/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 72 Determine the altitude \(h\) (in... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 72

Determine the altitude \(h\) (in kilometers) above the surface of the earth at which a satellite in a circular orbit has the same period, \(23.9344 \mathrm{h}\), as the earth's absolute rotation. If such an orbit lies in the equatorial plane of the earth, it is said to be geosynchronous, because the satellite does not appear to move relative to an earth-fixed observer.

Short Answer

Expert verified
The altitude \(h\) for a geosynchronous satellite is approximately 35,786 km above the Earth's surface.

Step by step solution

01

Identify the Known Values

The period of the satellite is given as 23.9344 hours, which is equivalent to the period of the Earth's rotation. You need to convert this period from hours to seconds for calculations: \ \[ T = 23.9344 \times 3600 \text{ seconds} \approx 86164 \text{ seconds} \] \
02

Use Kepler's Third Law

Kepler's Third Law states that the square of the orbital period \(T\) of a satellite is proportional to the cube of the semi-major axis \(a\) of its orbit: \ \[ T^2 = \frac{4\pi^2}{GM}a^3 \] \ where \(G\) is the gravitational constant \((6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2)\) and \(M\) is the mass of the Earth \((5.972 \times 10^{24} \, \text{kg})\). Substitute the known values to find \(a\).
03

Solve for Semi-major Axis

Rearrange the equation from Step 2 to solve for \(a\): \ \[ a^3 = \frac{GMT^2}{4\pi^2} \] \ Plug in the values for \(G\), \(M\), and \(T\) to find \(a^3\), then take the cube root: \ \[ a = \left(\frac{6.674 \times 10^{-11} \cdot 5.972 \times 10^{24} \cdot (86164)^2}{4\pi^2}\right)^{1/3} \] \ Calculate the value of \(a\).
04

Calculate the Altitude

The semi-major axis \(a\) calculated in Step 3 is the distance from the center of the Earth to the satellite. To find the altitude \(h\) above the Earth's surface, subtract the Earth's radius \(R_e \approx 6371 \, \text{km}\) from \(a\): \ \[ h = a - R_e \] \

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Orbital Mechanics
Orbital mechanics is a core concept in the study of celestial bodies and their paths around others. It's how we understand satellites' movements around Earth. This field uses physics to determine orbits and predict positions of satellites over time.
Key elements of orbital mechanics include:
  • Gravity: The force attracting satellites towards Earth.
  • Inertia: The tendency of a satellite to maintain its motion unless acted upon by another force.
  • Velocity: The speed and direction needed to maintain orbit.
Orbiting satellites are in a constant dance between gravity pulling them towards Earth and their inertia carrying them straight. Understanding this balance helps in planning satellite launches and predicting their paths, ensuring they stay in desired orbits.
Kepler's Third Law
Kepler's Third Law offers a crucial insight into the motion of planets and satellites. It says that the square of a satellite's orbital period (\(T^2\)) is proportional to the cube of the semi-major axis (\(a^3\)) of its orbit. Usually expressed as:\[T^2 = \frac{4\pi^2}{GM}a^3\]Here, \(G\) is the gravitational constant and \(M\) is the mass of the Earth. This formula connects the time it takes for a satellite to complete an orbit with the size of the orbit itself.
Understanding this law is key for determining orbits, as it allows us to calculate how far a satellite must be from Earth for a specific orbital period. This is vital in establishing geosynchronous orbits, where satellites match Earth's rotation.
Satellite Orbits
Satellites orbit the Earth in various paths depending on their mission. These orbits range from low Earth orbit (LEO) to geosynchronous orbits. Each type of orbit has unique characteristics and purposes.
Geosynchronous orbits are special because they allow satellites to stay over a fixed point on Earth's surface. This is ideal for communications and weather monitoring. Their orbital period matches Earth's rotation time of roughly 24 hours.
  • Geostationary orbit: A subset where the orbit is circular and directly above the equator.
  • Inclined geosynchronous orbit: Not perfectly circular or equatorial but still syncs with Earth's rotation.
Understanding satellite orbits helps in satellite deployment and operations, aiding everything from GPS to satellite TV.
Earth's Rotation
Earth's rotation is fundamental to many scientific concepts, influencing day and night cycles and affecting satellite orbits. The Earth completes one full rotation approximately every 23.9344 hours, which aligns closely with how we perceive a day.
When satellites are placed in geosynchronous orbit, they match Earth's rotation speed, allowing them to hover above a constant geographical location. This makes them very useful for communications and broadcast services, ensuring consistent, reliable coverage.
Such synchronization with Earth's rotation involves complex calculations that account for gravitational forces and orbital mechanics. Understanding how the Earth's rotation affects satellite paths is crucial in designing satellite missions and ensuring their operational success.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Freight car \(A\) of mass \(m_{A}\) is rolling to the right when it collides with freight car \(B\) of mass \(m_{B}\) initially at rest. If the two cars are coupled together at impact, show that the fractional loss of energy equals \(m_{B} /\left(m_{A}+m_{B}\right)\).

The hollow tube assembly rotates about a vertical axis with angular velocity \(\omega=\dot{\theta}=4 \quad \mathrm{rad} / \mathrm{s}\) and \(\dot{\omega}=\ddot{\theta}=-2 \operatorname{rad} / \mathrm{s}^{2} .\) A small \(0.2-\mathrm{kg}\) slider \(P\) moves inside the horizontal tube portion under the control of the string which passes out the bottom of the assembly. If \(r=0.8 \mathrm{m}, \dot{r}=-2 \mathrm{m} / \mathrm{s},\) and \(\ddot{r}=4 \mathrm{m} / \mathrm{s}^{2}\) determine the tension \(T\) in the string and the horizontal force \(F_{\theta}\) exerted on the slider by the tube.

The block of mass \(m\) is attached to the frame by the spring of stiffness \(k\) and moves horizontally with negligible friction within the frame. The frame and block are initially at rest with \(x=\) \(x_{0},\) the uncompressed length of the spring. If the frame is given a constant acceleration \(a_{0},\) determine the maximum velocity \(\dot{x}_{\max }=\left(v_{\mathrm{rel}}\right)_{\max }\) of the block relative to the frame.

It is experimentally determined that the drive wheels of a car must exert a tractive force of \(560 \mathrm{N}\) on the road surface in order to maintain a steady vehicle speed of \(90 \mathrm{km} / \mathrm{h}\) on a horizontal road. If it is known that the overall drivetrain efficiency is \(e_{m}=0.70,\) determine the required motor power output \(P\).

Two barges, each with a displacement (mass) of \(500 \mathrm{Mg},\) are loosely moored in calm water. A stunt driver starts his 1500 -kg car from rest at \(A\), drives along the deck, and leaves the end of the \(15^{\circ}\) ramp at a speed of \(50 \mathrm{km} / \mathrm{h}\) relative to the barge and ramp. The driver successfully jumps the gap and brings his car to rest relative to barge 2 at \(B\). Calculate the velocity \(v_{2}\) imparted to barge 2 just after the car has come to rest on the barge. Neglect the resistance of the water to motion at the low velocities involved.
See all solutions

Recommended explanations on Physics Textbooks

Measurements

Read Explanation

Thermodynamics

Read Explanation

Energy Physics

Read Explanation

Electromagnetism

Read Explanation

Quantum Physics

Read Explanation

Electrostatics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.