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Chapter 5: Problem 49

An Earth Satellite An Earth satellite moves in a circular orbit \(640 \mathrm{~km}\) above Earth's surface with a period of \(98.0 \mathrm{~min}\). What are (a) the speed and (b) the magnitude of the centripetal acceleration of the satellite?

Short Answer

Expert verified
The speed of the satellite is approximately 7480 m/s, and the centripetal acceleration is approximately 7.98 m/s虏.

Step by step solution

01

- Determine the orbital radius

Add the Earth's radius (\r{e}=6371\text{k}\text{m}) to the altitude of the satellite (h=640\text{km}). The orbital radius (r) is the distance from the center of the Earth to the satellite.\[ r = r_e + h = 6371 \text{km} + 640 \text{km} = 7011 \text{km} = 7.011 \times 10^6 \text{m} \]
02

- Convert the orbital period to seconds

Convert the orbital period from minutes to seconds. Given the period (T) is 98.0 minutes:\[ T = 98.0 \text{ min} \times 60 \text{ sec/min} = 5880 \text{ sec}\]
03

- Calculate the orbital speed

The speed (v) of the satellite can be found using the formula for the circumference of a circle and the period. \[ v = \frac{2 \pi r}{T} = \frac{2 \pi (7.011 \times 10^6 \text{ m})}{5880 \text{ s}} 鈮� 7480 \text{ m/s} \]
04

- Calculate the centripetal acceleration

The centripetal acceleration (a) can be calculated using the formula: \[ a = \frac{v^2}{r} = \frac{(7480 \text{ m/s})^2}{7.011 \times 10^6 \text{ m}} 鈮� 7.98 \text{ m/s}^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Orbital Radius
The **orbital radius** is the distance from the center of the Earth to the satellite in its orbit. It can be found by adding the Earth's radius to the altitude of the satellite.
Given:
- Earth's radius \((r_e)\): 6371 km
- Satellite's altitude above Earth's surface \((h)\): 640 km
The formula to find the orbital radius \((r)\) is:
\[r = r_e + h\]
Using the given values:
\[r = 6371\text{ km} + 640\text{ km} = 7011\text{ km} = 7.011 \times 10^6 \text{ m}\]
This formula helps us determine the complete distance from Earth's center to the satellite in its orbit.
Orbital Period
The **orbital period** is the time it takes for the satellite to complete one full orbit around the Earth.
Given its period is 98.0 minutes, we need to convert this to seconds, as standard units in physics are important for consistent calculations.
- \(1\text{ min} = 60\text{ s}\)
Thus:
\[T = 98.0\text{ min} \times 60\text{ s/min} = 5880\text{ s}\]
This calculation converts the orbital period from minutes into seconds.
Centripetal Acceleration
The satellite is in circular motion around the Earth, which means it is constantly changing direction, requiring a continuous force directed towards the center of its circular path. The **centripetal acceleration** reflects how quickly the satellite changes direction.
It can be calculated using the formula:
\[a = \frac{v^2}{r}\]
Where **\(v\)** is the orbital speed, and **\(r\)** is the orbital radius.
From previous calculations:
- \(v = 7480\text{ m/s}\)
- \(r = 7.011 \times 10^6\text{ m}\)
Substitute these values into the formula:
\[a = \frac{(7480\text{ m/s})^2}{7.011\times 10^6\text{ m}} \approx 7.98\text{ m/s}^2\]
This acceleration keeps the satellite in its circular orbit.
Orbital Speed
The **orbital speed** is the constant speed at which the satellite travels along its path. It can be calculated by considering the distance it covers (the circumference of the orbit) and the time it takes (the orbital period).
Use the formula:
\[v = \frac{2\pi r}{T}\]
Where:
- **\(2\pi r\)** is the circumference of the orbit
- **\(T\)** is the orbital period
Given previously:
- \(r = 7.011 \times 10^6\text{ m}\)
- \(T = 5880\text{ s}\)
Substitute these into the formula:
\[v = \frac{2 \pi (7.011 \times 10^6 \text{ m})}{5880\text{ s}} \approx 7480\text{ m/s}\]
This calculation shows the speed at which the satellite travels to maintain its orbit.

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Most popular questions from this chapter

An astronaut is rotated in a horizontal centrifuge at a radius of \(5.0 \mathrm{~m}\). (a) What is the astronaut's speed if the centripetal acceleration has a magnitude of \(7.0 \mathrm{~g}\) ? (b) How many revolutions per minute are required to produce this acceleration? (c) What is the period of the motion?

A rotating fan completes 1200 revolutions every minute. Consider the tip of a blade, at a radius of \(0.15 \mathrm{~m}\). (a) Through what distance does the tip move in one revolution? What are (b) the tip's speed and (c) the magnitude of its acceleration? (d) What is the period of the motion?

\(A\) is \(90 \mathrm{~km}\) west of oasis \(B\). A camel leaves oasis \(A\) and during a \(50 \mathrm{~h}\) period walks \(75 \mathrm{~km}\) in a direction \(37^{\circ}\) north of east. The camel then walks toward the south a distance of \(65 \mathrm{~km}\) in a \(35 \mathrm{~h}\) period after which it rests for \(5.0 \mathrm{~h}\). (a) What is the camel's displacement with respect to oasis \(A\) after resting? (b) What is the camel's average velocity from the time it leaves oasis \(A\) until it finishes resting? (c) What is the camel's average speed from the time it leaves oasis \(A\) until it finishes resting? (d) If the camel is able to go without water for five days \((120 \mathrm{~h})\), what must its average velocity be after resting if it is to reach oasis \(B\) just in time?

A cat rides a merry-go-round while turning with uniform circular motion. At time \(t_{1}=2.00 \mathrm{~s}\), the cat's velocity is $$ \vec{v}_{1}=(3.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(4.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}} $$ measured on a horizontal \(x y\) coordinate system. At time \(t_{2}=5.00 \mathrm{~s}\), its velocity is $$ \vec{v}_{2}=(-3.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(-4.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}} $$ What are (a) the magnitude of the cat's centripetal acceleration and (b) the cat's average acceleration during the time interval \(t_{2}-t_{1}\) ?

What is the magnitude of the acceleration of a sprinter running at \(10 \mathrm{~m} / \mathrm{s}\) when rounding a turn with a radius of \(25 \mathrm{~m}\) ?
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