Classical Mechanics

Copernican Revolution

The Copernican Revolution was the paradigm shift from the Ptolemaic model of the heavens, which described the cosmos as having Earth stationary at the centre of the universe (right) , to the heliocentric model with the Sun at the centre of the Solar System (left). Beginning with the publication of Nicolaus Copernicus’s De revolutionizes orbium coelestium, contributions to the “revolution” continued until finally ending with Isaac Newton’s work over a century later.

Apparent_retrograde_motionIn the Ptolemaic (earth centred) model of the universe 'epicycles' needed to be added to explain the apparent motion of the planets as observed from earth. With the Sun at the centre, these 'epicycles' were no longer required.


Galileo's laws of Motion

Aside from his numerous inventions, Galileo also laid down the first accurate laws of motion for masses. Galileo realized that all bodies accelerate at the same rate regardless of their size or mass. Everyday experience tells you differently because a feather falls slower than a cannonball. Galileo's genius lay in spotting that the differences that occur in the everyday world are in incidental complication (in this case, air friction) and are irrelevant to the real underlying properties (that is, gravity). He was able to abstract from the complexity of real-life situations the simplicity of an idealized law of gravity.

Key among his investigations are:

  • developed the concept of motion in terms of velocity (speed and direction) through the use of inclined planes.
  • developed the idea of force, as a cause for motion.
  • determined that the natural state of an object is rest or uniform motion, i.e. objects always have a velocity, sometimes that velocity has a magnitude of zero = rest.
  • objects resist change in motion, which is called inertia.

Friction1Galileo also showed that objects fall with the same speed regardless of their mass. The fact that a feather falls slowly than a steel ball is due to amount of air resistance that a feather experiences (alot) versus the steel ball (very little).
Much of this thinking dealt with objects on the Earth. Galileo didn't extend his ideas to beyond the Earth's surface, that was for an astronomer named Kepler.


Keplers Laws of Planetary Motion

Kepler developed, using Tycho Brahe's observations, the first kinematic description of orbits. Newton will develop a dynamic description that involves the underlying influence (gravity) 1st law (law of elliptic orbits): Each planet moves in an elliptical orbit with the Sun at one focus. Ellipses that are highly flattened have high eccentricity.

Ellipses that are close to a circle have low eccentricity. 2nd law (law of equal areas): a line connecting the Sun and a planet (called the radius vector) sweeps out equal areas in equal times. Objects travel fastest at the low point of their orbit, and travel slowest at the high point of their orbit.

3rd law (law of harmonics): The square of a planet's orbital period is proportional to its mean distance from the Sun cubed. The 3rd law is used to develop a "yardstick'' for the Solar System, expressing the distance to all the planets relative to Earth's orbit by just knowing their period (timing how long it takes for them to go around the Sun).

newton2 copyThe mathematical way to describe Kepler's 3rd law is: P2 α R3 where the α symbol means 'proportional to.' Proportions are expressions that imply there exists some constant, k, that relates the period, P, and the radius, R, such that P2 = kR3 We can determine k by expressing the formula in units of the Earth and its orbit around the Sun, such that: (1 yr)2 = k (1 A.U.)3 so k is equal to one, as long as we use units of years and A.U.'s (the Astronomical Unit, i.e . the distance from the Earth from the Sun). With k=1, then kepler's 3rd law becomes P2 = R3


Newton's Laws of Motion

Example: from Newton's 1st law we know that an object travels in a straight line unless acted upon by an external force. A circular orbit is clearly not a straight line, what is the force? Newton showed that the planets are acted on by the force of gravity arising from the Sun. Each orbit is a constantly changing velocity where gravity adds a small "delta-vee'' at each moment. This "delta-vee'' is what produces the elliptical curvature that is the orbit.

Example: from Newton's 2nd law when a baseball player throws a ball he applies a force, F, to the ball of mass m. Let's say he throws a tennis ball of mass of one-tenth the mass of a regular baseball (1/10m). What is the resulting acceleration? Ten times the acceleration of a regular baseball and, therefore, ten times the final velocity and ten times the distance hit.

Example: from Newton's 3rd law when a rocket is launched accelerating gases from the rocket engine push outwards from the rear. This inturn propels the rocket forward. This same principle can be seen when letting the air out of a party balloon.

delta_veeaddgateway_graphfma3rocket

Newtons Laws of Gravitational Attraction

Newton determines that gravity controls the motion of all objects in the Universe (i.e. Newton's apple)

Objects in the Universe attract each other with a force that varies directly as the product of their masses and inversely as the square of their distances

All masses, regardless of size, attract other masses with gravity. You don't notice the force from nearby objects because their mass is so small compared to the mass of the Earth.

gravfieldgrav 

 


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