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 22a

Chapter 13, Problem 22a

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?

Verified step by step guidance

First, understand that the satellite is in a circular orbit around the Earth. To find the time it takes to complete one orbit, we need to calculate the orbital period.

The orbital period can be found using Kepler's Third Law, which relates the period of orbit to the radius of the orbit. The formula is:

T = \frac{2 π}{\sqrt{\frac{r^{3}}{GM}}}

, where

r

is the radius of the orbit,

G

is the gravitational constant, and

M

is the mass of the Earth.

Calculate the radius of the orbit. The radius is the sum of the Earth's radius and the altitude of the satellite. Earth's average radius is approximately 6371 km, so the orbital radius

r

6371 + 705

km.

Substitute the values into the formula. Use

G

6.674 \times 10^- 11

m³ kg^-1 s^-2 and

M

5.972 \times 10^24

kg.

Convert the orbital period from seconds to hours by dividing by 3600 seconds per hour. This will give you the time it takes for the satellite to complete one orbit in hours.

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Key Concepts

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

Circular Orbit

A circular orbit is a path followed by an object around a celestial body where the distance from the center of the body remains constant. In this context, the satellite maintains a constant altitude of 705 km above Earth's surface, implying a uniform circular motion. Understanding circular orbits is crucial for calculating orbital periods and velocities.

Gravitational Force

Gravitational force is the attractive force between two masses, such as a satellite and Earth. It provides the necessary centripetal force to keep the satellite in orbit. The balance between gravitational pull and the satellite's inertia determines its orbital speed and period, which are essential for calculating how long it takes to complete one orbit.

Orbital Period

The orbital period is the time taken for a satellite to complete one full orbit around a celestial body. It depends on the altitude of the orbit and the mass of the central body. For a satellite in a circular orbit, the period can be calculated using Kepler's third law, which relates the orbital radius to the period, providing insights into the satellite's motion.

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