A geostationary satellite orbits the earth at a height of nearly 36000km from the surface of the earth. What is the potential due to earth's gravity at the site of this satellite?

Question

Answer

Mass of the Earth, M = 6.0 × 1024 kg
Radius of the Earth, R = 6400 km = 6.4 × 106 m
Height of a geostationary satellite from the surface of the Earth,
h = 36000 km = 3.6 × 107 m
Gravitational potential energy due to Earth’s gravity at height h,
= -GM / (R + h)
= – 6.67 × 10-11 × 6 × 1024 / (3.6 × 107 + 0.64 × 107)
= -9.4 × 106 J/kg.

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Answer

Potential Due to Earth's Gravity at Geostationary Satellite Orbit

At a height of nearly 36000km from the surface of the earth, a geostationary satellite orbits the earth. Let's find out the potential due to earth's gravity at the site of this satellite.

Formula for Potential Due to Earth's Gravity

The formula to calculate the potential due to earth's gravity is:

V(r) = - G M / r

Where:

V(r) is the potential at a distance r from the center of the earth

G is the universal gravitational constant

M is the mass of the earth

r is the distance from the center of the earth

Calculation

Using the above formula, we can calculate the potential at a height of 36000km from the surface of the earth as follows:

V(r) = - G M / r

V(r) = - (6.67 × 10^-11 N m^2/kg^2) × (5.97 × 10^24 kg) / (6.378 × 10^6 m + 3.6 × 10^7 m)

V(r) = - 2.66 × 10^7 J/kg

Explanation

The potential due to earth's gravity at the site of a geostationary satellite orbit is negative because it is a measure of the work that needs to be done to bring a unit mass from infinity to that point against the force of gravity. At this height, the gravitational force is balanced with the centrifugal force due to the satellite's orbital velocity, which keeps the satellite in a fixed position relative to the earth's surface. This height is known as the geostationary orbit, and satellites placed in this orbit are used for communication, weather monitoring, and other applications.

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