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Ch 13: Gravitation
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 13: GravitationProblem 27
Chapter 13, Problem 27

Two satellites are in circular orbits around a planet that has radius 9.00 × 106 m. One satellite has mass 68.0 kg, orbital radius 7.00 × 107 m, and orbital speed 4800 m/s. The second satellite has mass 84.0 kg and orbital radius 3.00 × 107 m. What is the orbital speed of this second satellite?

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1
Identify the relevant physics principle: The gravitational force provides the necessary centripetal force for a satellite in circular orbit. This can be expressed as: F_gravity = F_centripetal.
Write the formula for gravitational force: F_gravity = (G * M * m) / r^2, where G is the gravitational constant, M is the mass of the planet, m is the mass of the satellite, and r is the orbital radius.
Write the formula for centripetal force: F_centripetal = (m * v^2) / r, where m is the mass of the satellite, v is the orbital speed, and r is the orbital radius.
Set the gravitational force equal to the centripetal force: (G * M * m) / r^2 = (m * v^2) / r. Notice that the mass of the satellite (m) cancels out from both sides of the equation.
Solve for the orbital speed (v) of the second satellite: v = sqrt((G * M) / r). Substitute the given orbital radius of the second satellite into this equation to find its orbital speed.

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Key Concepts

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

Gravitational Force and Circular Motion

In circular orbits, the gravitational force provides the necessary centripetal force to keep a satellite in orbit. This relationship is expressed as F_gravity = F_centripetal, where F_gravity = G * (m1 * m2) / r^2 and F_centripetal = m * v^2 / r. Understanding this balance is crucial for determining the orbital speed of a satellite.
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Orbital Speed Formula

The orbital speed of a satellite in a circular orbit can be derived from the balance of gravitational and centripetal forces, resulting in v = sqrt(G * M / r), where G is the gravitational constant, M is the mass of the planet, and r is the orbital radius. This formula allows us to calculate the speed needed to maintain a stable orbit at a given radius.
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Kepler's Third Law

Kepler's Third Law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. For circular orbits, this implies that the orbital speed is inversely proportional to the square root of the orbital radius, which helps compare the speeds of satellites at different radii around the same planet.
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Related Practice
Textbook Question

In 2004 astronomers reported the discovery of a large Jupiter-sized planet orbiting very close to the star HD 179949 (hence the term 'hot Jupiter'). The orbit was just 1/9 the distance of Mercury from our sun, and it takes the planet only 3.09 days to make one orbit (assumed to be circular). How fast (in km/s) is this planet moving?

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Textbook Question

In its orbit each day, the International Space Station makes 15.65 revolutions around the earth. Assuming a circular orbit, how high is this satellite above the surface of the earth?

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Textbook Question

For a satellite to be in a circular orbit 890 km above the surface of the earth, what orbital speed must it be given?

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Textbook Question

On July 15, 2004, NASA launched the Aura spacecraft to study the earth's climate and atmosphere. This satellite was injected into an orbit 705 km above the earth's surface. Assume a circular orbit. How many hours does it take this satellite to make one orbit?

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Textbook Question

A planet orbiting a distant star has radius 3.24 × 106 m. The escape speed for an object launched from this planet’s surface is 7.65 × 103 m/s. What is the acceleration due to gravity at the surface of the planet?

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Textbook Question

The star Rho1 Cancri is 57 light-years from the earth and has a mass 0.85 times that of our sun. A planet has been detected in a circular orbit around Rho1 Cancri with an orbital radius equal to 0.11 times the radius of the earth's orbit around the sun. What are (a) the orbital speed and (b) the orbital period of the planet of Rho1 Cancri?