1. Kepler's Laws of Planetary Motion

Topic #8. Universal Gravitation

2. Universal Gravitation

3. Newton's Test of the Inverse Square Law

4. Cavendish Experiment

5. Law of Universal Gravitation and Weight

6. Gravitational Fields

7. Einstein's Concept of Gravity

Notes should include:

Kepler's Laws of Planetary Motion: Back in the second half of the 16th century (the 1500's), a man by the name of Tycho Brahe had a strong interest in astronomy. He also was well connected at court and King Frederick II of Denmark provided him with an island and the means to build what was considered among the best observatories of the time. Though telescopes were not available, there was equipment much like very large protractors mounted on mechanically operated machines that allowed for visible objects to be found and tracked with a great degree of precision. Brahe carefully recorded the positions of the visible planets and stars for over 20 years, recording all of his observations in the form of measurements that described the location of each object in the sky. An assistant of his by the name of Johannes Kepler was an astute mathematician as well as being a polish monk. He took Brahe's data and worked with it for about 22 years. His work resulted in the formulation of three laws.

Kepler's Three Laws of Planetary Motion:

1st law. The paths that planets follow are in the shape of ellipses.

2nd law. An imaginary line drawn from any planet to the sun sweeps out equal areas in

equal amounts of time.

3rd law. The ratio of the average of a planet's orbit's radius about the sun cubed and the planet's period squared is a constant for all planets in the solar system. Mathematically the equation is k = r3 / T2.

Universal Gravitation: Brahe's observations and Kepler's mathematics was the basis for Isaac Newton's theory of universal gravitation. Whether it is an actual event or not, it is said that Newton observed an apple to fall from a tree. He recognized that this occurred because an unbalanced force (his second law) must be acting on the apple. He asked the question as to whether all objects exert a force of gravity. He hypothesized that the moon was kept in its orbit because of the force of gravity and was actually falling towards the earth all of the time just as the apple fell to earth, but unlike the apple, the moon had forward motion, which kept it from actually colliding with the earth. The moon moved forward in proportion to the amount of distance it fell each second. The following equation shows what Newton was thinking. He redefined the equation for centripetal force as the equation Fc = m K / r2, where m is the mass of an object, K is a numerical constant, and r is the radius of the orbit around which the object is traveling. This equation tells us that the force varies directly with the mass of an orbiting object and the force varies inversely with the square of the radius of the orbit.

In terms of planetary movement and the force acting between the sun and a planet orbiting the sun, he realized that both the sun and the planet exerted gravitational force on one another, although the sun's greater mass had a greater impact on the force between the sun and the planet than did the mass of the planet. Newton generalized from his thinking about planets to formulate his law of universal gravitation. The law says that every body in the universe attracts every other body in the universe with a force that varies directly with the product of the masses and inversely with the square of the distance between them. Mathematically, he saw the situation in terms of the following relationship: Fg m1 m2 / d2. (, when used as a math symbol, means proportional to)

Newton's Test of the Inverse Square Law: Though equipment did not exist as yet to test his idea experimentally, Newton decided to apply his idea to the moon. According to his idea the gravitational acceleration of the earth on the moon should vary inversely with the square of the distance between the earth and the moon. Because the moon's orbit is close to being circular (remember orbits tend to be elliptical, though usually with very low eccentricities) it is logical to think that the earth's gravitational acceleration at a distance equal to the average of the moon's orbit's radius should equal the centripetal acceleration experienced by the moon as it travels around the earth. Newton did the computations and found a close agreement between the value he predicted by using the inverse square law and the value obtained by using the centripetal acceleration equation. This is hard evidence supported the inverse square law portion of his universal law of gravitation.

Cavendish Experiment: Almost one hundred years after Newton introduced the concept of the inverse square law, a scientist by the name of Henry Cavendish confirmed Newton's law of universal gravitation. He did this by inventing a device to measure small amounts of gravitational force. He suspended a rod from a support by means of a thin wire. Attached to the ends of the rods were two lead spheres. He then carefully measured the amounts of forces necessary to rotate the wire through various angles. Next he took two large lead spheres and placed one each near one of the lead spheres mounted on the rod. The two small spheres on the ends of the rods were attracted to the large lead spheres. This caused the wire to twist. By measuring the angles through which the wire twisted for various trials, he was able to determine the force between these lead spheres. By substituting the values for force, mass, and distance into the law of universal gravitation, he figured out the equation for calculating gravitational force. The equation Fg = G m1 m2 / d2 has a value for G of 6.67 x 10-11 N m2 / kg2, when mass is measured in kilograms, and distance is measured in meters. G is referred to as the proportionality constant for Newton’s Law of Universal Gravity. Newton wasn’t able to determine this value because at the time there was no device for measuring actual gravitational force. 100 years later Cavendish invented such a device and Newton’s proportionality became an equation.

Law of Universal Gravitation and Weight: The force that causes an object to fall towards earth is called gravitational force. We usually call this force weight. This complies with the law of universal gravitation. Near the earth's surface the weight is equal to the gravitational force between an object and the earth. Thus W = Fg = G m1 m2 / d2. And so, because W = m g, we can rewrite the equation so it becomes m1 g = G m1 m2 / d2. The mass of the object is m1 and the mass of the earth is m2. If we solve this equation for g , we get g = G m2 / d2. For bodies very near the earth's surface the constant, G, the mass of the earth, and the distance to the earth is essentially constant, so we say that g, the acceleration due to gravity, is a constant. Its value is often reported as being 9.80 m/s2. However, in general, as you move away from the earth the distance does change, so you would use the more general equation to find the acceleration due to gravity. The equation is g' = G m2 / d2 or g' = G me / d2. We would expect g' to decrease as the square of d increased.

Examples of the g value:

Gravitational Fields: To describe forces acting at a distance, science defined the term field. A field describes an effect experienced by an object because of its position with respect to some reference point. For example, if your reference point is the sun, theoretically every position around the sun as you move outward through three dimensional space has a g' vector. The value of this acceleration due to gravity vector is a function of the sun's mass and the distance between the sun and the position for which you are expressing the g' value. The equation can best be stated as the expression g' = F / m = G msun / d2.

Einstein's Concept of Gravity: The law of universal gravitation does not explain the nature of gravitational force. It really only tells us how to use information to calculate the strength of the force. Einstein hypothesized that gravity is a phenomenon of space and time. He suggested that mass, when present, somehow alters space itself. Einstein's general theory of relativity suggests that mass be considered a disturbance in the fabric of space and time rather than a force. Astronomers have found it easier to explain astronomical phenomenon in terms of Einstein's theory than to try to explain things in terms of Newton's law of universal gravity, where the presence of mass, produces a force. An idea for model of how mass is a disturbance in space and yet creates an observable phenomenon, which manifests itself as a force comes to us in the form a stretched rubber sheet. Imagine a large rubber sheet stretched tightly over a wood or metal frame. Now imagine a mass like a bowling ball being placed on the rubber sheet. You would see a depression in the sheet caused by the bowling ball. The more massive the bowling ball the deeper and wider the depression. Now imagine taking a much less massive ball like a billiard (pool) ball and rolling it across the rubber sheet so it moves across the depression caused by the bowling ball. Imagine for a minute what might happen.

Assuming no significant amount of friction was present, the billiard ball might:

1. Have enough velocity to roll across the depression, being turned to some extent, but have enough velocity to leave the depression and continue traveling.

2. Have too little velocity to escape the depression, but enough velocity to keep circling (orbiting) the bowling ball.

3. Have too little velocity to even keep circling the bowling ball and spiral down into the depression until it finally collides with the bowling ball.

Much in the same way that the billiard ball in statement two continues to orbit the bowling ball as it moves with fairly constant velocity (it might have entered an elliptical orbit, where its velocity changes periodically) around the bowling ball because of the depression the mass caused in the surface of the sheet, so too the earth and other planets orbit the sun, because of the distortion caused in space by the masses of the sun and the planets. It has been observed that gravity bends light rays. This is evidence that mass bends space around an object and the light merely traveling through space has to follow the curved path produced by the distortion of space by the presence of mass. An extreme case of mass distortion of space by mass is a black hole. The mass appears to be so dense in a black hole that the disturbance has no bottom. To express it another way, the disturbance is so great that the "depression" created by the mass that formed the black hole is infinitely deep. This is supported by the fact that light passing into the "depression" of the black hole is curved and changes direction; it appears to disappear into the black hole, never to emerge.

Vocabulary: Kepler’s First Law of Planetary Motion, Kepler’s Second Law of Planetary Motion, Kepler’s Third Law of Planetary Motion, Law of Universal Gravitation, gravitational force, inertial mass, gravitational mass, ellipse, eccentricity, planet, moon, asteroid, comet, meteor, star, black hole, galaxy.

Skills to be learned:

Solve problems involving Kepler's Laws of Planetary Motion

Solve problems involving the Law of Universal Gravitation

Solve problems involving the Law of Universal Gravitation and Weight

Solve problems involving g' values at different positions in a gravity field

Solve problems involving the launch velocity of a satellite

Assignments:

Textbook: Read / Study / Learn Chapter 8 Kepler’s Laws, Newton’s Law of Universal Gravitation and Einstein’s Theory of Gravity

Workbook Exercises: PS#8-1 and PS#8-2

Activities: TBA

Resources:

This Handout and the Overhead and Board Notes discussed in class

Textbook: Chapter 8

WB Lessons and Problem Sets

- “Study of Universal Gravitation”

 / Universal Gravitation