Ch 13: Gravitation

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 13: Gravitation

Problem 13a

Chapter 13, Problem 13a

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?

Verified step by step guidance

Start by recalling the formula for gravitational acceleration at the surface of a celestial body: \( g = \frac{G \cdot M}{R^2} \), where \( G \) is the gravitational constant, \( M \) is the mass of the body, and \( R \) is its radius.

Identify the given values: Titania has 1/8 the radius of Earth (\( R_T = \frac{1}{8} R_E \)) and 1/1700 the mass of Earth (\( M_T = \frac{1}{1700} M_E \)).

Substitute the expressions for Titania's mass and radius into the gravitational acceleration formula: \( g_T = \frac{G \cdot (\frac{1}{1700} M_E)}{(\frac{1}{8} R_E)^2} \).

Simplify the expression: \( g_T = \frac{G \cdot M_E}{1700} \times \frac{64}{R_E^2} \).

Recognize that \( g_E = \frac{G \cdot M_E}{R_E^2} \) is the gravitational acceleration on Earth, and use it to express \( g_T \) as \( g_T = \frac{64}{1700} g_E \).

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Key Concepts

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

Gravitational Acceleration

Gravitational acceleration is the acceleration of an object due to the force of gravity exerted by a massive body, such as a planet or moon. It is calculated using the formula g = G * M / R^2, where G is the gravitational constant, M is the mass of the body, and R is its radius. This concept is crucial for determining the acceleration due to gravity on Titania's surface.

Mass and Radius Relationship

The mass and radius of a celestial body are key factors in calculating gravitational acceleration. Titania's mass is 1/1700 that of Earth, and its radius is 1/8 that of Earth. These ratios are used to adjust the gravitational acceleration formula, allowing us to compare Titania's gravity to Earth's by considering how these differences affect gravitational pull.

Universal Gravitational Constant

The universal gravitational constant (G) is a fundamental constant in physics that quantifies the strength of the gravitational force between two masses. Its value is approximately 6.674 × 10^-11 N(m/kg)^2. This constant is essential for calculating gravitational forces and accelerations, including the gravity on Titania, by applying it in the formula for gravitational acceleration.

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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 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.)

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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

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

At what distance above the surface of the earth is the acceleration due to the earth's gravity 0.980 m/s² if the acceleration due to gravity at the surface has magnitude 9.80 m/s² ?

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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.