A satellite has a mass of 115 kg and is located at 1.95 ✕ 106 m above the surface of Earth.(a) What is the potential energy associated with the satellite at this location? Your response is off by a multiple of ten. J(b) What is the magnitude of the gravitational force on the satellite? Your response is off by a multiple of ten. N
Question
A satellite has a mass of 115 kg and is located at 1.95 ✕ 106 m above the surface of Earth.(a) What is the potential energy associated with the satellite at this location? Your response is off by a multiple of ten. J(b) What is the magnitude of the gravitational force on the satellite? Your response is off by a multiple of ten. N
Solution
(a) The potential energy (PE) of an object is given by the formula:
PE = m * g * h
where: m = mass of the object = 115 kg g = acceleration due to gravity = 9.8 m/s² (approximately, at the surface of the Earth) h = height above the surface of the Earth = 1.95 x 10^6 m
However, because the satellite is significantly far from the Earth's surface, we should use the formula for gravitational potential energy:
PE = - G * m * M / r
where: G = gravitational constant = 6.674 x 10^-11 N(m/kg)² m = mass of the satellite = 115 kg M = mass of the Earth = 5.972 x 10^24 kg r = distance from the center of the Earth = radius of the Earth + height above the surface = 6.371 x 10^6 m + 1.95 x 10^6 m = 8.321 x 10^6 m
Substituting these values in:
PE = - (6.674 x 10^-11 N(m/kg)²) * (115 kg) * (5.972 x 10^24 kg) / (8.321 x 10^6 m) PE = -2.05 x 10^10 J
The negative sign indicates that work would have to be done to remove the satellite from the Earth's gravitational field.
(b) The magnitude of the gravitational force (F) on the satellite is given by the formula:
F = G * m * M / r²
Substituting the same values in:
F = (6.674 x 10^-11 N(m/kg)²) * (115 kg) * (5.972 x 10^24 kg) / (8.321 x 10^6 m)² F = 2.72 x 10^3 N
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