Kepler's Third Law and Newton's Law of Universal Gravitation (a) Use Newton's Universal Law of Gravitation and what you know about centripetal acceleration/force to derive Kepler's Third Law for a planet in a circular orbit about the sun: T² = Kr³ K = constant = 4²/GM where T is the orbital period of the planet (the time for one complete orbit), r is the radius of the planet's orbit, M is the mass of the sun, and G is the universal gravitational constant. (b) Determine the metric system units of K and show that they make the units of T² = Kr³ work out correctly. (c) The earth orbits the sun once per year (365 days) and its average orbital radius is 1.50 x 10¹¹ m. Use this information and Kepler's Third Law to estimate the mass of the sun in kilograms. [answer: about 2 x 10³⁰ kg] (d) The radius of the sun is about 7 x 108 m. Use this radius and the mass of the sun estimated in part (c) to estimate the acceleration of an object near the surface of the sun. [answer: about 300 m/s²] F₂ =G Mplanet maun/r² Fnet = Fz = mac ac =v²/r G = 6.674 x 10-¹1 N-m²/kg² d=vt: d = circumference of the orbit - 2r and t = time for one orbit = T

Kepler's Third Law and Newton's Law of Universal Gravitation (a) Use Newton's Universal Law of Gravitation and what you know about centripetal acceleration/force to derive Kepler's Third Law for a planet in a circular orbit about the sun: T² = Kr³ K = constant = 4²/GM where T is the orbital period of the planet (the time for one complete orbit), r is the radius of the planet's orbit, M is the mass of the sun, and G is the universal gravitational constant. (b) Determine the metric system units of K and show that they make the units of T² = Kr³ work out correctly. (c) The earth orbits the sun once per year (365 days) and its average orbital radius is 1.50 x 10¹¹ m. Use this information and Kepler's Third Law to estimate the mass of the sun in kilograms. [answer: about 2 x 10³⁰ kg] (d) The radius of the sun is about 7 x 108 m. Use this radius and the mass of the sun estimated in part (c) to estimate the acceleration of an object near the surface of the sun. [answer: about 300 m/s²] F₂ =G Mplanet maun/r² Fnet = Fz = mac ac =v²/r G = 6.674 x 10-¹1 N-m²/kg² d=vt: d = circumference of the orbit - 2r and t = time for one orbit = T

Physics for Scientists and Engineers: Foundations and Connections

1st Edition

ISBN:9781133939146

Author:Katz, Debora M.

Publisher:Katz, Debora M.

Chapter7: Gravity

Section: Chapter Questions

Problem 65PQ

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