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A particular satellite was placed in a circular orbit about 328 mi above Earth.(a) Determine the orbital speed of the satellite. m/s(b) Determine the time required for one complete revolution.

Question

A particular satellite was placed in a circular orbit about 328 mi above Earth.(a) Determine the orbital speed of the satellite. m/s(b) Determine the time required for one complete revolution.

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Solution

(a) To determine the orbital speed of the satellite, we first need to find the radius of the orbit. The radius of the Earth is approximately 3960 miles, so the radius of the orbit is 3960 miles + 328 miles = 4288 miles. Convert this to meters (since we want the speed in m/s) using the conversion factor 1 mile = 1609.34 meters. So, the radius of the orbit is 4288 miles * 1609.34 m/mile = 6907715.52 meters.

The formula for the speed of an object in circular orbit around the Earth is given by v = sqrt(GM/r), where G is the gravitational constant (6.674 * 10^-11 m^3 kg^-1 s^-2), M is the mass of the Earth (5.972 * 10^24 kg), and r is the radius of the orbit.

Substituting the given values, we get v = sqrt((6.674 * 10^-11 m^3 kg^-1 s^-2 * 5.972 * 10^24 kg) / 6907715.52 m) = 7670.34 m/s. So, the orbital speed of the satellite is approximately 7670.34 m/s.

(b) The time required for one complete revolution, or the period of the orbit, is given by the formula T = 2πr/v, where r is the radius of the orbit and v is the speed of the satellite.

Substituting the given values, we get T = 2π * 6907715.52 m / 7670.34 m/s = 5651.14 seconds. Convert this to minutes by dividing by 60 (since there are 60 seconds in a minute), we get T = 5651.14 s / 60 = 94.19 minutes. So, the time required for one complete revolution is approximately 94.19 minutes.

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