## Newtonian Mechanics

Using Kepler’s deductions about planetary orbits, Isaac Newton created his scheme of dynamics and gravitation, published in his ‘Philosophiae Naturalis Principia Mathematica’ in 1687.

He assumed the world existed with a 3-dimensional Euclidian stage, motions occurring within time, which flows constantly ever forward unaffected by all around.

The position of a particle at time *t*, relative to some origin, is described by a vector * x(t)*. At a later time

*t*+ Δ

*t*the particle has moved to

**+ Δ**

*x***at the velocity**

*x***= Δ**

*v***/Δ**

*x**t*and as Δ

*t*→ 0 the velocity vector becomes tangential to the trajectory at point

**,**

*x***≡**

*v**d*

**x**/dtThe magnitude of the velocity vector is called the speed (distance/time).

If the velocity is not constant, then the acceleration is ** a** = Δ

**/Δ**

*v**t*and as

*t*→ 0,

**=**

*a**d*(speed/time = distance/time

**v**/dt^{2})

Newton also described the Linear Momentum vector (mass x speed) of the particle, * p = mv*.

**The Law of Inertia
**

Aristotle thought that the natural state of any body is ‘at rest’. Galileo’s experiments with inclined planes suggested otherwise and that an object would move forever at a constant speed; its motion only being retarded by frictional forces. Rene Descartes later formulated this principle into a form that Newton used as his first law: ‘The velocity of a body remains constant (in both magnitude and direction) unless a net force acts upon the body’.

The modern form of Newton’s first law is the **Law of Conservation of Linear Momentum**:

** P** =

*m*= constant, i.e.

**v***d*= 0

**P**/dt**Definition of Force
**

The concept of **Force** was defined in Newton’s second law of motion (the Force Law): ‘The acceleration imparted to a body is proportional to and in the direction of the force applied and inversely proportional to the mass of the body’, i.e. **F** = m**a**

The modern statement of the second law is * F = dP/dt*. In dynamics, mass represents the

**Inertia**of a body, the body’s resistance to change in its state of motion.

Completing his theory of dynamics, Newton developed his third law, the law of action-reaction: ‘For every force acting on a body (in a closed system), there is an equal and opposite force exerted by that body’. The modern version of the third law is the law of the **Conservation of Total Linear Momentum**, **P** = **P**_{1} + **P**_{2} = constant.

**Newton’s Law of Universal Gravitational
**

For a body moving in a circular orbit of radius *r*, the speed *v* of the body will be constant, but the direction of the velocity vector is constantly changing,

** a** = Δ

**/Δ**

*v**t*=

*v*^{2}/

*r*

A force that points directly towards the centre of the circle (**Centripetal Acceleration**).

*F*_{cent} = *m a* =

*m*

**v**^{2}/

*r*

And if *P* is the orbital period, then the speed of the body is, *v* = 2π*r*/*P
*

Kepler’s third law, relates the period and the orbital radius,

*P*^{2} = *kr*^{2}

⇒ F = 4π^{2}*m*/*kr*^{2}

i.e. the force maintaining the orbit is inversely proportional to the radius of the orbit. The body at the centre of the orbit must feel an equal and opposite force, which must be proportional to its mass. Redefining the constant of proportionality,

*F*_{grav} = *GMm*/*r*^{2 } (G has a measured value of 6.67×10^{-11}m^{3}/kg s^{2})

Which is Newton’s Law of Universal Gravitation.