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3.11 Newton's Law of Gravity

Newton's most brilliant step was to unite motions on Earth and in the sky. He knew that a force must cause an object like an apple to fall to the Earth. He also knew that the Moon must have a force acting on it to make it travel in a curved path as it orbits the Earth. Could these be the same force? (It is not known if the story of Newton getting his insight from watching a falling apple is true, but there is an apple orchard outside his childhood home in England!) Newton's universal law of gravitation is one of the most important discoveries in the history of science.
 


Isaac Newton. Click here for original source URL.

Newton expressed his law mathematically. In words, we can write: every particle in the universe attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. We therefore define gravity as a universal attractive force; it is a property of every object in the universe. This expression of gravity means that if you could double the mass of the Sun, the Sun's gravitational attraction to the Earth would double; but if you doubled the distance between them, the force on the Earth would decrease by a factor of four (the square of two). Similarly, if you tripled the Sun’s mass, the force would triple, but if you tripled the distance, the force would decrease by a factor of nine.
 


Halley's comet in 1986. Click here for original source URL.

Gravity is a force that follows an inverse square law. That is, the strength diminishes with the square of the distance from the object. Newton was able to show that if gravity followed an inverse square law, then the planet orbits that are predicted must be ellipses with the Sun at one focus. Here at last was the elegant explanation for Kepler's work. Newton's gravity was used to successfully predict the return of Halley’s comet on its highly elliptical orbit of the Sun — a stunning confirmation of the theory.

Newton's law of gravity applies to two single particles. How then can we calculate the force of gravity on a person, when the attraction due to all the different parts of the Earth must be added together? Newton invented the calculus to solve this problem - with the result that a spherical object like a planet behaves as if it had all its mass concentrated at the center. Newton did many of these important calculations and then buried them on his desk, as his voracious mind found new problems to study. Spurred on by Edmund Halley, the famous astronomer, he finally published his work in a masterful book called "Principia," in 1687.
 


Illustration of the inverse square law of gravity. An object twice the distance "r" from object "S" will experience four times less gravitational force from object S. Similarly, and object three times the distance "r" from object "S" will experience nine times less gravitational force from object S, and so on. Click here for original source URL.

Newton's universal law of gravitation is a common type of law called an inverse square law. The force of gravitydue to some object depends on the inverse square of the distance. This dependence is typical of many forces that emanate from a point — in fact, it is a property of the three dimensional space we inhabit. We can imagine lines of gravity force emerging from an object in all directions. The concentration of the lines of force is then a measure of the strength of the gravity. The surface area of any sphere surrounding the object is 4πD2, where D is the distance from the object. The lines of force spread out more and more as we move away from the object. This decreasing concentration corresponds to the decreasing force of gravity. The total number of lines of force does not moving away from the object, but they are spread over an area that increases proportional to D2. Therefore the concentration or number of line of force in any particular area decreases with the inverse square of the distance, i.e. proportional to 1 / D2.

Electric and magnetic forces obey the same relationship with distance as gravity. You can also understand how the inverse square law would apply to the radiation from a light bulb, reaching surfaces at different distances. Imagine rays of light moving out from a light source. If at a certain distance, the light emitted in a certain direction is spread over one unit, then at twice that distance it is spread over four units. The intensity of the light at twice the distance is thus 1/4 as much. In the same way, the intensity at three times the initial distance is only 1/9 as much. So light, like gravity, obeys an inverse square law.

Here is Isaac Newton's reasoning as he used the inverse square law of gravity to connect the falling apple with the orbit of the Moon. We will use simple geometry to get the answer roughly — calculus is required for a more accurate result. The distance from the Moon to the center of the Earth is 60 times the radius of the Earth. So by the inverse square law the Earth's gravity at the distance of the Moon should be 602 times less than at the Earth's surface. The acceleration of the Moon that causes it to deviate from a straight path and curve around the Earth is 9.8 / 602 = 0.0027 meters per second per second. In each second of its orbit, the moon falls 0.0027 / 2 = 0.0014 meters or only 1.4 millimeters towards the Earth.

How far does the Moon travel in its orbit in one second? This is just the circumference of the orbit divided by the orbital time in seconds. The orbit time is 27.3 days or 27.3 x 24 x 3600 = 2.36 x 106 seconds (Note that the time for the Moon to complete a cycle of phases — 29.5 days — is longer than the orbit time because the Earth moves in its own orbit in a month). So in one second the Moon travels (2 π 384,000) / 2.36 × 106 = 1.02 kilometers. For such a small piece of the orbit we can approximate the curved path as a straight line and use the small angle equation, which gives a / 206,265 = d / D, so a = 206,265 (0.0014 / 1020) = 0.3 arc seconds. This is the angle by which the Moon deviates in its orbit each second due to the gravity of the Earth. The angle by which it rotates in its orbit is a / 206,265 = d / D, so a = 206,265 (1.02 / 384,000) = 0.5 arc seconds, a similar number. So we can see that an inverse square law of gravity describes the Moon’s orbit.

It is actually easier to write the law of gravitation in mathematical terms than to spell it out in words. If we have two masses MA and MB, separated by a distance R, then Newton's law of gravity gives the force between mass A and mass B of

F = G MA MB / R2

The number G is the gravitational constant, a fundamental constant of nature. If we measure mass in kilograms, distance in meters, and force in the normal units of Newtons (N), G = 6.67 × 10-11 N m2 kg2. The constant G is a tiny number - gravity is actually a very weak force. It is only the presence of enormous amounts of matter that gives gravity a sizable force.

The mass of the Earth, 6 × 1024 kg, gives us a downward acceleration of 9.8 meters/sec2. But everything with mass attracts everything else with mass, so what about other objects? Suppose two ocean liners of mass 10,000 tons (107 kg) were sitting in the water separated by 100 meters. The force between them would be (6.67 × 10-11 x 107 x 107) / (100)2 = 0.67 Newtons. Using Newton's second law of motion, acceleration = force / mass, the acceleration on each ship would be 6.7 × 10-8 meters/sec2. This is clearly too tiny to measure — the gravity of astronomical objects clearly overwhelms the gravity of everyday objects!

If we want to compare the gravity due to two different objects, we can take ratios and not need to use the gravitational constant. To compare the relative gravitational force of objects B and C on object A, we divide

FB on A / FC on A = (MB / MC) x (RC to A / RB to A)2

The gravitational constant and the mass of A cancel out. We are pinned to the Earth by its gravity. Let's see how the gravity of other objects compares. Using the equation above where B is the Sun and C is the Earth, the relative force is (2 × 1030 / 6 × 1024) x (6400 / 1.5 × 108)2 = 6 × 10-4. So the Sun exerts less than a tenth of a percent as much gravity on you as the Earth. What about two people sitting in a room? If they weigh 50 kg and are 1 meter apart, the relative force is (50 / 6 × 1024) × (6400 / 0.001)2 = 2 × 10-9. This is a billionth of the force that keep us down to Earth. Whatever else might attract two people, gravity has very little part in it!

Newton’s gravity incorporates his laws of motion. Gravity is the force that keeps the Moon in its curving motion around the Earth, or the Earth in its curving motion around the Sun. In the absence of gravity, the Earth would just fly off straight through space, just as a stone whirled overhead on a string would fly if the string broke. In the vacuum of space there is no friction or air resistance, so the solar system can maintain its motions for a very long time. Also, gravity is a mutual force that illustrates Newton's third law of motion. The Earth exerts a gravitational force on you, but you exert an equal gravitational force on the Earth! Gravity has a long reach. It declines in strength as the distance increases but never becomes zero. Using this logic Newton was sure that gravity was a universal force.

It is important to keep in mind the difference between mass and weight. Mass is a fundamental property of an object or particle. It is the amount of "stuff" or the number of atoms in something, measured in units of kilograms. This measure determines the force of gravity. Weight depends on your location in space. On the Earth's surface, the acceleration due to gravity is 9.8 meters/sec2 — usually given the symbol g (to distinguish it from the universal constant G). This is an increase in speed of 9.8 meters per second for every second of falling. On the smaller and less massive Moon, the acceleration due to gravity is only 1.6 meters/sec2. If you weighed 60 kg on Earth you would weigh 10 kg on the Moon. In orbit around the Earth, you would be weightless. This is because you and the spacecraft experience the same gravity force, and there is no net force between you and the spacecraft. You would also be weightless in deep space, far from the gravity of any star or planet. In all of these situations your mass is the same. But your weight depends on the local gravity force.

Newton the man was a bundle of contradictions. He could be humble when thinking of his predecessors. He said "If I have seen any further than others, it is because I have stood on the shoulders of giants." Yet he could be brutal with colleagues and rivals. The man who gave birth to the rational and mechanistic view of a "clockwork" universe spent much of his effort on alchemy. After he died his estate was found to contain thousands of pages of detailed analysis of the Bible. Newton the scientist is easier to judge. He used one simple law to understand a huge variety of seemingly unrelated effects: your weight, an apple falling from a tree, the Moon moving around the Earth, the arcing path of a comet, or the planets moving around the Sun. The English writer, Alexander Pope, got it right when he penned this clever verse upon Newton’s death:

"Nature and Nature’s laws lay hid in night;
God said "Let Newton be! and all was light."