AP Physics 1 Lesson 10.a Law of Universal Gravitation Homework Outcomes

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1 AP Physics 1 Lesson 10.a Law of Universal Gravitation Homework Outcomes 1. Use Law of Universal Gravitation to solve problems involving different masses. 2. Determine changes in gravitational and kinetic energy for orbiting bodies 3. Analyze changes to the gravitational force using the linear relationship between the force and the product of the masses and the inverse relationship between the force and the distance between the masses. Name: Date: 1. Two sacks contain the same number of identical apples and are separated by a distance r. The two sacks exert a gravitational force on each other that is opposite in direction but with the same magnitude, F. Half the apples are removed from one sack and placed in the second sack. What is the magnitude of the gravitational force between the two sacks in terms of the original force, F? A) ½ F B) ¾ F C) F D) 3/2 F 2. An object experiences an acceleration, g, when it is on the surface of a planet of radius R. What will be the acceleration on the object after it has been moved to a distance of 4R from the center of the planet? A) 16 g B) 4 g C) ¼ g D) 1/16 g 3. The radius of an asteroid s orbit about the Sun is approximately two times the radius of Earth s orbit about the Sun. Measured in Earth years, the time for the asteroid to make one complete revolution about the Sun is most nearly A) 1.4 years B) 2.0 years C) 2.8 years D) 8.0 years 4. A student whirls a stopper tied to the end of a string in a horizontal circle of radius, r, for ten complete revolutions. Which of the following diagrams best represents the directions of the velocity and the acceleration of the stopper at the instant that it is at position A, as shown? 5. Most commercial satellites operate in geostationary orbits. A geostationary orbit is an orbit whose position in the sky remains the same for a stationary observer on Earth. However, some satellites operate at medium Earth orbits or low Earth orbits, and have periods that are less than 24 hours. Four satellites are in a circular orbit around Earth, well outside the atmosphere. The mass (m) and orbital radius (r) of each satellite are given in the provided data table. Which satellite has the greatest speed? 1

2 A) A B) B C) C D) D 6. The two spheres pictured have equal densities and are subject only to their mutual gravitational attraction. Which of the following quantities must have the same magnitude for both spheres? A) Acceleration B) Velocity C) Inertia D) Gravitational Force 7. Two planets have the same size but different masses, and no atmospheres. Which of the following would be the same for objects with equal mass on the surfaces of the two planets? A) The rate at which each object would fall freely B) The gravitational force exerted on each object C) The amount of mass each object would balance on an equal-arm balance D) The weight of the objects 8. A student wishes to calculate the acceleration of a stopper tied to the end of a string. She whirls the stopper at constant speed in a horizontal circular path of radius R. She measues and records the value of R. The stopper completes ten complete revolutions in a time interval Δt. The student calculates the time period, T, for one complete revolution. What additional information is needed to calculate the centripetal acceleration of the stopper? A) No additional information B) The mass of the stopper C) The tension in the string D) The frequency of the stopper 9. A person weighing 800 N on Earth travels to another planet that has twice the mass and twice the radius of Earth. The person's weight on this other planet is most nearly 2

3 10. Satellite A has mass M. Satellites A and B have circular orbits around a planet with distances of R and 2R, respectively. The gravitational force between the planet and satellite A is F. The mass of satellite B is 3M. In terms of F, what is the gravitational force between the planet and satellite B? A) ¼ F B) 2/3 F C) ¾ F D) 3/2 F 11. The planet Mars is host to five functioning spacecraft, three in orbit about the planet and two on the surface of the planet. Thanks to those spacecraft, we know that the planet Mars has a mass that is 0.11 times that of Earth and a radius that is 0.53 times that of Earth. The acceleration of an object in free-fall near the surface of Mars is most nearly what in terms of the local value of g on Earth? A) Zero B) 0.10 g C) 0.19 g D) 0.39 g 12. A satellite is moving around a planet in a circular orbit, radius R, measured from the center of the planet. The satellite falls out of this orbit and establishes a new orbit at radius 1/4 R. What additional information is needed to calculate the speed of the satellite in its new orbit? A) No additional information needed B) The planet s radius C) The planet s mass D) The satellite s mass 3

4 13. Two spacecraft are 45,500 km apart and heading toward a large planet. The spacecraft plan to rendezvous and orbit the planet to determine its mass and size. Spacecraft Alpha, with a mass of 3400 kg, and spacecraft Beta, with a mass of 1700 kg, plan a circular orbit around the planet at a radius of m from the planet s center. When both spacecraft are in the same circular orbit around the planet (select two answers.), A) Alpha experiences twice the centripetal acceleration as Beta. B) Alpha experiences the same centripetal acceleration as Beta C) Alpha experiences twice the gravitational force as Beta. D) Alpha experiences the same gravitational force as Beta. 14. In which of the following situations would the object be accelerated? Select two answers. A) A block slides down an inclined plane at a constant speed. B) A toy car moves in a straight line at constant speed. C) A satellite moves in a circle at constant speed around the Earth. D) A ball moves in a parabolic path in a gravitational field with neglibible air resistance. 15. A racing car is moving around the circular track of radius 300 m as shown in the diagram. At the instant when the car's velocity is directed due east, its acceleration is directed due south and has a magnitude of 3 m/s 2. When viewed from above, during one complete lap around the track the car is moving (select two answers.) A) Clockwise B) Counterclockwise C) With a changing speed D) With a constant speed 16. A carnival merry-go-round rotates around a vertical axis at a constant rate. Two horses are attached to the rotating merry-go-round with the red horse placed at a shorter distance from the center of the merry-go-round than the blue horse. Which of the following statements are true concerning both horses? Select two answers. A) The blue horse has a greater period than the red horse. B) The blue horse and the red horse have equal periods. C) The blue horse has a greater speed than the red horse. D) The blue horse and the red horse have equal speeds. 17. A motorcycle passes over the top of a hill that has a radius of curvature of 100 m. The mass of the motorcycle plus rider is 300 kg. The motorcycle is moving at a speed of 30 m/s. The surface exerts a normal force of magnitude Fn on the motorcycle. The motorcycle passes over the top of the hill again but now is moving at a speed of 33 m/s. How does the normal force exerted on the motorcycle compare to Fn? A) greater than Fn B) less than Fn C) equal to Fn D) greater or less depending on the gravitational force 4

5 18. A student wishes to calculate the acceleration of a stopper tied to the end of a string. She whirls the stopper at constant speed in a horizontal circular path of radius R. She measues and records the value of R. The stopper completes ten complete revolutions in a time interval Δt. The student calculates the time period, T, for one complete revolution. What additional information is needed to calculate the centripetal acceleration of the stopper? A) No additional information B) The mass of the stopper C) The tension in the string D) The frequency of the stopper Use the following information for questions 19. And 20. A ball on the end of a string is being swung in a vertical circle, rotating clockwise as shown in the diagram. Position I is at the top of the circle and position III is at the bottom of the circle. 19. Rank the magnitudes of the centripetal force exerted on the ball at locations I, II, and III. A) I > II > III B) II > I = III C) III > II > I D) I = II = III 20. The ball has a mass m and a speed v as it moves around the vertical circle of radius r. Which of the following expressions can be used to find the minimum speed of the ball at position I such that the circular path is maintained? Two spherical planets of radii R 1 and R 2 such that R 2 = 2 R 1 and average densities ϱ 1 and ϱ 2. The accelerations due to gravity a 1 and a 2 on the two planets are related by which of the following? 5

6 Two objects are separated by a distance r. If the gravitational force on the two is decreased to onethird its original magnitude without changing either mass, then the separation distance must become A coin of mass m is placed on a vinyl stereo record of radius R and moves at a constant tangential speed v. If the frictional force between the coin and the vinyl record is at its maximum value, then which of the following expressions can be used to find the value of the coefficient of friction between the coin and the vinyl record? 6

7 Free Response 1. (a) Qualitatively discuss why the satellite is continually in free-fall as it orbits the planet. Justify your answer in a clear, coherent, paragraph-length explanation. (b) Quantitatively determine the radius of revolution, r, of the satellite in terms of the given quantities and any fundamental constants. (c) What is the relationship between the radius of revolution and the mass of the satellite that is in freefall around the planet? Justify your response. (d) Assume the acceleration due to gravity g at a distance r from the center of the planet of mass M is 9 m/s 2. In terms of the radius of revolution r, what would the speed of the satellite have to be to remain in a circular orbit around this planet at this distance? 7

8 (e) The satellite of mass 1000kg travels in a circular orbit a distance of 1.7x10 6 m above the Earth s surface. M e =5.98x10 24 kg and r e =6.3x10 6 m. What is the potential energy of the satellite while in this orbit? (f) Determine the total mechanical energy of the satellite while in this orbit. (Assume that the gravitational potential energy is zero at an infinite distance from the Earth.) Suppose the satellite is launched from a location on the Earth s equator. (g) Determine the speed that the satellite has due to the Earth s rotation while it is located at the Earth s equator. (Remember the period of the Earth must be in seconds.) (h) Determine the kinetic energy that the satellite has due to the Earth s rotation while it is located at the Earth s equator. (i) Determine the potential energy of the satellite when it is located at the Earth s equator. (j) Determine the total mechanical energy of the satellite at the Earth s equator. (k) Compare the total mechanical energy of the satellite while in orbit with its energy when located at the Earth s equator. (Compare your answers to f. with j.) Is energy conserved? Justify your answer. (l) Why are satellites launched near the equator rather than further north or south toward the poles of the Earth? 8

9 (m) Qualitatively discuss what it would mean if the total mechanical energy of the satellite was greater than or equal to zero. 2. The figure shown represents the path of a planet as it orbits the Sun. The planet follows an elliptical path, but unlike the ellipse followed by a pendulum, the Sun is at a focal point of the ellipse and not at the center of the elliptical path. The planet is moving from point A to point E as shown in the figure. (a) On the figure provided, draw an arrow showing the direction of the net force on the planet at points A, B, C, D, and E. If the net force is zero at any point, label the point with 0. (b) Rank the magnitudes of the gravitational force on the planet at points A, B, C, D, and E. Justify your answer. (c) Rank the magnitude of the velocities on the planet at points A, B, C, D, and E. Describe the motion of the planet from point A to point E. Justify your answer. 9

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