Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Problem 47

The asteroid Vesta has a diameter of \(530 \mathrm{km}\) and a mass of \(2.7 \times 10^{20} \mathrm{kg}\) a. Calculate the density (mass/volume) of Vesta. b. The density of water is \(1,000 \mathrm{kg} / \mathrm{m}^{3},\) and that of rock is about \(2,500 \mathrm{kg} / \mathrm{m}^{3} .\) What does this difference tell you about the composition of this primitive body?

Short Answer

Expert verified
Vesta's density can be calculated and compared to known densities of water and rock to infer its composition.

Step by step solution

01

- Calculate the Volume of Vesta

Given the diameter of Vesta is 530 km, first convert it to meters: 530 km = 530,000 m. Assume Vesta is a spherical body. The volume of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \] where \(r\) is the radius. Given the diameter, the radius \(r\) would be: \[ r = \frac{530,000}{2} = 265,000 \space m \] Thus, the volume can be calculated as: \[ V = \frac{4}{3} \pi (265,000)^3 \] Perform the calculation to find the volume.
02

- Calculate the Density of Vesta

Density is given by the formula: \[ \text{Density} = \frac{\text{mass}}{\text{volume}} \] Given mass of Vesta as \(2.7 \times 10^{20} \mathrm{kg}\) and the previously calculated volume, substitute these values into the formula to find the density.
03

- Compare the Density of Vesta with Water and Rock

The density of water is \(1,000 \mathrm{kg} / \mathrm{m}^3\) and that of rock is \(2,500 \mathrm{kg} / \mathrm{m}^3\). Compare Vesta's density to these values to infer the basic composition of Vesta. A density closer to rock suggests a rocky composition, while a lower density suggests a mix of rock and other materials or voids.

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.

Volume Calculation
To calculate the volume of Vesta, we need to use its given diameter. First, we convert the diameter from kilometers to meters, which will make our calculations easier and more precise. Given diameter: 530 km = 530,000 meters.

Next, we consider Vesta as a perfect sphere. The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius. The radius is half of the diameter:
\( r = \frac{530,000}{2} = 265,000 \) meters.

Now we plug the radius into our formula for volume:
\[ V = \frac{4}{3} \pi (265,000)^3 \]

By calculating this, we get the volume of Vesta. It’s a straightforward method if we break it down into simple steps.
Density Formula
Once we have the volume, we can easily find the density of Vesta. Density tells us how much mass is packed into a given volume. The formula is:
\[ \text{Density} = \frac{\text{mass}}{\text{volume}} \]

From the problem, we know that the mass of Vesta is given as \( 2.7 \times 10^{20} \) kilograms. Using the volume calculated earlier, we substitute the mass and volume into the formula to find the density:
\[ \text{Density} = \frac{2.7 \times 10^{20}}{\text{calculated volume}} \]

This gives us the density of Vesta in kilograms per cubic meter (kg/m^3). Breaking it down like this helps us see that calculating density involves understanding the relationship between mass and volume.
Comparison to Water and Rock
We now compare Vesta’s density to the densities of water and rock to understand its composition better. The density of water is 1,000 kg/m^3, while the density of rock is about 2,500 kg/m^3.

By comparing these values with Vesta’s density:
  • If Vesta’s density is closer to 2,500 kg/m^3, it suggests that Vesta is primarily made of rock.
  • If Vesta’s density is significantly lower, it may contain a mix of rock and other less dense materials, or it could have voids (empty spaces) inside.

These comparisons help us infer that an object with a density similar to rock is rock-like in composition, while a lower density may suggest a more mixed or porous composition. This step shows how densities provide clues to the internal structure and materials of celestial bodies like Vesta.

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

since angular momentum is conserved, an ice-skater who throws her arms out will a. rotate more slowly. b. rotate more quickly. c. rotate at the same rate. d. stop rotating entirely.

How does the law of conservation of angular momentum control a figure-skater's rate of spin?

If the radius of an object's orbit is halved, what must happen to the speed so that angular momentum is conserved? a. It must be halved. b. It must stay the same. c. It must be doubled. d. It must be squared.

Using the exoplanet catalogs: a. Go to the "Catalog" Web page (http://exoplanet.edu/ catalog) of the Extrasolar Planets Encyclopaedia and click on "All Planets detected." Look for a star (in the left column that has multiple planets. Make a graph showing the distances of the planets from their star, and note the masses and sizes of the planets. Put the Solar System planets on the same axis. How does this extrasolar planet system compare with the Solar System? b. Go to the "Exoplanets Data Explorer" website (http:// exoplanets.org \(),\) and click on "Table." This website lists planets that have detailed orbital data published in scientific journals, and it may have a smaller total count than the site in (a). Pick a planet that was discovered this year or last, as specified in the "First Reference" column. What is the planet's minimum mass? What is its semimajor axis and the period of its orbit? Is its orbit circular or more elliptical? Click on the star name in the first column to get more information. Is there a radial velocity curve for this planet? Was it observed in transit, and if so, what is the planet's radius and density? Is it more like Jupiter or more like Earth?

Look under your bed for "dust bunnies." If there aren't any, look under your roommate's bed, the refrigerator, or any similar place that might have some. Once you find them, blow one toward another. Watch carefully and describe what happens as they meet. What happens if you repeat this action with additional dust bunnies? Will these dust bunnies ever have enough gravity to begin pulling themselves together? If they were in space instead of on the floor, might that happen? What force prevents their mutual gravity from drawing them together into a "bunny-tesimal" under your bed?
See all solutions

Recommended explanations on Physics Textbooks

Force

Read Explanation

Atoms and Radioactivity

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Thermodynamics

Read Explanation

Physical Quantities And Units

Read Explanation

Energy 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