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. Find the value of g at this altitude.

Verified step by step guidance

1. Step 1: Recall the formula for gravitational acceleration at a distance from the center of the Earth: $g=\frac{GM}{r^2}$, where $G$ is the gravitational constant, $M$ is the mass of the Earth, and $r$ is the distance from the center of the Earth.
2. Step 2: Determine the total distance $r$ from the center of the Earth to the satellite. This is the sum of the Earth's radius ($R=6.37\times {10}^6 m$) and the altitude of the geosynchronous orbit ($3.58\times {10}^7 m$).
3. Step 3: Substitute the values of $G$ ($6.674\times {10}^{-} N{m}^2/{\mathrm{kg}}^2$), $M$ ($5.972\times {10}^{24} \mathrm{kg}$), and $r$ (calculated in Step 2) into the formula for $g$.
4. Step 4: Perform the necessary calculations to evaluate $g$. This involves squaring the value of $r$, multiplying $G$ and $M$, and dividing by $r^2$.
5. Step 5: Interpret the result. The value of $g$ at this altitude will be significantly smaller than the gravitational acceleration at the Earth's surface ($9.8 m/s^2$) due to the increased distance from the Earth's center.

Key Concepts

Gravitational Acceleration

Gravitational acceleration, denoted as 'g', is the acceleration experienced by an object due to the gravitational force exerted by a massive body, such as Earth. At the surface of the Earth, this value is approximately 9.81 m/s². However, as altitude increases, 'g' decreases due to the inverse square law of gravitation, which states that gravitational force diminishes with the square of the distance from the center of the mass.

Geosynchronous Orbit

A geosynchronous orbit is a circular orbit around the Earth where a satellite's orbital period matches the Earth's rotation period, approximately 24 hours. This allows the satellite to remain fixed over a specific point on the equator. The altitude required for a geosynchronous orbit is about 35,786 kilometers (22,236 miles), where the gravitational pull and the centripetal force balance perfectly.

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 satellites in orbit, this force is provided by the gravitational attraction between the satellite and the Earth. The balance between gravitational force and the required centripetal force determines the satellite's stable orbit, which is crucial for maintaining its position relative to the Earth's surface.