Predict/Calculate Satellite A has a mass of 1000 kg and satellite B has a mass of 2000 kg . (a)

step 1 Two satellites are orbiting the Earth at exactly one Earth radius above the surface of the Earth

Predict/Calculate Satellite A has a mass of 1000 kg and...

Predict/Calculate Satellite A has a mass of 1000 $\mathrm{kg}$ and satellite $\mathrm{B}$ has a mass of 2000 $\mathrm{kg}$ . (a) When each satellite orbits at one Earth radius above the surface of the Earth, will the period of satellite $\mathrm{B}$ be longer, shorter, or the same as the period of satellite A? (b) Calculate the orbital periods of satellites A and B.

Key Concepts

Kepler's Third Law

Kepler's Third Law relates the orbital period of a satellite to the radius of its orbit, showing that the square of the period is proportional to the cube of the orbital radius. This principle is fundamental in orbital mechanics and is used to predict how changes in orbital distance affect the period.

Gravitational Force and Centripetal Force Equivalence

In circular orbital motion, the gravitational force acting on a satellite provides the necessary centripetal force to keep it in orbit. By setting the gravitational force equal to the centripetal force, one can derive key formulas governing orbital motion, such as the orbital period expression.

Mass Independence of Orbital Period

A critical result in orbital mechanics is that for satellites orbiting a much more massive central body, such as Earth, the satellite’s mass cancels out in the equations. This means that the orbital period depends only on the distance from the center of the Earth and the Earth's mass, not the mass of the satellite itself.

Orbital Period Formula

The orbital period of a satellite in a circular orbit can be calculated using the formula T = 2??(r³/GM), where T is the period, r is the orbital radius (distance from the center of the Earth), G is the gravitational constant, and M is the mass of the Earth. This formula is essential for computing the time taken for one complete orbit.

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