Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Problem 40

Global Positioning System (GPS) satellites circle Earth at altitudes of approximately \(20,000 \mathrm{km},\) where the gravitational acceleration has \(5.8 \%\) of its surface value. To the nearest hour, what's the orbital period of the GPS satellites?

Short Answer

Expert verified
The orbital period of the GPS satellites, rounded to the nearest hour, is 4 hours.

Step by step solution

01

Compute the radius of the orbit

Start by calculating the actual radius of the orbit, which is the Earth's radius plus the altitude of the GPS satellite. Using the Earth's radius as approximately \(6371 \mathrm{km}\) and the altitude \(20000 \mathrm{km}\), the orbital radius \(R = 26371 \mathrm{km} = 2.6371 \times 10^{7} \mathrm{m}\).
02

Identify the centripetal acceleration

In the problem it was stated that the gravitational acceleration at the altitude of the GPS satellites has \(5.8 \%\) of its surface value. So, the centripetal acceleration \(a\) is \(5.8\%\) of \(9.8 \mathrm{m/s^2}\), which gives us \(a = 0.058 * 9.8 \mathrm{m/s^2}\) rounding up to \(a = 0.569 \mathrm{m/s^2}\).
03

Apply the equation for centripetal acceleration

The formula for centripetal acceleration is \(a = \frac{v^2}{R}\), where \(v\) is the velocity and \(R\) is the radius. We can rearrange this to solve for \(v = \sqrt{aR}\). Substituting the values of \(a\) and \(R\) found earlier to find the velocity, \(v = \sqrt{(0.569 \mathrm{m/s^2})(2.6371 \times 10^{7} \mathrm{m})} = 3445.51 \mathrm{m/s}\).
04

Compute the orbital period using the found velocity

Now that we have the velocity, we can calculate the time it takes for the GPS satellite to complete one orbit. That is the orbital period \(T\). The formula for the period is \(T = \frac{2\pi R}{v}\). Substituting our values for \(R\) and \(v\), we find that \(T = \frac{2\pi (2.6371 \times 10^{7} \mathrm{m})}{3445.51 \mathrm{m/s}}\). After calculating this, the orbital period is roughly \(15251 \mathrm{s}\) or about \(4.23 \mathrm{hours}\). Rounding this to the closest hour, we get \(4 \mathrm{hours}\).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

A biologist looking through a microscope sees a bacterium at \( \vec{r}_{1}=2.2 \hat{\imath}+3.7 \hat{\jmath}-1.2 \hat{k} \mu \mathrm{m} .\) After \(6.2 \mathrm{s},\) it's at \(\vec{r}_{2}=4.6 \hat{\imath}+\) \(1.9 \hat{k} \mu \mathrm{m} .\) Find \((\mathrm{a})\) its average velocity, expressed in unit vectors, and (b) its average speed.

A ferryboat sails between towns directly opposite each other on a river, moving at speed \(v^{\prime}\) relative to the water. (a) Find an expression for the angle it should head at if the river flows at speed \(V .\) (b) What's the significance of your answer if \(V>v^{\prime} ?\)

Let \(A=15^{T}-40 \mathrm{J}\) and \(B=31 \mathrm{J}+18 \mathrm{k} .\) Find \(C\) such that \(\vec{A}+\vec{B}+\vec{C}=\overrightarrow{0}\)

You're heading an international effort to save Earth from an asteroid heading toward us at \(15 \mathrm{km} / \mathrm{s}\). Your team mounts a rocket on the asteroid and fires it for 10 min, after which the asteroid is moving at \(19 \mathrm{km} / \mathrm{s}\) at \(28^{\circ}\) to its original path. In a news conference, what do you report for the acceleration imparted to the asteroid?

How fast would a car have to round a 75 -m-radius turn for its acceleration to be numerically equal to that of gravity?
See all solutions

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Read Explanation

Mechanics and Materials

Read Explanation

Physical Quantities And Units

Read Explanation

Physics of Motion

Read Explanation

Astrophysics

Read Explanation

Engineering Physics

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.

Sign-up for free