to predict motion we soon run into problems. It suggests that objects under a constant force move with a fixed velocity but while gravity definitely feels like a constant force it clearly doesn't make objects move with constant velocity. A thrown ball can even reverse direction, under the influence of gravity alone.

Eventually, people started looking for a view of dynamics that actually worked. Newton found the answer, partially inspired by the heavens.

### Newton

In contrast to earthly behavior, the motions of celestial objects seem effortless. No obvious forces act to keep the planets in motion around the sun. In fact, it appears that celestial objects simply coast along at constant velocity unless something acts on them.

This Newtonian view of dynamics — objects change their velocity rather than their position when a force is exerted on them — is expressed by Newton's second law:

${\displaystyle {\vec {F}}=m{\vec {a}}}$

where ${\displaystyle {\vec {F}}}$  is the force exerted on a body, ${\displaystyle m}$  is its mass, and ${\displaystyle {\vec {a}}}$  is its acceleration. Newton's first law, which states that an object remains at rest or in uniform motion unless a force acts on it, is actually a special case of Newton's second law which applies when ${\displaystyle {\vec {F}}={\vec {0}}}$  .

It is no wonder that the first successes of Newtonian mechanics were in the celestial realm, namely in the predictions of planetary orbits. It took Newton's genius to realize that the same principles which guided the planets also applied to the earthly realm as well.

In the Newtonian view, the tendency of objects to stop when we stop pushing on them is simply a consequence of frictional forces opposing the motion. Friction, which is so important on the earth, is negligible for planetary motions, which is why Newtonian dynamics is more obviously valid for celestial bodies.

Note that the principle of relativity is closely related to Newtonian physics and is incompatible with pre-Newtonian views. After all, two reference frames moving relative to each other cannot be equivalent in the pre-Newtonian view, because objects with nothing pushing on them can only come to rest in one of the two reference frames!

Einstein's relativity is often viewed as a repudiation of Newton, but this is far from the truth — Newtonian physics makes the theory of relativity possible through its invention of the principle of relativity. Compared with the differences between pre-Newtonian and Newtonian dynamics, the changes needed to go from Newtonian to Einsteinian physics constitute minor tinkering.

## Newton's first law:

"An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an external force".

He gave this law assuming the body or the system to be isolated. If we look upon into our daily life we find that this law is applicable in reality like a bicycle stops slowly when we stop moving the pedals, this is because in our daily life there are 2 external forces which opposes the motion, these are frictional force and air resistance ( These forces aren't included in the course of an isolated system). But if these two forces are absent the above law is applicable, this can be observed in space.

## Newton's second law:

"The rate change of linear momentum of an object is directly proportional to the external force on the object."

The concept behind this is - "Forces arise due to interaction between the bodies." With the help of this law we can derive this formula: ${\displaystyle F=ma}$  . In this formula, ${\displaystyle F}$  means the force exerted on the object, ${\displaystyle m}$  means the mass of the object, and ${\displaystyle a}$  means the acceleration of the object. Also the second law of motion is universal law in nature, i.e. it consists of both the laws. Newton assumed the body to be an isolated one, thus according to second law, if there is no interaction between bodies of two different systems( means not to be an isolated one), then there is no force that can stop or shake the object from its state of being.

Suppose two bodies are interacting with each other, then each of them would be exerting force on the other( Look from each side and apply second law of motion), it is third law.

## Newton's third law:

"Every action has an equal and opposite reaction"

This means if body A exerts a force on body ${\displaystyle B}$  , then body ${\displaystyle B}$  exerts a force on body ${\displaystyle A}$  that is equal in magnitude but opposite in direction from the force from ${\displaystyle A}$  . ${\displaystyle F_{AB}=-F_{BA}}$

• Negative sign shows that the force is in opposite direction.