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 24

Chapter 13, Problem 24

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?

Verified step by step guidance

First, understand that the International Space Station (ISS) completes 15.65 revolutions around the Earth in one day. This information can be used to find the orbital period (T) of the ISS.

Calculate the orbital period (T) by dividing the total time in one day (24 hours) by the number of revolutions (15.65). Convert the time into seconds for consistency in units.

Use Kepler's Third Law, which relates the orbital period (T) to the radius of the orbit (r). The formula is:

T^{2} = \frac{4 π^{2}}{G M} r^{3}

, where G is the gravitational constant and M is the mass of the Earth.

Rearrange the formula to solve for the radius (r) of the orbit:

r = {\frac{G M T^{2}}{4 π^{2}}}^{\frac{1}{3}}

Finally, subtract the Earth's radius from the calculated orbital radius to find the height of the ISS above the Earth's surface. The Earth's average radius is approximately 6,371 kilometers.

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Key Concepts

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

Orbital Mechanics

Orbital mechanics involves the study of the motion of objects in space under the influence of gravitational forces. For a satellite like the International Space Station (ISS), understanding its orbit requires knowledge of how gravitational forces from Earth dictate its path and speed, ensuring it remains in a stable orbit.

Circular Orbit

A circular orbit is a type of orbit where the satellite maintains a constant distance from the center of the Earth, resulting in a circular path. This simplifies calculations as the radius of the orbit remains constant, allowing for easier determination of orbital parameters such as altitude and velocity.

Gravitational Force

Gravitational force is the attractive force between two masses, such as the Earth and the ISS. It is crucial for maintaining the satellite's orbit, as it provides the necessary centripetal force to keep the ISS moving in its circular path. The strength of this force depends on the mass of the Earth and the distance between the Earth and the satellite.

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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?

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