Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 39

Earth's mean radius is \(6,371 \mathrm{km}\), and its mass is \(6.0 \times 10^{24} \mathrm{kg}\) The Moon's mean radius is \(1,738 \mathrm{km},\) and its mass is \(7.2 \times 10^{22} \mathrm{kg}\) a. Calculate Earth's average density. Show your work; do not look this value up. b. The average density of Earth's crust is \(2,600 \mathrm{kg} / \mathrm{m}^{3}\). What does this value tell you about Earth's interior? c. Compute the Moon's average density. Show your work. d. Compare the average densities of the Moon, Earth, and Earth's crust. What do these values tell you about the Moon's composition compared to that of Earth and of Earth's crust?

Short Answer

Expert verified
Earth's density is higher than both the Moon's and Earth's crust. This suggests Earth's interior contains denser materials.

Step by step solution

01

Calculate Earth's volume

To calculate the volume of Earth, use the formula for the volume of a sphere: \( V = \frac{4}{3} \pi R^3 \), where \( R \) is the radius of the Earth. Using Earth's mean radius \( 6,371 \text{ km} = 6,371,000 \text{ m} \), \( V_E = \frac{4}{3} \pi (6,371,000)^3 \)
02

Calculate Earth's density

Density \( \rho \) is given by \( \rho = \frac{M}{V} \). Using Earth's mass \( 6.0 \times 10^{24} \text{ kg} \) and the volume calculated in Step 1, \( \rho_E = \frac{6.0 \times 10^{24}}{\frac{4}{3} \pi (6,371,000)^3} \)
03

Interpret Earth's crust density

Given that Earth's crust has an average density of \( 2,600 \text{ kg} / \text{m}^3 \), this suggests that Earth's interior must be composed of materials with a higher average density than the crust. This implies the presence of denser materials like metals in the interior.
04

Calculate the Moon's volume

To calculate the volume of the Moon, use the formula for the volume of a sphere: \( V = \frac{4}{3} \pi R^3 \). Using the Moon's mean radius \( 1,738 \text{ km} = 1,738,000 \text{ m} \), \( V_M = \frac{4}{3} \pi (1,738,000)^3 \)
05

Calculate the Moon's density

Density \( \rho \) is given by \( \rho = \frac{M}{V} \). Using the Moon's mass \( 7.2 \times 10^{22} \text{ kg} \) and the volume calculated in Step 4, \( \rho_M = \frac{7.2 \times 10^{22}}{\frac{4}{3} \pi (1,738,000)^3} \)
06

Compare the densities

Compare the average densities of Earth, the Moon, and Earth's crust. Earth's average density is higher than that of Earth's crust, suggesting denser materials in Earth's interior. Compare this to the Moon's density to understand the differences in composition.

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!

Key Concepts

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

Earth's density
Understanding Earth's density is crucial in astronomy. Density is defined as mass divided by volume and is denoted by the formula \( \rho = \frac{M}{V} \). To find Earth's density, we first need its volume, which is calculated using the formula for the volume of a sphere: \( V = \frac{4}{3} \pi R^3 \). Given Earth's mean radius (\( R = 6,371 \text{ km} \) or \( 6,371,000 \text{ m} \)), we substitute this into the formula to get the volume. After calculating, we use Earth's mass (\( 6.0 \times 10^{24} \text{ kg} \)) and divide it by the volume to find the density.
Moon's density
The Moon's density is calculated in a similar fashion to Earth's. Given the Moon's mean radius (\( R = 1,738 \text{ km} \) or \( 1,738,000 \text{ m} \)), we use the volume formula for a sphere to find its volume. Using the Moon's mass (\( 7.2 \times 10^{22} \text{ kg} \)), we then find the Moon's density using \( \rho = \frac{M}{V} \). This density gives us insight into the materials composing the Moon and helps us compare it with Earth.
Crust density
The average density of Earth's crust is \( 2,600 \text{ kg} / \text{m}^3 \). This value is crucial as it indicates the materials forming the crust are less dense compared to the whole Earth. Since the Earth's overall density is higher, it implies that materials in the interior of the Earth are denser than those in the crust. This suggests the presence of heavier elements and metals like iron and nickel in the Earth's core.
Volume of a sphere
The volume of a sphere is calculated using the formula \( V = \frac{4}{3} \pi R^3 \). This formula is essential for determining the volumes of celestial bodies like Earth and the Moon. By knowing their radii, we can use this formula to find their volumes, which is a cornerstone in further calculations such as determining density. Understanding this formula helps in deriving important characteristics of planets and their satellites.
Density comparison
Density comparisons between Earth, the Moon, and Earth's crust can reveal important information about their compositions. Earth's average density is higher than the crust's, suggesting a denser interior. The Moon's density is lower than Earth's but similar to the crust's, indicating possibly similar surface materials. These comparisons help us understand not only the structure but also the history and formation of these celestial bodies.
  • Earth's density: higher overall, indicating heavier core materials.
  • Moon's density: lower, similar to Earth's crust, suggesting less dense overall.
  • Crust's density: lower, indicating lighter surface materials.

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

Scientists know the history of Earth's magnetic field because a. the magnetic field hasn't changed since Earth formed. b. they see today's changes and project backward in time. c. the magnetic field becomes frozen into rocks, and plate tectonics spreads those rocks apart. d. they compare the magnetic fields on other planets to Earth's.

Suppose an earthquake occurs on an imaginary planet. Scientists on the other side of the planet detect primary waves but not secondary waves after the quake. This suggests that a. part of the planet's interior is liquid. b. all of the planet's interior is solid. c. the planet has an iron core. d. the planet's interior consists entirely of rocky materials. e. the planet's mantle is liquid.

Explain how scientists know that rock layers at the bottom of the Grand Canyon are older than those found on the rim.

Lava flows on the Moon and Mercury created large, smooth plains. We don't see similar features on Earth because a. Earth has less lava. b. Earth had fewer large impacts in the past. c. Earth has plate tectonics that recycle the surface. d. Earth is large compared to the size of these plains, so they are not as noticeable. e. Earth rotates much faster than either of these other worlds.

Earth has fewer craters than Venus. Why? a. Earth's atmosphere protects better than Venus's. b. Earth is a smaller target than Venus. c. Earth is closer to the asteroid belt. d. Earth's surface experiences more erosion.
See all solutions

Recommended explanations on Physics Textbooks

Magnetism

Read Explanation

Astrophysics

Read Explanation

Quantum Physics

Read Explanation

Linear Momentum

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Electric Charge Field and Potential

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