Question

A planet has a mass of M1, a radius of R1, and a density of Q1 A second planet has a mass of M2, a radius of R2, and a density of g2. This problem will explore the relationships between the surface gravities (g1 and g2) of the planets depending on the relative sizes of their masses, radii, and densities. Assume that planet 2 has X times the mass of planet 1, or M2 - XM1. The densities of both planets arc the same. Write an expression for the ratio of the surface gravity of planet 2 to planet 1 in terms of X. This would be true if the radii of the planets would be the same. But in this question we have two planets with the same densities. Suppose now the radius of the second planet is Y times the size of the radius of the first planet, or R2 = YRj. Write an expression for the ratio of the surface gravities, g2lg} in terms of Y assuming the densities arc the same. Suppose now M2 - 2Mj and q2 = 2qj. What is the ratio of g2/gj now (here we want the actual number; because you arc writing a ratio, the number will be unites? Now suppose R2 = 3Rj and q2 = 3qj. Find the ratio of g2/gj (again as a number and again the number will be unitless).

Answer 1

Part (A) Given M, XM, and the denstities are the same. We have, M, R 1/3 2/3 GM We know that, g-n g, M, |-|- - Y13 2/3 Hence, Part (B) Given R -YR, and the densities are the same Hence, 2Y. 81 Y

Part (C) Given M,-204 and Q2 Wehave. M Hence, R R 201 g, M, R Part (D) Given R, - 3R, and Wehave. Hence, 2 81 8 M Therefore, 2 -_2 81