Universal Gravitation

The origins and motions of the heavenly bodies have been a source of wonder and concern since prehistory and likely since humankind first took notice of the sky and its contents. The ideas and mathematics of Galileo, Newton and others have linked the motions of stars and planets and comets with that of balls and arrows and the tumblers in carnivals, the unknown with the commonplace. Gravity is one of the four fundamental forces in nature and it operates over unimaginably huge distances. In effect, it holds the entire universe together.

Kepler's Laws of Planetary Motion Understanding often begins with descriptions which lead to explanations and so, easier than explaining how a certain star came to be is recording its motions across the night sky. Johannes Kepler is noted for both the analysis of astronomical data collected by others, e.g., his employer Tycho Brahe, and for his record keeping. As a result of his analyses, he developed his laws summarizing three basic aspects of the motions of the planets in our solar system. Although his reasoning is now recognized as faulty, his insightful statements still are a key introduction to the motions of objects within our solar system. His laws are:

1. The planets orbit the Sun in ellipses (ovals) not circles.

2. The planets move fastest when they are closest to the Sun and slowest when they are farthest from it. This is the same idea as a figure skater's spin rate being controlled by the spread of their arms -- tight in against their chest for a fast spin and extended for a slower spin. (Platform divers control the rate of their spin by making their body compact or elongated.) By the way, if we imagine a line joining a planet to the Sun and look at the motion of this line as the planet zips around in its orbit, a wedge shape is created. Because the planet moves quickly when it is close to the Sun, the wedge shape on that side of the planet's orbit ellipse is short and broad. The wedge created in the same time period on the opposite side of the orbit ellipse is long and narrow. But, the area of each wedge is the same. What does this tell us? That the angular momentum of the planet is constant during its voyage around the Sun.

3. The orbital periods of the planets are indirectly related to their orbital radii from the Sun. This idea can be used to compare the motions of natural or manmade satellites or planets orbiting circling a common center. Use: (Ta / Tb ) ² = (ra / rb) ³ where the T is the orbital period and r is the orbital radius.

Find the period of a mystery planet in a 6.7 * 10 ⁷ km orbit around its sun if another planet takes 250 days to move once around its 9.3 * 10 ⁸ km orbit.

Ta ² = Tb ² ra ³ / rb ³

250 ² (6.7 * 10 ⁷ km) ³ / (9.3 * 10 ⁸ km) ³

23.27 days ² so Ta = 4.83 days.

Universal Gravitation Newton looked at Kepler's work and deduced an equation describing the variations in the gravitational force that would keep everything moving in accordance with his laws. Newton's equation can be used to calculate the gravitational attraction between any pair of masses. It is:

F = G m1m2 / d ². The G is a universal constant, 6.67 * 10 ^{-- 11} Nm ² / kg ². (We know the value of G due to the work of Cavendish, below.)

Find the gravitational attraction between a 3 000 kg asteroid and a 5 500 kg asteroid 1 250 m apart.

F = G m1m2 / d ²

6.67 * 10 ^{-- 11} Nm ² / kg ² * 3 000 kg * 5 500 kg / (1 250 m) ² = 7.04 * 10 ^{-- 10} N

Weighing the Earth About 100 years after Newton's universal gravitational equation, Cavendish created an unusual type of balance that allowed him to calculate the value of G. He attached a mass to each end of a small horizontal bar and, on a fine wire, suspended it sideways between two huge masses. Even though the force of gravity between the masses was very small, it was enough to make the bar pivot on the wire; gravity was trying to get all four masses were lined up. Now look at Newton's equation. Cavendish knew the size of the masses and the distance between them and the amount of twist in the wire told him the amount of gravitational force. So, he was able to solve the rearranged equation for G. The really neat thing is that you can use your own mass and weight with this same equation to find the mass of the Earth itself! The F is your weight (from W =mg), the d is the radius of the Earth (since you are standing on its surface), one m is your mass, you know the G and so, rearrange and solve for the other mass, that of the Earth.

Velocities and Periods Recalling the direction of the gravitational force in all these orbital motions is inward, i.e., centripetal, a bit of substitution for the centripetal force from the previous unit on two dimensional motion gets us to a couple of related equations, one to find the orbital velocity and another to find the orbital period. The equation to find the orbital velocity is: v = (GMe / r). To find the orbital period, use: T = 2 ( r ³ / GMe)

Find the orbital velocity of a communications satellite orbiting at an altitude of 750 km.

v = (GMe / r)

(6.67 * 10 ^{-- 11} Nm ² / kg ² * 5.98 * 10 ²⁴ kg / 6.37 * 10 ⁶ m + 7.50 * 10 ⁵ m)

(3.99 * 10 ¹⁴ / 7.12 * 10 ⁶) = 7 485.94 m/s

Find the orbital period of a weather satellite looking down from an altitude of 800 km.

T = 2 ( r ³ / GMe)

2 ( 6.37 * 10 ⁶ m + 8.00 * 10 ⁵ m) ³ / 6.67 * 10 ^{-- 11} Nm ² / kg ² * 5.98 * 10 ²⁴ kg)

2 (3.68 * 10 ²⁰ / 3.99 * 10 ¹⁴)

2 9.22 * 10 ⁵ = 6034.17 s

Weightlessness Are astronauts weightless? When the Earth's gravity pulls us down against an unyielding surface, our muscles give us a sense of weight. Think of your tired legs and aching feet after a long day of standing. In a situation where the supporting surface can move away from us, the effect of gravity is less and our sense of weight is less. On carnival rides that descend rapidly, the pull of gravity seems less because the supporting surface is moving down quickly. We notice our suddenly reduced weight as a sensation in our stomach. Astronauts have a sense of weightlessness because their space craft or space station is in constant freefall, i.e., their supporting surface is constantly falling down from under them and so they have no surface on which to stand and feel the sensation of weight. The same applies to parachutists during the freefall phase of their descent. We can feel just a bit of this weightlessness for the few seconds it takes an elevator to begin its descent. No gravity in space is a myth -- gravity is always present, although in the deepest reaches of space its effect would be essentially zero. To find the gravitational acceleration at some point in space above the Earth, use: g = GMe / d ².

Find the acceleration of gravity at a point 1 200 km above the Earth.

g = GMe / d ²

6.67 * 10 ^{-- 11} Nm ² / kg ² * 5.98 * 10 ²⁴ kg / 6.37 * 10 ⁶ m + 1.20 * 10 ⁶ m

3.99 * 10 ¹⁴ / (7.57 * 10 ⁶) ² = 6.96 m/s²

By the way, are astronauts massless in space? Recall mass is a measure of the amount of matter, so no, astronauts have the same mass everywhere.

The Impact of Gravity Gravity is the universe's most far reaching force and keeps everything moving in its predictable path. Computers can look back to see the origins of comets and ahead to their ultimate fate bashing into the surface of some planet or star. One way to model the effect of gravity is to imagine space as a rubber sheet. Placed here and there on the sheet are balls whose size depends on the mass of the matter they represent. And, the size of the dimple they create in the rubber sheet, called a gravity well, also depends on the mass of the matter they represent. So, massive objects create deep dimples while small masses hardly make a dent. Now, roll a ball over the sheet, perhaps it is a comet travelling through space, and see where it goes. If it passes by one of the deeper gravity wells, it may become trapped in an ever smaller orbit and eventually crash. Passing by a smaller gravity well may just alter its speed and direction (velocity). Measuring the amount a distant star bends the path of light from even more distant stars beyond indicates its mass. A star bending light in this way is acting as a gravity lens. Black holes are the ultimate gravity traps -- they are so massive, all surrounding matter and energy seem to be sucked into them, makings them even more massive. What happens to the matter and energy then?