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Gravitation HW 1. A person weighing 800 newtons on Earth travels to another planet with twice the mass and twice the radius of Earth. The person's weight on this other planet is most nearly 2. Mars has a mass 1/10 that of Earth and a diameter 1/2 that of Earth. The acceleration of a falling body near the surface of Mars is most nearly (A) 0.25 m/s2 (B) 0.5 m/s2 (C) 2 m/s2 (D) 4 m/s2 (E) 25 m/s2 3. An object has a weight W when it is on the surface of a planet of radius R. What will be the gravitational force on the object after it has been moved to a distance of 4R from the center of the planet? (A) 16W (B) 4W (C) W (D) 4 (E) 1/16 W 4. A new planet is discovered that has twice the Earth's mass and twice the Earth's radius. On the surface of this new planet, a person who weighs 500 N on Earth would experience a gravitational force of (A) 125 N (B) 250 N (C) 500 N (D) 1000 N (E) 2000 N 5. A newly discovered planet has twice the mass of the Earth, but the acceleration due to gravity on the new planet's surface is exactly the same as the acceleration due to gravity on the Earth's surface. The radius of the new planet in terms of the radius R of Earth is 6. A spacecraft orbits Earth in a circular orbit of radius R, as shown above. When the spacecraft is at position P shown, a short burst of the ship's engines results in a small increase in its speed. The new orbit is best shown by the solid curve in which of the following diagrams? 7. The escape speed for a rocket at Earth's surface is ve. What would be the rocket's escape speed from the surface of a planet with twice Earth's mass and the same radius as Earth? 8. Two artificial satellites, 1 and 2, are put into circular orbit at the same altitude above Earth’s surface. The mass of satellite 2 is twice the mass of satellite 1. If the period of satellite 1 is T, what is the period of satellite 2? (A) T/4 (B) T/2 (C) T (D) 2T (E) 4T 9. What happens to the force of gravitational attraction between two small objects if the mass of each object is doubled and the distance between their centers is doubled? (A) It is doubled (B) It is quadrupled (C) It is halved (D) It is reduced fourfold (E) It remains the same 10. Kepler’s Second Law about “sweeping out equal areas in equal time” can be derived most directly from which conservation law? (A) energy (B) mechanical energy (C) angular momentum (D) mass (E) linear momentum 11. Two stars orbit their common center of mass as shown in the diagram below. The masses of the two stars are 3M and M. The distance between the stars is d. What is the value of the gravitational potential energy of the two-star system? 12. When two stars are very far apart, their gravitational potential energy is zero; when they are separated by a distance d, the gravitational potential energy of the system is U. What is the gravitational potential energy of the system if the stars are separated by a distance 2d? (A) U/4 (B) U/2 (C) U (D) 2U (E) 4U 13. Two artificial satellites I and II have circular orbits of radii R and 2R, respectively, about the same planet. The orbital velocity of satellite I is v. What is the orbital velocity of satellite II? 14. A student is given the set of orbital data for some of the moons of Saturn shown below and is asked to use the data to determine the mass M of Saturn. Assume the orbits of these moons are circular. a. Write an algebraic expression for the gravitational force between Saturn and one of its moons. b. Use your expression from part (a) and the assumption of circular orbits to derive an equation for the orbital period T of a moon as a function of its orbital radius R. c. Which quantities should be graphed to yield a straight line whose slope could be used to determine Saturn's mass? d. Complete the data table by calculating the two quantities to be graphed. Label the top of each column, including units. e. Plot the graph on the axes below. Label the axes with the variables used and appropriate numbers to indicate the scale. f. Using the graph, calculate a value for the mass of Saturn. 15. In March, 1999 the Mars Global Surveyor (GS) entered its final orbit about Mars, sending data back to Earth. Assume a circular orbit with a period of 1.18 x 10 2 minutes = 7.08 x 103 s and orbital speed of 3.40 x 103 m/s. The mass of the GS is 930 kg and the radius of Mars is 3.43 x 10 6 m. (a) Calculate the radius of the GS orbit. (b) Calculate the mass of Mars. (c) Calculate the total mechanical energy of the GS in this orbit. (d) If the GS was to be placed in a lower circular orbit (closer to the surface of Mars), would the new orbital period of the GS be greater than or less than the given period? _________Greater than _________ Less than Justify your answer. (e) In fact, the orbit the GS entered was slightly elliptical with its closest approach to Mars at 3.71 x 10 5 m above the surface and its furthest distance at 4.36 x 10 5 m above the surface. If the speed of the GS at closest approach is 3.40 x 103 m/s, calculate the speed at the furthest point of the orbit. 16. A spacecraft of mass 1,000 kilograms is in an elliptical orbit about the Earth, as shown above. At point A the spacecraft is at a distance rA = 1.2 x 107 meters from the center of the Earth and its velocity, of magnitude vA = 7.1 x 103 meters per second, is perpendicular to the line connecting the center of the Earth to the spacecraft. The mass and radius of the Earth are M E = 6.0 X 1024 kilograms and rE = 6.4 X 106 meters, respectively. Determine each of the following for the spacecraft when it is at point A. a. The total mechanical energy of the spacecraft, assuming that the gravitational potential energy is zero at an infinite distance from the Earth. b. The magnitude of the angular momentum of the spacecraft about the center of the Earth. Later the spacecraft is at point B on the exact opposite side of the orbit at a distance r B = 3.6 X 107 meters from the center of the Earth. c. Determine the speed vB of the spacecraft at point B. Suppose that a different spacecraft is at point A, a distance r A = 1.2 X 107 meters from the center of the Earth. Determine each of the following. d. The speed of the spacecraft if it is in a circular orbit around the Earth e. The minimum speed of the spacecraft at point A if it is to escape completely from the Earth