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# Chapter 9: Newton's Universal Law of Gravity

Difficulty Level: At Grade Created by: CK-12

Credit: Courtesy of NASA
Source: http://commons.wikimedia.org/wiki/File:Solar_system.jpg

We know gravity as the force pulling downwards on everyday objects.  The principles of gravity, however, apply on a much larger scale, and were discovered from study of the solar system.  This chapter will cover Kepler's Laws of Planetary Motion, Newton's Universal Law of Gravity, and the mechanics of circular orbits.

## Chapter Outline

### Chapter Summary

1. Kepler’s Three Laws of Planetary Motion are:
1. The orbital paths of the planets about the sun are ellipses with the sun at one focus.
2. If an imaginary line is drawn from the sun to a planet as the planet orbits the sun, this line will sweep out equal areas in equal times. (A planet moves faster when it is closer to the sun and slower when it is farther away from the sun.)
3. The square of the time T2\begin{align*}T^2\end{align*} for the orbital period of a planet about the sun is proportional to the cube of the average distance r3\begin{align*}r^3\end{align*} between the sun and the planet. That is, T2r3\begin{align*}T^2 \propto r^3\end{align*} or T2=kr3\begin{align*}T^2 = kr^3\end{align*} where k\begin{align*}k\end{align*} equals 4π2GM\begin{align*}\frac{4\pi^2}{GM}\end{align*} and M\begin{align*}M\end{align*} is the mass of the central body (T2=4π2Gmr3)\begin{align*}\left(T^2 = \frac{4\pi^2}{Gm} r^3\right)\end{align*}. If T\begin{align*}T\end{align*} is expressed in years and r\begin{align*}r\end{align*} in astronomical units than k=1\begin{align*}k = 1\end{align*} and T2=r3\begin{align*}T^2 = r^3\end{align*}
2. The Universal Law of Gravity The force F\begin{align*}F\end{align*} between two objects is directly proportional to the product of their masses, m1m2\begin{align*}m_1m_2\end{align*}, and inversely proportional to the square of the distance, r2\begin{align*}r^2\end{align*}between their centers:

F=Gm1m2r2

where G\begin{align*}G\end{align*} is the universal gravitational constant equal to G=6.67×1011Nm2kg2\begin{align*}G = 6.67 \times 10^{-11} \frac{N*m^2}{kg^2}\end{align*}.

3. The gravitational acceleration near a massive body of mass m\begin{align*}m\end{align*} is g=Gmr2\begin{align*}g = \frac{Gm}{r^2}\end{align*} where m\begin{align*}m\end{align*} is the mass that creates the gravitational acceleration and r\begin{align*}r\end{align*}is the distance from the center of the planet to a point outside the planet.
4. The electrostatic force between two charged bodies is F=kq1q2r2\begin{align*}F = \frac{kq_1q_2}{r^2}\end{align*}.

1. [1]^ Credit: Courtesy of NASA; Source: http://commons.wikimedia.org/wiki/File:Solar_system.jpg; License: CC BY-NC 3.0

Jun 27, 2013