Ch 13: Gravitation

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 13: Gravitation

Problem 13b

Chapter 13, Problem 13b

Titania, the largest moon of the planet Uranus, has 1/8 the radius of the earth and 1/1700 the mass of the earth. What is the average density of Titania? (This is less than the density of rock, which is one piece of evidence that Titania is made primarily of ice.)

Verified step by step guidance

Start by recalling the formula for density, which is defined as mass divided by volume: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \).

To find the volume of Titania, use the formula for the volume of a sphere: \( V = \frac{4}{3} \pi r^3 \). Since Titania has 1/8 the radius of Earth, its radius \( r_T \) is \( r_T = \frac{1}{8} r_E \), where \( r_E \) is the radius of Earth.

Substitute \( r_T = \frac{1}{8} r_E \) into the volume formula: \( V_T = \frac{4}{3} \pi \left( \frac{1}{8} r_E \right)^3 \). Simplify this expression to find the volume of Titania in terms of Earth's volume.

Given that Titania has 1/1700 the mass of Earth, express Titania's mass \( M_T \) as \( M_T = \frac{1}{1700} M_E \), where \( M_E \) is the mass of Earth.

Finally, substitute the expressions for \( M_T \) and \( V_T \) into the density formula: \( \text{Density of Titania} = \frac{\frac{1}{1700} M_E}{\frac{4}{3} \pi \left( \frac{1}{8} r_E \right)^3} \). Simplify this expression to find the average density of Titania in terms of Earth's mass and radius.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Density

Density is a measure of mass per unit volume, typically expressed in kilograms per cubic meter (kg/m³). It is calculated by dividing the mass of an object by its volume. Understanding density is crucial for determining the composition of celestial bodies, as it provides insights into their internal structure and material makeup.

Volume of a Sphere

The volume of a sphere is calculated using the formula V = (4/3)πr³, where r is the radius of the sphere. This formula is essential for determining the volume of spherical objects like moons and planets, which is a necessary step in calculating their density when the mass is known.

Gravitational Scaling

Gravitational scaling involves understanding how changes in size and mass affect gravitational properties. For celestial bodies, knowing how the radius and mass compare to Earth helps in calculating other properties like density. In this context, Titania's radius and mass relative to Earth are used to find its volume and subsequently its density.

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Textbook Question

Jupiter's moon Io has active volcanoes (in fact, it is the most volcanically active body in the solar system) that eject material as high as 500 km (or even higher) above the surface. Io has a mass of 8.93 × 10²² kg and a radius of 1821 km. For this calculation, ignore any variation in gravity over the 500-km range of the debris. How high would this material go on earth if it were ejected with the same speed as on Io?

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Textbook Question

Ten days after it was launched toward Mars in December 1998, the Mars Climate Orbiter spacecraft (mass 629 kg) was 2.87 × 10⁶ km from the earth and traveling at 1.20 × 10⁴ km/h relative to the earth. At this time, what were (a) the spacecraft's kinetic energy relative to the earth and (b) the potential energy of the earth–spacecraft system?

Textbook Question

Use the results of Example 13.5 (Section 13.3) to calculate the escape speed for a spacecraft (a) from the surface of Mars and (b) from the surface of Jupiter. Use the data in Appendix F. (c) Why is the escape speed for a spacecraft independent of the spacecraft's mass?

Textbook Question

The mass of Venus is 81.5% that of the earth, and its radius is 94.9% that of the earth. If a rock weighs 75.0 N on earth, what would it weigh at the surface of Venus?

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Textbook Question

Titania, the largest moon of the planet Uranus, has 1/8 the radius of the earth and 1/1700 the mass of the earth. What is the acceleration due to gravity at the surface of Titania?

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Textbook Question

The mass of Venus is 81.5% that of the earth, and its radius is 94.9% that of the earth. Compute the acceleration due to gravity on the surface of Venus from these data.