Chapter 8.1 Notes
Kepler's laws and Universal Gravitation
Kepler's Three Laws of Planetary Motion
Johannes Kepler (1571-1630), using data he stole from his one-time mentor Tycho Brahe, waged his own private "war on Mars" and discovered that the shape of Mars' orbit was not a perfect circle, but an ellipse. He then managed to deduce that the orbits of all the planets were also ellipses. It was fortunate that Kepler worked with Mars first, since its orbit was the most eccentric of the planets known at the time, and its secret was therefore the most discoverable! This concept became Kepler's First Law, the Law of Ellipses.
Further, Kepler determined that as a planet moves through its orbit, the area of the ellipse swept out by the radius of orbit in a given time period is equal to the area swept out by the radius of orbit at any point in the orbit during an equal amount of time. This became Kepler's Second Law, the Law of Equal Areas.
Finally Kepler determined that the ratio of the Period of Orbit (T) squared to the Radius of Orbit (r) cubed is a constant for any satellite of a central mass in an orbital system. For example,
1. using the period T of the Earth's orbit around the Sun (1 year) and the average radius of the Earth's orbit around the Sun (1.49 x 1011 meters) produces the same value (3.0 x 10-34 yr2/m3)as when the orbital information for Jupiter is used!
2. , and so on. So, for any system, = k, where k is a constant for that system.
This is Kepler's Third Law, the Law of Harmonies. We will see the derivation of the mathematical representation of this law shortly.
If you know the Period and radius of orbit for one planet, and the Period of orbit for another planet, you can easily find the radius of orbit for the second.
Similarly, when you use data for satellites of the Earth itself, a constant will also be obtained for any satellite of the Earth.
This will be a different constant from that obtained for the Sun.
Video: Kepler's Laws Of Motion, from the Mechanical Universe Series.
Activity: The Orbit, page 179. Come to class with an ellipse already drawn as shown in the activity and the video.
Homework: Set 1: Chapter 8 questions 1-8 set #2 problems 31-32, 34-39
Newton realized that the force that caused apples to fall toward the earth also causes the moon to fall toward the earth.
The force the earth exerts on the apple and the force the earth exerts on the moon is given by the same equation, known as Newton's Law of Universal Gravitation.. The law is expressed mathematically as
Now, if we combine Newton's Law of Universal Gravitation with his Second Law of Motion as it applies to objects in centripetal motion, we come up with something quite interesting. Remember that planets around the Sun, for example, are displaying centripetal motion. That means that they are being centripetally accelerated according to
6. just like the rubber stopper in chapter 7.
But in this case, the planet is not connected by a string to the Sun. Soooo... what is supplying the necessary centripetal force that keeps the planets in their orbits?
GRAVITY...gravity according to Newton's Universal Law of Gravitation!
7. and the centripetal force on an object in circular motion was (from chapter 7) is
8. where m was the mass of the object moving in the circle , r is the radius of its orbit, and T is its period of orbit. This tells us that
Lets talk specifically about the Earth orbiting around the Sun here. On the left side, m is still the mass of the object in orbit, the Earth. There are two masses on the right side. Let m1 be the mass of the Sun and m2 be the mass of the Earth. Remember... the Universal Law of Gravitation is the gravitational force of attraction between 2 masses, and here this force is supplying the centripetal force keeping Earth in its orbit around the Sun. Also, r and d represent the same value, the radius of orbit of the Earth around the Sun.
Notice from equation 9. above that the mass of the Earth cancels out of the equation, leaving (and I'm replacing d with r here, since they represent the same value) this equation:
10. . (note that this gives us a new value for ac, which we will use later.
With a little rearranging I get
Now.. before we go any further.. back to the Law of Universal Gravitation (formula #5 above) for a second. The constant G does not depend at all upon what masses are being considered, where they are in relationship to one another, or what their state of motion is. Think about a person standing on the Earth's surface. Is there a force of gravitational attraction between the Earth and the person (masses m1 and m2, respectively)? Of course there is, and the Universal Law can be used to calculate that force.
You try: Calculate the force of attraction between the Earth and you!
Henry Cavendish (1731-1810) knew about this result also. He knew that
12. = mg. m is the same as m2 in the Universal Law equation, and m2 cancels out of both sides leaving
The value of g (-9.8 m/s2)was already known at Cavendish's time, thanks to Gallileo, and Eratosthenes had discovered the radius of the Earth a long time before. Cavendish himself determined the value of G experimentally. The only unknown in the expression is m1, the mass of the Earth, which Cavendish simply solved for.
You try: "Weigh the Earth" by solving formula 13 for m1. Compare your answer to the value stated in your textbook.
Chapter 8 activities:
Videos: "Kepler's Laws
Activity: The Orbit.