Question

You are on a recently discovered planet in our solar system that has the same mass

density as earth, but twice the radius.

A. Do you weigh more or less on this planet than you do on earth? By what factor does

your weight change?

B. What is the escape speed of the planet?

C. What is the shortest possible period for a satellite orbiting this planet?

D. If the mass of the sun is Ms = 1:99  1030 kg, how far away must this new planet be

to have the same period as earth?

Answer 1

A. By Newton's law of universal gravitation:

$$mg=\frac{GMm}{R^2},\\\space\\ mg=\frac{G(4/3·\pi R^3\rho)m}{R^2}=\frac 43G\pi R\rho m,\\\space\\ g=\frac 43G\pi R\rho.$$

My weight would increase twice.

B. Escape speed:

$$v_e=\sqrt\frac{2G(4/3\pi R^3\rho)}{R}=R\sqrt{\frac83G\pi\rho}.$$

The escape speed would increase twice.

C. The shortest possible period for a satellite orbiting this planet:

$$T=2\pi\sqrt{\frac{R^3}{GM}},\\\space\\ T=2\pi\sqrt{\frac{R^3}{G(4/3·\pi\rho R^3)}},\\\space\\ T=\pi\sqrt{\frac{3}{G\pi\rho}}=85\text{ min}$$

(same as Earth).

D. The period is independent on mass.