Figure 4.30 shows a print depicting this order. Good metaphors for the Medieval world would be a "garden" or a "kingdom". With Newton this "human-scale" view of the universe changed rapidly and radically. Newton was able to provide precise mathematical descriptions of for the natural world. Forces and motion could be described elegantly and accurately with only a few key principles. Planetary orbits could be calculated with great precision and far into the future or back into the past. A more appropriate world-metaphor was the "world as machine" or the "world as clock". While we might feel "out of place" in Galileo's medieval world we would probably be at home in Newton's mechanical universe.

Figure4.30 Woodcut depicting the Aristotelian ordered cosmos.

Newtonian Mechanics and the Calculus

Newton's name will forever be linked with two of the greatest ideas of western culture, one physical and one mathematical. Newton captured the essential ideas that could describe virtually all physical phenomena in three simple statements or Laws of Motion which are summarized in Table 4.3. By carefully applying these Laws one can not only construct a detailed analysis of the orbit of a comet but also determine the conditions of stability of a bridge or the speed with which a sound wave will vibrate through air. To do any of this, however, a new mathematical way of thinking - The Calculus - was also needed. Both Newton and his contemporary Leibniz are consider the co-founders of calculus. Calculus is the mathematics of change. With calculus and the laws of motion, the ideas of Galileo's Law of Falling Bodies or Kepler's Laws of Planetary motion are all subsumed under one powerful system usually known as Newtonian Mechanics. It was the emergence of this powerful new way of thinking about the world that irrevocably changed how we view the universe and our place in it.

The Theory of Universal Gravitation

Story has it that it was the fall of an apple that prompted Newton to recognize that an apple, falling to Earth and the Moon in its orbit shared a common connection - they were both accelerating toward the centre of the Earth! While it is not difficult to see that an apple could be accelerating toward the Earth it is far less obvious that the Moon is accelerating toward the Earth. This is where Newton's Laws of Motion come in. According to Newton's first Law an object (the Moon) will continue to travel in a straight line unless acted upon by a force. Another way to say this is that only way to get an object to travel in a curved path like an orbit is to continuously apply a force pointing inward toward the centre of it motion. Figure 4.31a,b illustrate this. In Figure 4.31b imagine attaching a apple to a string and swinging it in a circle (an "orbit"?). The string must supply a constant force (called the Tension Force) that pulls the apple in a circular path. In each animation, click the play button several times to progress through the animation.

Figure 4.31a: The Moon and an apple both experience a force directed toward the centre of the Earth. (Click on image to download/play applet.)	Figure 4.31b The Moon in its orbit and an apple swung on a string share a common property - an inward force is needed to cause the motion that you see. (Click on image to download/play applet.)

Newton made a great leap of genius at this point! He knew how to measure the rate at which both the apple and the moon accelerate! It turns out that the Moon's acceleration is only 1/3600 as great as the acceleration of the apple and both accelerations are pointing to the centre of the Earth. Newton also knew that the moon is 60 times farther away from the centre of the Earth than the apple. Is there a "simple" mathematical connection between 1/3600 and 60? YES!

Newton also reasoned that the size of the force attracting one object to another depends on the mass of each object. All of this led to a precise (and testable) mathematical formula for the gravitational force between objects:

There are several features of this equation to notice. First, the negative sign may seem odd. This indicates that the force will cause the separation between objects to decrease - in other words it tells you that the force is attractive. The second is the "one-over-r-squared". This is know as a inverse-square law relationship and tells us a great deal about the nature of gravity. To appreciate this better consider the following example:

Figure 4.32 Graphing applet showing how an inverse-square law relationship works. In this case how Gravitational Force relates to changing Distance.

Orbital Motion

You may have wondered what keeps the moon (or the space shuttle) in its orbit. It may surprise that we have understood how orbits work for more than 300 years!

Newton's scientific masterpiece The Principia (1687) is one of the most profound scientific books ever written. In this book there appears a remarkable drawing. Figure 4.33 shows the cover page on The Principia and a drawing that anticipates the idea of an orbiting satellite!

Figure 4.33 Cover of Newton's Principia and an illustration within the principia depicting the motion of a cannon ball fired from a high mountain.

In "Newton's cannon" we see both a thought experiment and a visual "essay" on what orbital motion is. In this illustration Newton is showing us that orbital motion is just falling motion! Imagine firing a cannon from a high mountain. With just a little gun powder the cannon ball leaves the gun with a relatively low speed and may land a few kilometers away. More gun powder! Now the ball travels perhaps several hundred km's and we begin to see how the motion of the cannon ball and the curvature of the Earth come into play. You can simulate this effect with the applet given in Figure 4.34. To help see the cannon better you can increase the height the mountain by adjusting the lower right slider. Once you have picked a height you like, begin "firing" the cannon. Start small - pick 1 km/s or so and then slowly work your way up!

Figure 4.34 Newton's cannon. (Click to download/play applet.)

In all cases once the cannon ball is fired it begins to fall toward the earth. If, however, the ball is moving fast enough then as it moves down range the ground (because of the curvature of the Earth) will fall away at the same rate and, although the cannon ball is falling it never gets any closer to the Earth! It is in orbit.

So - what force keeps the Moon or the Space Shuttle in orbit? Gravity - constantly tugging the object toward the Earth.

What is Weightlessness?

Figure 4.35 shows a video clip of a NASA astronaut in a state of weightlessness. Weightlessness does not mean that the astronaut is no longer attracted by the Earth. Quite the contrary - the gravitational tug of the Earth on him is about the same as if he were on the ground. The reason he appears to be weightless is that he and everything else in the Space Station are orbiting the Earth and hence falling toward the Earth at the same rate. This is called free-fall. In order to experience the sensation of weight you must be pressing into an object at rest with respect to the Earth. If you step on a bathroom scale then you press down on the scale and the spring in the scale compresses and registers your weight. If you put the same spring scale in an elevator and then let the elevator drop you and the scale would both accelerate at the same rate. You would no longer compress the spring and your scale reading would be zero - you are weightless! A better term to use to describe this is free-fall.

Figure 4.35 Nasa video clip showing an astronaut in a state of "weightlessness".

Tides

One of the mysteries of the natural world before Newton was the phenomenon of tides. For millennia people had a vague understanding of tides and that they were somehow related to the moon but there was no clear understanding of why this should be so. Galileo, in correctly, argued that tides were caused by the Earth's rotation. It was Newton's Law of Universal Gravitation that provided the answer.

Figure 4.36 is a roll-over image showing the Earth and Moon (not drawn to scale). As you saw earlier, the gravitational force becomes much weaker with distance and this means that the gravitational tug that the Moon exerts is larger on the side nearest the Moon and least on the opposite side of the Earth. This causes the shape of the Earth to distort and since water is able to flow the distortion is produced by water forming a tidal bulge on each side of the Earth. These are the locations of high tides. At right angles to the bulge is the location of the low tides. As the Earth rotates the tidal bulge still points toward the moon but different regions of the globe pass in and out of the bulge and hence the time of high tide can be determined by determining when your location is pointing directly toward or away from the Moon.

The strongest tides - called Spring Tides - occur when the Moon and the Sun are in line. This means the strongest tides occur at either the New or Full phase of the moon. The weakest tides, called Neap Tides, occur when the Moon is at right angles to the Earth-Sun line, in other words at either First or Third Quarter phase.