Book edition 1st Edition

Author(s) Paul Peter Urone

Pages 1272 pages

ISBN 9781938168000

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Short Answer

A geosynchronous Earth satellite is one that has an orbital period of precisely 1 day. Such orbits are useful for communication and weather observation because the satellite remains above the same point on Earth (provided it orbits in the equatorial plane in the same direction as Earth’s rotation). Calculate the radius of such an orbit based on the data for the moon in Table 6.2.

The geosynchronous satellite's orbital radius is $4.23 \times 10^{7} m$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of Satellite

A satellite is an item that is placed into orbit on purpose, and those are known as artificial satellites. There are natural satellites that orbit other planets. For example, the moon orbits around the earth, and the earth orbits the sun.

Step 2: Given data and concept used

Kepler's third law, written in mathematical form in, relates the period, or time, for one orbit to the radius of the orbit,

$\frac{{T_{1}}^{2}}{{T_{2}}^{2}} = \frac{{r_{1}}^{3}}{{r_{2}}^{3}}$

For the Moon, we'll use the subscript "m", and for the satellite, we'll use the subscript "s".

The Orbital radius of the moon from the Table 6.2, $r_{m} = 3.84 \times 10^{5} km = 3.84 \times 10^{8} m$ .
Time taken by the moon to complete one orbit is, $T_{m} = 0.07481 y$ .
The Orbital radius of the geosynchronous earth satellite, $r_{s} =?$ .
The Orbital Period of the geosynchronous earth satellite, $T_{s} = 1 day = 1 day \times \frac{1 y}{365 day} = \frac{1}{365} y$ .

Step 3: Calculating the radius of the geosynchronous earth satellite’s orbit

Applying Kepler’s third law,

$\begin{array}{l} \frac{{T_{s}}^{2}}{{T_{m}}^{2}} = \frac{{r_{s}}^{3}}{{r_{m}}^{3}} \\ {r_{s}}^{3} = \frac{{T_{s}}^{2}}{{T_{m}}^{2}} \times {r_{m}}^{3} \\ r_{s} = {(\frac{{T_{s}}^{2}}{{T_{m}}^{2}})}^{\frac{1}{3}} \times r_{m} \\ r_{s} = {(\frac{T_{s}}{T_{m}})}^{\frac{2}{3}} \times r_{m} \end{array}$

Putting the values $T_{s}, T_{m} and r_{m}$ in the above expression, and we get,

$\begin{array}{l} r_{s} = {(\frac{\frac{1}{365} y}{0.07481 y})}^{\frac{2}{3}} \times 3.84 \times 10^{8} m \\ r_{s} = 4.23 \times 10^{7} m \end{array}$

Hence, the orbital radius of the geosynchronous satellite is ${4.23×10}^{7} m$ .

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College Physics (Urone)

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Short Answer

Step by Step Solution

Step 1: Definition of Satellite

Step 2: Given data and concept used

Step 3: Calculating the radius of the geosynchronous earth satellite’s orbit

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College Physics (Urone)

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Short Answer

Step by Step Solution

Step 1: Definition of Satellite

Step 2: Given data and concept used

Step 3: Calculating the radius of the geosynchronous earth satellite’s orbit

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