The Moon orbits the Earth once every 27.3 days. Suppose that a crazy billionaire

Question

The Moon orbits the Earth once every 27.3 days. Suppose that a crazy billionaire puts a satellite into orbit around the Earth, so that the satellite completes exactly 5 orbits of the Earth during every 3 orbits of the Moon. (Assume that both orbits are circular.) a. Calculate the ratio of the orbital radii of the satellite and the Earth's Moon. Which one is closer to Earth? Show your work completely. Express your final answer to 3 significant figures. b. Calculate the ratio of the orbital speeds of the satellite and the Earth's Moon. Which one moves faster? Show your work completely. Express your final answer to 3 significant figures.

Explanation / Answer

(5) T_moon = 27.3 days

w_moon = 2 pi / T_moon

5 T = 3 T_moon

T = 3 x 27.3 / 5 = 16.38 days

and w = 2pi / T

For circular orbit,

G M m / r^2 = m w^2 r

r^3 = G M_earth /w^2 = G M_earth T^2 / 4 pi^2

( r_satellite / r_moon ) ^3 = (16.38 / 27.3)^2

r_satellite / r_moon ) = 0.711

satellite is closer.

(B) v = w / r

v_satellite / v_moon = T_moon r_moon / T_satellite r_satellite

= (27.3 / 16.38) ( 1/0.711 )

= 2.34

satellite moves faster.

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