A 5,000 kg satellite is orbiting the Earth in a circular path. The height of the satellite above the surface of the Earth is 800 km. The time it takes for the satellite to travel around the Earth is (M_E = 5.98 × 10²⁴ kg, R_E = 6.37 × 10⁶ m, G = 6.67 × 10⁻¹¹ N·m²/kg²) A) 0.750 hours. B) 0.950 hours. C) 1.25 hours. D) 1.68 hours. E) 2.01 hours.

Question

Quizplus · Accepted Answer

The time it takes for a satellite to orbit the Earth once can be found using Kepler's Third Law:

$\frac{T^2}{r^3}=\frac{4\pi^2}{GM_{Earth}}$

where $T$ is the period of the satellite (the time it takes to orbit once), $r$ is the distance between the center of the Earth and the satellite, $G$ is the gravitational constant, and $M_{Earth}$ is the mass of the Earth.

First, we need to convert the altitude of the satellite above the surface of the Earth from 800 km to the total distance from the center of the Earth to the satellite:

$r = 800	ext{ km} + 6,370	ext{ km} = 7,170	ext{ km}$

Next, we can solve for the period of the satellite:

$T = 2\pi\sqrt{\frac{r^3}{GM_{Earth}}}$

$T = 2\pi\sqrt{\frac{(7,170	ext{ km}\ +\ 6,370	ext{ km})^3}{6.67 	imes 10^{-11} 	ext{ N}\cdot	ext{m}^2\cdot	ext{kg}^{-2} \cdot 5.98 	imes 10^{24} 	ext{ kg}}}$

$T = 1.68	ext{ hours}$

Therefore, the answer is D) 1.68 hours.

A 5,000 Kg Satellite Is Orbiting the Earth in a Circular