Final answer:
To calculate the radius of a geosynchronous Earth orbit, we use the gravitational law based on Kepler's Third Law. The radius of the orbit where the geosynchronous satellite will precisely match the orbital period of one day is approximately 42,164 kilometers from the center of the Earth.
Step-by-step explanation:
The student has asked to calculate the radius of a geosynchronous Earth orbit, which is an orbit where the satellite has an orbital period of exactly one day. To calculate this radius, we can use the gravitational law of orbital motion, derived from Kepler's Third Law, which relates the square of the orbital period (T) of a satellite to the cube of the radius (r) of its orbit around a central body (in this case, the Earth). We have the orbital period of one day (T = 24 hours) and need to solve for the radius 'r'.
The formula we use takes the form: T² = (4π²/GM)r³, where G is the gravitational constant, M is the mass of the Earth, and π is the pi constant. For the Earth, the value of GM (the geocentric gravitational constant) is approximately 3.986 x 10⁵ km³/s². Rearranging the formula to solve for 'r' gives us r³ = (GMT²)/(4π²). After substituting the known values and solving, we find that the radius 'r' is approximately 42,164 kilometers from the center of the Earth.
This radius allows the geosynchronous satellite to maintain a fixed position relative to the Earth's surface. Such satellites are crucial for applications like communication, broadcasting, and weather observations.