PHY 2048C General Physics I with lab Spring 2011 CRNs 11154, & Dr. Derrick Boucher Assoc. Prof. of Physics Session 18, Chapter 13

Chapter 13 Homework Due Tuesday midnight was Monday 3/21

Chapter 13 Practice Problems 13, 15, 17, 19, 27, 29, 31, 39, 47, 49, 51 Unless otherwise indicated, all practice material is from the “Exercises and Problems” section at the end of the chapter. (Not “Questions.”)

Chapter 13. Newton’s Theory of Gravity The beautiful rings of Saturn consist of countless centimeter- sized ice crystals, all orbiting the planet under the influence of gravity. Chapter Goal: To use Newton’s theory of gravity to understand the motion of satellites and planets.

Topics: History and Newton Newton’s Law of Gravity Little g and Big G Gravitational Potential Energy Satellite Orbits and Energies Chapter 13. Newton’s Theory of Gravity

Newton proposed that every object in the universe attracts every other object. Newton’s Law of Gravity

The constant G, called the gravitational constant, is a proportionality constant necessary to relate the masses, measured in kilograms, to the force, measured in newtons. In the SI system of units, G has the value 6.67 × 10 −11 N m 2 /kg 2. Newton’s Law of Gravity

Suppose an object of mass m is on the surface of a planet of mass M and radius R. The local gravitational force may be written as Little g and Big G where we have used a local constant acceleration: On earth near sea level it can be shown that g surface = 9.80 m/s 2.

Example, Problem 12-22, p. 405

When two isolated masses m 1 and m 2 interact over large distances, they have a gravitational potential energy of Gravitational Potential Energy where we have chosen the zero point of potential energy at r = ∞, where the masses will have no tendency, or potential, to move together. Note that this equation gives the potential energy of masses m 1 and m 2 when their centers are separated by a distance r. Not on equation sheet (will be test time)

Example, Problem 12-30, p. 405

Example, Problem 12-28a, p. 405

Clicker Problem 12-38, p. 406 A binary star system has two stars, each with the same mass as our sun, separated by 1.0x10 12 m. A comet is very far away and essentially at rest, when it is attracted to the stars. Suppose the comet travels along a line that bisects the distance between the two stars. What is its speed as it passes between them?

Example, Problem 12-47, p. 406

Clicker Problem Using the planetary data from the back cover of the textbook, find the location where a spacecraft experiences an equal but opposite gravitational force from the Earth and the Moon. Give your answer as a ratio of the distance from the from the center of the Earth divided by the distance from the center of the Moon.

Special applications: orbits All these are special case equations: use for homework but always try to start from “scratch” and use energy and force concepts with the new quantities U G, F G

EXAMPLE 13.2 Escape speed Escape speed ASSUMES that the object leaves the surface of the planet at that speed and receives no other propulsion after that initial kick.

Satellite Orbits The mathematics of ellipses is rather difficult, so we will restrict most of our analysis to the limiting case in which an ellipse becomes a circle. Most planetary orbits differ only very slightly from being circular. If a satellite has a circular orbit, its speed is

Orbital Energetics We know that for a satellite in a circular orbit, its speed is related to the size of its orbit by v 2 = GM/r. The satellite’s kinetic energy is thus But −GMm/r is the potential energy, U g, so If K and U do not have this relationship, then the trajectory will be elliptical rather than circular. So, the mechanical energy of a satellite in a circular orbit is always:

Chapter 13. Clicker Questions

A satellite orbits the earth with constant speed at a height above the surface equal to the earth’s radius. The magnitude of the satellite’s acceleration is A. g on earth. B. g on earth. C. g on earth. D. 4g on earth. E. 2g on earth.

A. one quarter as big. B. half as big. C. the same size. D. twice as big. E. four times as big. The figure shows a binary star system. The mass of star 2 is twice the mass of star 1. Compared to, the magnitude of the force is

A planet has 4 times the mass of the earth, but the acceleration due to gravity on the planet’s surface is the same as on the earth’s surface. The planet’s radius is A. R e. B. R e. C. 4R e. D. R e. E. 2R e.

In absolute value: A. U e > U d > U a > U b = U c B. U b > U c > U d > U a > U e C. U e > U a = U b = U d > U c D. U e > U a = U b >U c > U d E. U b > U c > U a = U d > U e Rank in order, from largest to smallest, the absolute values |U g | of the gravitational potential energies of these pairs of masses. The numbers give the relative masses and distances.

Two planets orbit a star. Planet 1 has orbital radius r 1 and planet 2 has r 2 = 4r 1. Planet 1 orbits with period T 1. Planet 2 orbits with period A. T 2 = T 1. B. T 2 = T 1 /2. C. T 2 = 8T 1. D. T 2 = 4T 1. E. T 2 = 2T 1.

Gravity simulators: physik.de/medienserver/Mediafiles/Applets/xchaos5/anders/Gravitation. html