2 Suppose The Distance Between The Planet And The Sun Is R And Consider The Circular Orbit R T R Cos Wt R Sin W 1 (237.41 KiB) Viewed 22 times
2 Suppose The Distance Between The Planet And The Sun Is R And Consider The Circular Orbit R T R Cos Wt R Sin W 2 (112.37 KiB) Viewed 22 times
(2) Suppose the distance between the planet and the Sun is R, and consider the circular orbit r(t) = (R cos(wt), R sin(wt)) (a) Find constraints on w (in terms of G, M and R) so that r(t) satisfies the differential equation 1.1. (b) Use part (a) to deduce a version of Kepler's Third Law for this orbit. 40 R3 GM (Hint: What is the relationship between w and T?) T2 = ( (3) A satellite orbiting above the equator of the Earth is called geosynchronous if T = 24 hours. Assuming that the satellite has a circular, geosynchronous orbit, use Kepler's third law to find the distance from the satellite to the surface of the Earth. (Given that the Earth has mass M ~ 5.974 x 1024 kg, and radius 6371 km).
As a consequence of Newton's law of gravitation, combined with Newton's second law of motion, it turns out that there is a relationship between the motion of the planet, r(t), and its second derivative r"(t) (acceleration). This kind of relationship between a function and its derivatives is called a differential equation. Precisely stated, r(t) satisfies the differential equation GM ||r(t)||2 (1.1) "(t) = (- 12MP) = er(t)
Join a community of subject matter experts. Register for FREE to view solutions, replies, and use search function. Request answer by replying!