Assignment 10 Due: 11:59pm on Sunday, March 21, 2010 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View] Properties of Circular Orbits Learning Goal: To teach you how to find the parameters characterizing an object in a circular orbit around a much heavier body like the earth. The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit--a circular one. This problem concerns the properties of circular orbits for a satellite orbiting a planet of mass . For all parts of this problem, where appropriate, use for the universal gravitational constant. Part A Find the orbital speed for a satellite in a circular orbit of radius . Hint A.1 Find the force Find the radial force on the satellite of mass . (Note that will cancel out of your final answer for .) Express your answer in terms of , , , and . Indicate outward radial direction with a positive sign and inward radial direction with a negative sign. ANSWER: = Correct Hint A.2 A basic kinematic relation Find an expression for the radial acceleration for the satellite in its circular orbit. Express your answer in terms of and . Indicate outward radial direction with a positive sign and inward radial direction with a negative sign. ANSWER: = Answer Requested Hint A.3 Newton's 2nd law Apply to the radial coordinate. Express the orbital speed in terms of , , and . ANSWER: = Correct Part B Find the kinetic energy of a satellite with mass in a circular orbit with radius . Express your answer in terms of , , , and . ANSWER: = Correct Part C Express the kinetic energy in terms of the potential energy . Hint C.1 Potential energy What is the potential energy of the satellite in this orbit? Express your answer in terms of , , , and . ANSWER: = Correct ANSWER: = Correct MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?ass... 1 of 8 4/17/10 5:31 PM This is an example of a powerful theorem, known as the Virial Theorem. For any system whose motion is periodic or remains forever bounded, and whose potential behaves as , Rudolf Clausius proved that , where the brackets denote the temporal (time) average. Part D Find the orbital period . Hint D.1 How to get started Hint not displayed Express your answer in terms of , , , and . ANSWER: = Correct Part E Find an expression for the square of the orbital period. Express your answer in terms of , , , and . ANSWER: = Correct This shows that the square of the period is proportional to the cube of the semi-major axis. This is Kepler's Third Law, in the case of a circular orbit where the semi-major axis is equal to the radius, . Part F Find , the magnitude of the angular momentum of the satellite with respect to the center of the planet. Hint F.1 Definition of angular momentum Recall that , where is the momentum of the object and is the vector from the pivot point. Here the pivot point is the center of the planet, and since the object is moving in a circular orbit, is perpendicular to . Express your answer in terms of , , , and . ANSWER: = Correct Part G The quantities , , , and all represent physical quantities characterizing the orbit that depend on radius . Indicate the exponent (power) of the radial dependence of the absolute value of each. Hint G.1 Example of a power law The potential energy behaves as , so depends inversely on . Therefore, the appropriate power for this is (i.e., ). Express your answer as a comma-separated list of exponents corresponding to , , , and , in that order. For example, -1,-1/2,-0.5,-3/2 would mean , , and so forth. ANSWER: -0.500,-1,-1,0.500 Correct Energy of a Spacecraft Very far from earth (at ), a spacecraft has run out of fuel and its kinetic energy is zero. If only the gravitational force of the earth were to act on it (i.e., neglect the forces from the sun and other solar system objects), the spacecraft would eventually crash into the earth. The mass of the earth is and its radius is . Neglect air resistance throughout this problem, since the spacecraft is primarily moving through the near vacuum of space. Part A Find the speed of the spacecraft when it crashes into the earth. Hint A.1 How to approach the problem Use a conservation-law approach. Specifically, consider the mechanical energy of the spacecraft when it is (a) very far from the earth and (b) at the surface of the earth. MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?ass... 2 of 8 4/17/10 5:31 PM Hint A.2 Total energy What is the total mechanical energy of the spacecraft when it is far from earth, at a distance ? ANSWER: = 0 Correct Hint A.3 Potential energy If the spacecraft has mass , what is its potential energy at the surface of the earth? Note that, at the surface of the earth, the spacecraft is a distance from the center of the earth. Hint A.3.1 Formula for the potential energy Hint not displayed Express your answer in terms of , , , and the universal gravitational constant . ANSWER: = Correct Express the speed in terms of , , and the universal gravitational constant . ANSWER: = Correct Part B Now find the spacecraft's speed when its distance from the center of the earth is , where . Hint B.1 General approach This problem is very similar to the problem that you've just done. Note that the potential energy of the spacecraft at a distance is different from its potential energy at the earth's surface. Hint B.2 First step in finding the speed Find the spacecraft's speed at in terms of , , , and . ANSWER: = Answer Requested Express the speed in terms of and . ANSWER: = Correct Gravitational Acceleration inside a Planet Consider a spherical planet of uniform density . The distance from the planet's center to its surface (i.e., the planet's radius) is . An object is located a distance from the center of the planet, where . (The object is located inside of the planet.) Part A Find an expression for the magnitude of the acceleration due to gravity, , inside the planet. Hint A.1 Force due to planet's mass outside radius Hint not displayed Hint A.2 Find the force on an object at distance Hint not displayed Hint A.3 Finding from Hint not displayed Express the acceleration due to gravity in terms of , , , and , the universal gravitational constant. ANSWER: = Correct MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?ass... 3 of 8 4/17/10 5:31 PM Part B Rewrite your result for in terms of , the gravitational acceleration at the surface of the planet, times a function of R. Hint B.1 Acceleration at the surface Note that the acceleration at the surface should be equal to the value of the function from Part A evaluated at the radius of the planet: . Express your answer in terms of , , and . ANSWER: = Correct Notice that increases linearly with , rather than being proportional to . This assures that it is zero at the center of the planet, as required by symmetry. Part C Find a numerical value for , the average density of the earth in kilograms per cubic meter. Use for the radius of the earth, , and a value of at the surface of . Hint C.1 How to approach the problem You already derived the relation needed to solve this problem in Part A: . At what distance is known so that you could use this relation to find ? Calculate your answer to four significant digits. ANSWER: = 5497 Correct Kepler's 3rd Law A planet moves in an elliptical orbit around the sun. The mass of the sun is . The minimum and maximum distances of the planet from the sun are and , respectively. Part A Using Kepler's 3rd law and Newton's law of universal gravitation, find the period of revolution of the planet as it moves around the sun. Assume that the mass of the planet is much smaller than the mass of the sun. Use for the gravitational constant. Hint A.1 Kepler's 3rd law Kepler's 3rd law states that the square of the period of revolution of a planet around the sun is proportional to the cube of the semi-major axis of its orbit. Try finding the period of a circular orbit and then using Kepler's 3rd law (which applies equally to circular and elliptical orbits) to extend your result to an elliptical orbit. Hint A.2 Find the semi-major axis Find the semi-major axis . Hint A.2.1 Definition of semi-major axis The semi-major axis of an ellipse is half of its major axis. The sun is at the focus of the elliptical orbit and the focus lies on the major axis. Express the semi-major axis in terms of and . ANSWER: = Answer Requested Hint A.3 Find the period of a circular orbit Hint not displayed Express the period in terms of , , , and . ANSWER: = Correct Weight on a Neutron Star Neutron stars, such as the one at the center of the Crab Nebula, have about the same mass as our sun but a much smaller diameter. Part A MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?ass... 4 of 8 4/17/10 5:31 PM If you weigh 660 on the earth, what would be your weight on the surface of a neutron star that has the same mass as our sun and a diameter of 20.0 ? Take the mass of the sun to be = 1.99×10 30 , the gravitational constant to be = 6.67×10 −11 , and the acceleration due to gravity at the earth's surface to be = 9.810 . Hint A.1 How to approach the problem "Weight" is the same as "the force of gravitational attraction." Use Newton's law of gravitation to calculate the gravitational force that would be exerted on you if you were standing on the surface of the star. Be careful when determining the values needed for the equation. Hint A.2 Law of universal gravitation The gravitational force exerted on a mass by a second mass is , where is the distance between the two masses, and 6.67×10 −11 is the universal gravitational constant. Hint A.3 Calculate your mass Calculate your mass if you weigh 660 on earth. Express your answer in kilograms. ANSWER: = 67.3 Correct Hint A.4 Calculate the distance between you and the star Calculate your distance from the center of the star if you are standing on its surface. Express your answer in meters. ANSWER: = 1.00×10 4 Correct Express your weight in newtons. ANSWER: = 8.93×10 13 Correct This is over times your weight on earth! You probably shouldn't venture there.... Exercise 12.4 Two uniform spheres, each with mass and radius , touch one another. Part A What is the magnitude of their gravitational force of attraction? Express your answer in terms of the variables , , and appropriate constants. ANSWER: Correct Exercise 12.41 Consider the ring-shaped body of the figure . A particle with mass is placed a distance from the center of the ring, along the line through the center of the ring and perpendicular to its plane. Part A Calculate the gravitational potential energy of this system. Take the potential energy to be zero when the two objects are far apart. MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?ass... 5 of 8 4/17/10 5:31 PM ANSWER: = Correct Part B Show that your answer to part (a) reduces to the expected result when is much larger than the radius of the ring. ANSWER: My Answer: Part C Use to find the magnitude and direction of the force on the particle. ANSWER: = Correct Part D Show that your answer to part (c) reduces to the expected result when is much larger than . ANSWER: My Answer: Part E What is the value of when ? ANSWER: = Correct Part F What is the value when ? ANSWER: = 0.00×10 0 Correct Part G Explain why the results for and make sense. ANSWER: My Answer: Problem 12.52 At a certain instant, the earth, the moon, and a stationary 1070 spacecraft lie at the vertices of an equilateral triangle whose sides are in length. Part A Find the magnitude of the net gravitational force exerted on the spacecraft by the earth and moon. ANSWER: = 2.91 Correct Part B Find the direction of the net gravitational force exerted on the spacecraft by the earth and moon. State the direction as an angle measured from a line connecting the earth and the spacecraft. ANSWER: 0.607 Correct Part C What is the minimum amount of work that you would have to do to move the spacecraft to a point far from the earth and moon? You can ignore any gravitational effects due to the other planets or the sun. MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?ass... 6 of 8 4/17/10 5:31 PM ANSWER: = 1.12×10 9 Correct Problem 12.81 Planets are not uniform inside. Normally, they are densest at the center and have decreasing density outward toward the surface. Model a spherically symmetric planet, with the same radius as the earth, as having a density that decreases linearly with distance from the center. Let the density be 2.00×10 4 at the center and 2700 at the surface. Part A What is the acceleration due to gravity at the surface of this planet? Express your answer using two significant figures. ANSWER: = 13 Correct Exercise 12.30: International Space Station The International Space Station makes 15.65 revolutions per day in its orbit around the earth. Part A Assuming a circular orbit, how high is this satellite above the surface of the earth? ANSWER: = 370 Correct Problem 12.85 A shaft is drilled from the surface to the center of the earth as in the example 12.10 (section 12.6), make the unrealistic assumption that the density of the earth is uniform. With this approximation, the gravitational force on an object with mass , that is inside the earth at a distance from the center, has magnitude (as shown in the example 12.10) and points toward the center of the earth. Part A Derive an expression for the gravitational potential energy of the object-earth system as a function of the object's distance from the center of the earth. Take the potential energy to be zero when the object is at the center of the earth. ANSWER: = Correct Part B If an object is released in the shaft at the earth's surface, what speed will it have when it reaches the center of the earth? ANSWER: = 7900 Correct Exercise 12.11 A particle of mass 3 is located 1.70 from a particle of mass . Part A Where should you put a third mass so that the net gravitational force on due to the two masses is exactly zero? ANSWER: = 1.08 Correct from the mass 3 Part B Is the equilibrium of at this point stable or unstable for points along the line connecting and 3 ? ANSWER: equilibrium is stable equilibrium is unstable Correct Part C Is the equilibrium of at this point stable or unstable for points along the line passing through and perpendicular to the line connecting and 3 ? MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?ass... 7 of 8 4/17/10 5:31 PM ANSWER: equilibrium is stable equilibrium is unstable Correct Exercise 12.19 Part A Calculate the earth's gravity force on a 90 astronaut who is repairing the Hubble Space Telescope 600 above the earth's surface Express your answer using two significant figures. ANSWER: = 740 Correct Part B Compare this gravity force with his weight at the earth's surface. Express your answer using two significant figures. ANSWER: = 0.84 Correct Part C In view of your result, explain why we say astronauts are weightless when they orbit the earth in a satellite such as a space shuttle.Is it because the gravitational pull of the earth is negligibly small? ANSWER: My Answer: Problem 12.79 A 3490-kg spacecraft is in a circular orbit a distance 2660 above the surface of Mars. Part A How much work must the spacecraft engines perform to move the spacecraft to a circular orbit that is 4270 above the surface? ANSWER: = 2.59×10 9 Correct Score Summary: Your score on this assignment is 100%. You received 140 out of a possible total of 140 points. MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?ass... 8 of 8 4/17/10 5:31 PM Ryker Huffman MasteringPhysics: Assignment Print View