Lecture 4 -- Newton and the Law of Gravity

In 1665 Edmund Halley and Robert Hooke took a trip from London to Cambridge, England. Their task was to convince a young scientist, Isaac Newton, to join the Royal Society and work on the problem of planetary motions. At the time, things were very confused. According to Aristole, bodies in motion would always come to rest. Galileo disputed this, saying that, if not for friction, a bodies in motion would continue moving forever in a circle. Kepler's laws, however, seemed to argue against both versions, since the planetary orbits were definitely not circles. In France, Descartes was arguing that that some force must be present that pushes the planets along in their orbit. Hooke, the president of Britain's Royal Society, had the idea that maybe a force from the Sun was somehow involved. Halley and Hooke explained all this to Newton; Newton listened, attended one or two meetings of the Society, and then stopped coming.

Three years later, Halley and Hooke rode to Cambridge to again try to persuade Newton to work on the problem of planetary motions. This time, when they arrived, Newton rummaged through his old papers, and showed Halley and Hooke the solution to the problem. (Well, he didn't show them the entire solution. To do the problem, Newton had to invent a whole new field of mathematics, which he called "flextures" and which we now call Calculus. Newton didn't want to share his tricks with the rest of the world, so he could only show Halley and Hooke part of the solution. As for the rest, Newton had to spend the next several years re-deriving everything he did in tedious geometry. And, if you ever read his book, Principia, you'll see how tedious the geometry can be!)

Newton began with three laws of motion. They concern mass, (how much stuff there is), velocity (the speed and direction of an object's motion, acceleration (the change in an object's velocity, and force (what it takes to change an object's motion). Newton's three laws of motion are

Law 1: Bodies in motion will continue in a straight line motion (at the same speed), and bodies at rest will stay at rest, unless acted upon by an outside force.

Law 2: A body's change in motion is proportional to the force acting on it, and will be in the direction of the force. In mathematical terms, F = m a , where F is the force, m is the mass, and a is the acceleration. The larger the mass of the object, the more force it takes to accelerate it.

Law 3: For every force, there exists an equal and opposite force. For example, when you push the floor, the floor pushes back at you. That is how you jump off the ground.

In addition, Newton stated his law of gravity. Mathematically, the law is F = G M m / r^2 . In other words, the force due to gravity is an attractive force between two bodies, which depends only on the mass of the two bodies (M and m) and inversely on the square of the separation between the two bodies. (If you double the mass of the earth, its gravitational force will become twice as big; if you get 3 times further away from the earth, its gravitational force will be 3 x 3 = 9 times weaker.)

Note that the force of gravity does not depend on the object's shape, density, or what it is made of. GRAVITY DEPENDS ON MASS AND DISTANCE AND NOTHING ELSE!

Newton's laws have many implications. First, they say that many of the motions we see every day are complex combinations of two motions. For instance, consider a baseball thrown in an arch. There is no force to stop it from going sideways, so it moves sideways at a constant speed. However, the earth's gravity pulls the ball downward. So at each moment, the motion of the ball is a combination of a constant sideways motion, and an acceleration downward towards the center of the earth.

Consider the orbit of the Moon about the earth. The gravitational attraction between the earth and Moon is causing the Moon to fall towards the earth. If the Moon were completely at rest, it would hit the earth. However, the Moon is also going sideways, so while the Moon is falling, it is also continuing to move sideways. As a result, it falls, but ``misses'' the earth. It is thus in orbit.

According to Newton's Law of Gravity, the attraction one body has on another body is F = G M m / r^2. This formula says that the attractive force never goes to zero --- unless you are at an infinite distance from a source, there is always some attraction.

Here's a simple question. You are standing on the earth. But, how far as you from the earth? The distance you are from soil you are standing on is essentially zero, but you are several thousand miles from the soil in Australia. What distance does the law of gravity refer to? One of Newton's triumphs was his mathematical proof to showed that, when you calculate the total gravity associated with a round object, it's as if all the mass of the object resided at the object's center. In other words, when you talk about the distance you are from something, you are talking about the distance to the center of the object. So, although you are now standing (or sitting) on the earth's surface, for purposes of gravitational calculations, you are about 4000 miles from the earth.

But this now presents a puzzle. When you stand on the earth, you are about 4000 miles from the earth's center. When astronauts orbit the earth, they are about 150 miles above the earth's surface. In other words, they are 4150 miles from the earth's center. That's not much of a difference. So how come you feel weight, while the astronauts float around weightless?

Even Jules Verne, in his novel From the Earth to the Moon, got that wrong. When you stand on the earth, the earth's gravity is pulling you down. But the ground doesn't let you move. You are pushing on the ground (due to gravity), and the ground is pushing on you (Newton's third law). The push of the ground causes you to feel weight.

Now consider an astronaut in space. The earth's gravity is pulling him down, just like it is pulling you down. However, he is in a spacecraft, and gravity is pulling the spacecraft down as well. Both are falling together, and nothing is pushing back. The astronaut is weightless.

Mass is also important in gravitational calculations. Consider the orbit of the earth. The earth is going around the Sun. However, the Sun is also going around the earth. In fact, both the Sun and the Earth are going around a center of mass. Because the Sun is so much more massive than the earth, the orbit it takes due to the earth is very small. However, if the earth had the same mass of the Sun, both would orbit a point half-way between the two.

This has an implication for Kepler's third law. In fact, the law is wrong. Instead of P^2 = a^3, where P is the orbital period and a is the semi-major axis of the orbit, the correct equation is (M + m) P^2 = a^3 , where M is the mass of the Sun, and m is the mass of the planet that is in orbit. In other words, the relation between period and orbital size depends also on the masses involved. In the solar system, the mass of the Sun is so much greater than the mass of any planet, that m in the above equation can be neglected. (Note that for ease of math, astronomers measure orbital periods in years, orbital sizes in astronomical units, and masses in solar masses.)

Now, suppose the Sun were expanded out to twice its size. If it contained the same amount of matter, then its gravitational force would remain the same. Similarly, if the Sun were squeezed into the size of a basketball, its gravitational pull would remain the same, as long as no mass was lost.

Finally, let's consider the topic of tides. The formula for gravity says that the force due to gravity depends on mass and distance. The Moon has mass, and it is some finite distance from the earth, so the earth feels its gravity. But consider: the side of the earth which faces the Moon is about 4000 miles closer to the Moon than the center of the earth. Therefore, it feels a greater gravitational force, and material on this near side of the earth is actually pulled away from the earth's center. Similarly, the center of the earth is about 4000 miles closer to the Moon than the side facing away from the Moon. So the center of the earth is pulled away from the material on the earth's far side. The result is that the earth becomes elongated, with tidal bulges on the sides toward and away from the Moon.

In fact, the Moon has relatively little mass and is moderately far away, so its ``tidal force'' isn't great enough to cause the rocks on earth to overcome friction. However, water can move much more easily. The result is that the water is continually pulled toward (and away) from the Moon. These are the tides. If you are on the side of the earth facing the Moon (or away from the Moon), that's where the water is, and you have high tide. If you are in between, you have low tide.

Any object with finite size can be affected by tides. If you do the math, you will find that the ``tidal'' force, i.e., the difference between the pull on one side of an object and that on the other is F(tide) = G M s / r^3 , where M is the mass of the body causing the tides, r is the distance to that body, and s is the size of the body being affected. So, if the Moon were twice as far away as it is now, the earth's tides would be 2 x 2 x 2 = 8 times less.

The Sun's gravity also pulls the earth, so it also causes tides. The Sun is about 30,000,000 times more massive than the Moon, but it is also 400 times further away. Accordingly, its gravitational force on the earth is almost 200 times greater than that of the Moon, but its tides are only half as big. When the Sun and Moon line up, so that they tidal bulges in the same direction, the tides are extra high and extra low. These are called spring tides. When the Moon and Sun are at right angles, their effects cancel, and we get neap tides.