Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Problem 22

Mars has two small satellites, Phobos and Deimos. Phobos circles Mars once every \(0.31891\) day at an average altitude of \(5980 \mathrm{~km}\) above the planet's surface. The diameter of Mars is \(6794 \mathrm{~km}\). Using this information, calculate the mass and average density of Mars.

Short Answer

Expert verified
The mass of Mars is approximately \(0.641 \times 10^{24}\) kg and the average density of Mars is approximately \(3933\) kg/m³.

Step by step solution

01

Understanding Kepler's Third Law

Kepler's Third Law states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit, which in this situation is the distance of the satellite from the center of the planet. Given the period \(P = 0.31891\) day and the radius of the orbit \(r = 5980 km + 6794 km / 2\), we have to find the constant of proportionality which will be used to find the mass of Mars.
02

Convert Units

We need to convert the units of the period from days to seconds to be consistent with the common usage in physics. 1 day is equivalent to 86400 seconds, so \(P = 0.31891 \times 86400 = 27567.84\) seconds.
03

Calculate Mass

The constant of proportionality in the equation of Kepler's Third Law is the Gravitational constant divided by 4pi² and the mass of the larger object, in this case, Mars. Rearranging Kepler's Third Law and substituting the given values gives us the mass of Mars, \(M = \frac{4\pi²r³}{GP²}\), where G is the universal gravitational constant \(6.674 \times 10^{-11} \, m³ kg⁻¹ s⁻²\). Substituting all known values and performing the calculation gives the mass of Mars.
04

Calculate Volume

The diameter of Mars is given as \(6794\) km. To find the volume of Mars, we convert this to meters (multiply by 1000), and then use the formula for the volume of a sphere \(V = \frac{4}{3}\pi r³\), where r is the radius of Mars, which is half the diameter.
05

Calculate Density

The average density of Mars can be found using the formula \(ρ = \frac{M}{V}\), where M is mass and V is volume. Substitute the calculated mass and volume of Mars into this equation and simplify to find the average density of Mars.

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

Use the Stamy Night Enthusiast TM program to examine the Jovian planets Jupiter, Saturn, Uranus, and Neptune. Select each of these planets from the Solar System submenu in the Favourites menu. If you desire, you can remove the image of the astronaut's feet by selecting Feet in the View menu. Position the mouse cursor over the planet and click and drag the image to examine the planet from different views. Describe each planet's appearance. Which has the greatest color contrast in its cloud tops? Which has the least color contrast? What can you say about the thickness of Saturn's rings compared to their diameter?

What are the differences in chemical composition between the terrestrial and Jovian planets?

What is one piece of evidence that impact craters are actually caused by impacts?

Use the Deep Space Explorer \({ }^{T M}\) program to examine the Jovian planets Jupiter, Saturn, Uranus, and Neptune. In the left-hand part of the window, under the heading Solar System select Explore. Then click on the name of each planet to view it in detail. You can zoom in and zoom out using the buttons at the upper left of the window (an upward-pointing triangle and a downward- pointing triangle). You can also rotate the planet by putting the mouse cursor over the image of the planet or asteroid, holding down the mouse button, and moving the mouse. (On a two-button mouse, hold down the left mouse button.) Describe each planet's appearance. Which has the greatest color contrast in its cloudtops? Which has the least color contrast? What can you say about the thickness of Saturn's rings compared to their diameter?

(a) Find the mass of a hypothetical spherical asteroid \(2 \mathrm{~km}\) in diameter and composed of rock with average density \(2500 \mathrm{~kg} / \mathrm{m}^{3}\). (b) Find the speed required to escape from the surface of this asteroid. (c) A typical jogging speed is \(3 \mathrm{~m} / \mathrm{s}\). What would happen to an astronaut who decided to go for a jog on this asteroid?
See all solutions

Recommended explanations on Physics Textbooks

Turning Points in Physics

Read Explanation

Medical Physics

Read Explanation

Force

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Translational Dynamics

Read Explanation

Measurements

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