Calculating the Acceleration Due to Gravity

Oct 13, 2025 by magicalceylontours.com 149 views

Introduction

The acceleration due to gravity is a fundamental concept in physics that describes the force exerted by the Earth on objects near its surface. However, as objects move further away from the Earth's surface, the acceleration due to gravity decreases. In this article, we will determine the acceleration due to gravity for a satellite in orbit 200 km above the surface of the Earth.

Given Information

Radius of Earth, $R_E = 6.38 \times 10^6 \text{ m}$
Acceleration due to gravity at Earth's surface, $g = 9.8 \text{ m/s}^2$

Understanding the Problem

The acceleration due to gravity at a distance $r$ from the center of the Earth is given by the formula:

g(r) = \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.

However, we are given the acceleration due to gravity at the Earth's surface, $g$ , and the radius of the Earth, $R_E$ . We need to find the acceleration due to gravity at a distance $r = R_E + 200 \text{ km}$ from the center of the Earth.

Calculating the Acceleration Due to Gravity

To calculate the acceleration due to gravity at a distance $r = R_E + 200 \text{ km}$ from the center of the Earth, we can use the formula:

g(r) = \frac{GM}{r^2}

We know that the mass of the Earth, $M$ , is related to the acceleration due to gravity at the Earth's surface, $g$ , by the formula:

g = \frac{GM}{R_E^2}

We can rearrange this formula to solve for $M$ :

M = \frac{gR_E^2}{G}

Now, we can substitute this expression for $M$ into the formula for $g(r)$ :

g(r) = \frac{G\frac{gR_E^2}{G}}{r^2}

Simplifying this expression, we get:

g(r) = \frac{gR_E^2}{r^2}

Now, we can substitute $r = R_E + 200 \text{ km}$ into this expression:

g(r) = \frac{gR_E^2}{(R_E + 200 \text{ km})^2}

Numerical Calculation

To calculate the acceleration due to gravity at a distance $r = R_E + 200 \text{ km}$ from the center of the Earth, we need to substitute the given values into the expression:

$g = 9.8 \text{ m/s}^2$
$R_E = 6.38 \times 10^6 \text{ m}$
$200 \text{ km} = 2 \times 10^5 \text{ m}$

Substituting these values into the expression, we get:

g(r) = \frac{9.8 \text{ m/s}^2 (6.38 \times 10^6 \text{ m})^2}{(6.38 \times 10^6 \text{ m} + 2 \times 10^5 \text{ m})^2}

Evaluating this expression, we get:

g(r) = 8.02 \text{ m/s}^2

Conclusion

In this article, we determined the acceleration due to gravity for a satellite in orbit 200 km above the surface of the Earth. We used the formula for the acceleration due to gravity at a distance $r$ from the center of the Earth and substituted the given values into the expression. We found that the acceleration due to gravity at a distance $r = R_E + 200 \text{ km}$ from the center of the Earth is $8.02 \text{ m/s}^2$ .

References

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Additional Resources

[1] NASA. (n.d.). Gravity. Retrieved from https://www.nasa.gov/subject/1236/gravity
[2] Khan Academy. (n.d.). Gravity. Retrieved from https://www.khanacademy.org/science/physics/one-dimensional-motion/gravity/v/gravity

Discussion

Q: What is the acceleration due to gravity for a satellite in orbit 200 km above the surface of the Earth?

A: The acceleration due to gravity for a satellite in orbit 200 km above the surface of the Earth is $8.02 \text{ m/s}^2$ .

Q: How does the altitude of the satellite affect the acceleration due to gravity?

A: The altitude of the satellite affects the acceleration due to gravity by decreasing it. As the satellite moves further away from the Earth's surface, the acceleration due to gravity decreases.

Q: What are some other factors that affect the acceleration due to gravity for a satellite in orbit?

A: Some other factors that affect the acceleration due to gravity for a satellite in orbit include:

The mass of the Earth: The mass of the Earth affects the acceleration due to gravity by increasing it.
The radius of the Earth: The radius of the Earth affects the acceleration due to gravity by decreasing it.
The distance from the center of the Earth: The distance from the center of the Earth affects the acceleration due to gravity by decreasing it.

Q: How does the acceleration due to gravity for a satellite in orbit compare to the acceleration due to gravity at the Earth's surface?

A: The acceleration due to gravity for a satellite in orbit is less than the acceleration due to gravity at the Earth's surface. At the Earth's surface, the acceleration due to gravity is $9.8 \text{ m/s}^2$ , while at an altitude of 200 km, the acceleration due to gravity is $8.02 \text{ m/s}^2$ .

Q: What are some real-world applications of the acceleration due to gravity for satellites in orbit?

A: Some real-world applications of the acceleration due to gravity for satellites in orbit include:

Space exploration: Understanding the acceleration due to gravity for satellites in orbit is crucial for space exploration, as it affects the trajectory of the satellite and the forces acting on it.
Satellite design: The acceleration due to gravity for satellites in orbit affects the design of the satellite, including its mass, shape, and propulsion system.
Navigation: The acceleration due to gravity for satellites in orbit affects the navigation of the satellite, including its position, velocity, and attitude.

Q: How can the acceleration due to gravity for a satellite in orbit be measured?

A: The acceleration due to gravity for a satellite in orbit can be measured using a variety of methods, including:

Gravimeters: Gravimeters are instruments that measure the acceleration due to gravity by detecting the force exerted on a test mass.
Accelerometers: Accelerometers are instruments that measure the acceleration of the satellite by detecting the force exerted on it.
Orbit determination: Orbit determination involves using a combination of measurements and calculations to determine the position, velocity, and attitude of the satellite.

Q: What are some common misconceptions about the acceleration due to gravity for a satellite in orbit?

A: Some common misconceptions about the acceleration due to gravity for a satellite in orbit include:

The acceleration due to gravity for a satellite in orbit is zero: This is incorrect, as the acceleration due to gravity for a satellite in orbit is actually less than the acceleration due to gravity at the Earth's surface.
The acceleration due to gravity for a satellite in orbit is constant: This is incorrect, as the acceleration due to gravity for a satellite in orbit varies with altitude and distance from the center of the Earth.

Q: What are some future directions for research on the acceleration due to gravity for satellites in orbit?

A: Some future directions for research on the acceleration due to gravity for satellites in orbit include:

Developing more accurate models of the acceleration due to gravity for satellites in orbit
Investigating the effects of the acceleration due to gravity on the behavior of satellites in orbit
Developing new technologies for measuring the acceleration due to gravity for satellites in orbit

Conclusion

In this article, we have answered some common questions about the acceleration due to gravity for a satellite in orbit. We have discussed the factors that affect the acceleration due to gravity, including the altitude of the satellite, the mass of the Earth, and the distance from the center of the Earth. We have also discussed some real-world applications of the acceleration due to gravity for satellites in orbit and some common misconceptions about it. Finally, we have identified some future directions for research on the acceleration due to gravity for satellites in orbit.