Communications satellites are placed in circular orbits where they stay directly over a fixed point on the equator as the Earth rotates. These are called geosynchronous orbits. The altitude of a geosynchronous orbit is 3.58 x 10⁷ m (approximately 22,00 miles). Astronomical data are inside the back cover of the book. What is the weight of a 2000 kg satellite in a geosynchronous orbit?

Verified step by step guidance

1. Step 1: Understand the problem. The weight of an object in orbit is the gravitational force acting on it. The formula for gravitational force is given by: $F=\frac{GmM}{r^2}$, where $G$ is the gravitational constant, $m$ is the mass of the satellite, $M$ is the mass of the Earth, and $r$ is the distance from the center of the Earth to the satellite.
2. Step 2: Calculate the distance $r$ from the center of the Earth to the satellite. This is the sum of the Earth's radius and the altitude of the geosynchronous orbit. Use the Earth's radius $R=6.37\times {10}^6 m$ and the altitude of the orbit $h=3.58\times {10}^7 m$. Add these values: $r=R+h$.
3. Step 3: Substitute the known values into the gravitational force formula. Use $G=6.674\times {10}^{-11} \frac{N}{m^2}$, $M=5.97\times {10}^{24} \mathrm{kg}$, and the mass of the satellite $m=2000 \mathrm{kg}$. The formula becomes: $F=\frac{6.674\times {10}^{-11}\times 2000\times 5.97\times {10}^{24}}{{6.37\times {10}^6+3.58\times {10}^7}^2}$.
4. Step 4: Simplify the denominator by squaring the total distance $r$. Then, simplify the numerator by multiplying the constants and masses together.
5. Step 5: Divide the simplified numerator by the simplified denominator to find the gravitational force $F$, which represents the weight of the satellite in the geosynchronous orbit.

Key Concepts

Weight and Gravitational Force

Weight is the force exerted on an object due to gravity, calculated as the product of mass and the acceleration due to gravity (W = mg). In a geosynchronous orbit, the gravitational force acting on the satellite is balanced by the centripetal force required to keep it in circular motion, allowing it to maintain a stable position relative to the Earth's surface.

Geosynchronous Orbit

A geosynchronous orbit is a circular orbit around the Earth where a satellite's orbital period matches the Earth's rotation period of approximately 24 hours. This allows the satellite to remain fixed over a specific point on the equator, making it ideal for communication purposes, as it can continuously cover the same area.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. For a satellite in geosynchronous orbit, this force is provided by the gravitational pull of the Earth, which must equal the required centripetal force to maintain the satellite's circular motion at that altitude.