Suppose the orbital radius of a satellite is quadrupled. (a)...

Suppose the orbital radius of a satellite is quadrupled.
(a) Does the period of the satellite increase, decrease, or stay the same?
(b) By what factor does the period of the satellite change?
(c) By what factor does the orbital speed change?

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Key Concepts

Gravitational Force and Centripetal Force Balance

Newton’s law of universal gravitation describes the attractive force between two masses, which, when balanced by the centripetal force required for circular motion, determines the satellite's orbital parameters. This balance is essential for understanding how changes in the orbital radius affect both the period and the speed of the orbiting satellite.

Kepler's Third Law

This law states that the square of the orbital period of a satellite is proportional to the cube of the semi?major axis of its orbit (T² ? r³). It provides a fundamental relationship between how far a satellite is from the planet (its orbital radius) and how long it takes to complete one orbit, meaning that an increase in orbital radius results in a longer orbital period in a predictable manner.

Orbital Velocity in Circular Orbits

For satellites in circular orbits, the orbital speed is determined by the balance of gravitational force and centripetal force, leading to the relation v = ?(GM/r). This indicates that as the orbital radius increases, the orbital speed decreases, following an inverse square root dependency on the radius.

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