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A satellite is placed between the Earth and the Moon, along a straight line that connects their...

Question:

A satellite is placed between the Earth and the Moon, along a straight line that connects their centers of mass. The satellite has an orbital period around the Earth that is the same as that of the Moon, {eq}27.3 {/eq} days. How far away from the Earth should this satellite be placed?

Orbital Period

Orbital Period refers to the amount of time an object needed to finish its orbit around another object.

According to Kepler's third law, the orbital period can be defined as,

{eq}T = \sqrt{\frac{4\pi ^2 r^3}{GM}} {/eq}

Here,

r = Radius

G = Universal Gravitational Constant

M = Mass of Earth

We will use this formula to determine the radius of the object with an orbit similar to that of the moon.

Answer and Explanation:

The definition of period is specified as,

{eq}T = \sqrt{\frac{4\pi ^2 r^3}{GM}} {/eq}

Here,

{eq}T = \text{period (Must be in seconds)}\\ G = \text{Universal Gravitational constant}\\ M = \text{Mass of Earth }\\ r = \text{Distance from center of Earth to object} {/eq}

Rearranging to find the orbital distance,

{eq}r = \sqrt[3]{\frac{T^2 GM}{4\pi^2}} (1) {/eq}

Time is defined in days, so we convert it in seconds,

{eq}T = 27.3 days (\frac{24 hours}{1day}) (\frac{60 min}{1 hour}) (\frac{60 s}{1 min})\\ T = 2358720 (s) {/eq}

employing the value of T in (1), we have,

{eq}r = \sqrt[3]{\frac{( 2358720)^2 (6.67\times 10^{-11} N\cdot m^2/kg^2)(5.97 \times 10^{24} kg)}{4\pi^2}}\\ \boxed{ r = 382852 km} {/eq}

The satellite should be place at the distance of 382852 km away from Earth.


Learn more about this topic:

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Kepler's Three Laws of Planetary Motion

from Basics of Astronomy

Chapter 22 / Lesson 12
47K

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