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Consider an object of mass m in a circular orbit of radius r and angular velocity w about a larger object of mass M and radius R, where M >> m. (a) What is the the potential energy of the mass-m in its circular orbit, assuming that r > R? (b) In 1. Vectors and kinematics, we showed that the acceleration in the radial direction of an object in a circular orbital of radius r and angular velocity w is w²r. Use this result, and the result that the velocity of such an object is wr to calculate the mass-m object's kinetic energy in terms of G, M, m, and r. 3.11. PROBLEMS (c) Calculate the corresponding velocity of the mass m object in terms of G, M, m, and r. 117

Consider an object of mass m in a circular orbit of radius r and angular velocity w about a larger object of mass M and radius R, where M >> m. (a) What is the the potential energy of the mass-m in its circular orbit, assuming that r > R? (b) In 1. Vectors and kinematics, we showed that the acceleration in the radial direction of an object in a circular orbital of radius r and angular velocity w is w²r. Use this result, and the result that the velocity of such an object is wr to calculate the mass-m object's kinetic energy in terms of G, M, m, and r. 3.11. PROBLEMS (c) Calculate the corresponding velocity of the mass m object in terms of G, M, m, and r. 117

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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### Circular Orbits and Gravitational Systems Consider an object of mass \( m \) in a circular orbit of radius \( r \) and angular velocity \( \omega \) about a larger object of mass \( M \) and radius \( R \), where \( M \gg m \). #### Problems (a) What is the potential energy of the mass-\( m \) in its circular orbit, assuming that \( r > R \)? (b) In _1. Vectors and kinematics_, we showed that the acceleration in the radial direction of an object in a circular orbital of radius \( r \) and angular velocity \( \omega \) is \( \omega^2 r \). Use this result, and the result that the velocity of such an object is \( \omega r \) to calculate the mass-\( m \) object's kinetic energy in terms of \( G, M, m, \) and \( r \). (c) Calculate the corresponding velocity of the mass \( m \) object in terms of \( G, M, m, \) and \( r \). (d) What is the total **mechanical** energy of the mass \( m \)-mass \( M \) system? (e) How do the potential, kinetic, and total energy compare in this gravitational bound state? (f) Now, suppose that the mass density of the mass \( M \) object at radius \( s \) is \( \rho(s) \). What is the total mass within radius \( r \)? (g) It turns out that for an object orbiting at radius \( r \), the relevant gravitational mass is solely the mass within radius \( r \). How is Newton’s Second Law modified in this case? What then is the corresponding velocity at radius \( r \)? --- No graphs or diagrams are included in this text.
Transcribed Image Text:### Circular Orbits and Gravitational Systems Consider an object of mass \( m \) in a circular orbit of radius \( r \) and angular velocity \( \omega \) about a larger object of mass \( M \) and radius \( R \), where \( M \gg m \). #### Problems (a) What is the potential energy of the mass-\( m \) in its circular orbit, assuming that \( r > R \)? (b) In _1. Vectors and kinematics_, we showed that the acceleration in the radial direction of an object in a circular orbital of radius \( r \) and angular velocity \( \omega \) is \( \omega^2 r \). Use this result, and the result that the velocity of such an object is \( \omega r \) to calculate the mass-\( m \) object's kinetic energy in terms of \( G, M, m, \) and \( r \). (c) Calculate the corresponding velocity of the mass \( m \) object in terms of \( G, M, m, \) and \( r \). (d) What is the total **mechanical** energy of the mass \( m \)-mass \( M \) system? (e) How do the potential, kinetic, and total energy compare in this gravitational bound state? (f) Now, suppose that the mass density of the mass \( M \) object at radius \( s \) is \( \rho(s) \). What is the total mass within radius \( r \)? (g) It turns out that for an object orbiting at radius \( r \), the relevant gravitational mass is solely the mass within radius \( r \). How is Newton’s Second Law modified in this case? What then is the corresponding velocity at radius \( r \)? --- No graphs or diagrams are included in this text.
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(d) What is the total mechanical energy of the mass m-mass M system? (e) How do the potential, kinetic, and total compare in this gravitational bound state?

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