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Chapter 9: Newton's Universal Law of Gravity

Difficulty Level: At Grade Created by: CK-12

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 \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 \begin{align*}r^3\end{align*} between the sun and the planet. That is, \begin{align*}T^2 \propto r^3\end{align*} or \begin{align*}T^2 = kr^3\end{align*} where \begin{align*}k\end{align*} equals \begin{align*}\frac{4\pi^2}{GM}\end{align*} and \begin{align*}M\end{align*} is the mass of the central body \begin{align*}\left(T^2 = \frac{4\pi^2}{Gm} r^3\right)\end{align*}. If \begin{align*}T\end{align*} is expressed in years and \begin{align*}r\end{align*} in astronomical units than \begin{align*}k = 1\end{align*} and \begin{align*}T^2 = r^3\end{align*}
  2. The Universal Law of Gravity The force \begin{align*}F\end{align*} between two objects is directly proportional to the product of their masses, \begin{align*}m_1m_2\end{align*}, and inversely proportional to the square of the distance, \begin{align*}r^2\end{align*}between their centers:

    \begin{align*}F = \frac{Gm_1m_2}{r^2}\end{align*}

    where \begin{align*}G\end{align*} is the universal gravitational constant equal to \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 \begin{align*}m\end{align*} is \begin{align*}g = \frac{Gm}{r^2}\end{align*} where \begin{align*}m\end{align*} is the mass that creates the gravitational acceleration and \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 \begin{align*}F = \frac{kq_1q_2}{r^2}\end{align*}.

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Jun 27, 2013
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Jun 07, 2016
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