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 radiusR, 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 ofradius r and angular velocity w is w²r. Use this result, and the result that the velocity of such an object is wr to calculate themass-m object's kinetic energy in terms of G, M, m, and r.3.11. PROBLEMS117(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 compare in this gravitational bound state?(f) Now, suppose that the mass density of the mass M object at radius s is p(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 isNewton's Second Law modified in this case? What then is the corresponding velocity at radius r?

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Answered: Consider an object of mass m in a…

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter8: Central-force Motion

Section: Chapter Questions

Problem 8.4P

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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
117
(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 compare in this gravitational bound state?
(f) Now, suppose that the mass density of the mass M object at radius s is p(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?

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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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