Determining the Altitude for a Given Gravitational Acceleration
Understanding where the gravitational acceleration on Earth's surface changes to a specific value is crucial in various scientific and engineering applications. In this article, we explore how to determine the altitude above the Earth's surface at which the gravitational acceleration is 9.71 m/s2. This specific value is often used as it is approximately two-thirds of the standard gravitational acceleration on Earth's surface (9.81 m/s2).
Understanding Gravitational Acceleration
The gravitational acceleration at any point on or above the Earth's surface is affected by the distance from the Earth's center. Gravitational force follows the inverse square law, which states that the gravitational acceleration is inversely proportional to the square of the distance from the Earth's center.
Theoretical Background
The gravitational acceleration ( g ) at a height ( h ) above the Earth's surface can be determined using the formula:
[g frac{GM}{(R h)^2}]where ( G ) is the gravitational constant (( 6.674 times 10^{-11} , mathrm{m^3 , kg^{-1} , s^{-2}} )), ( M ) is the mass of the Earth (( 5.972 times 10^{24} , mathrm{kg} )), and ( R ) is the radius of the Earth (( 6.371 times 10^6 , mathrm{m} )).
Calculating the Required Altitude
Given the gravitational acceleration on the Earth's surface ( g_0 9.81 , mathrm{m/s^2} ), we need to determine the altitude ( h ) above the Earth's surface where the gravitational acceleration is ( g 9.71 , mathrm{m/s^2} ).
The formula for gravitational acceleration at a height ( h ) is given by:
[g frac{g_0 R^2}{(R h)^2}]Substituting the given values, we get:
[9.71 frac{9.81 times (6.371 times 10^6)^2}{(6.371 times 10^6 h)^2}]Solving for ( h ), we find:
[(6.371 times 10^6 h)^2 frac{9.81 times (6.371 times 10^6)^2}{9.71}][(6.371 times 10^6 h) sqrt{frac{9.81 times (6.371 times 10^6)^2}{9.71}}]
[6.371 times 10^6 h 6.405 times 10^6]
[h 6.405 times 10^6 - 6.371 times 10^6 3.4 times 10^4 , mathrm{m} 34 , mathrm{kms}]
Conclusion
The altitude above the Earth's surface at which the gravitational acceleration is approximately 9.71 m/s2 is approximately 34 kilometers. This calculation is based on the principles of gravitational force and the inverse square law, as well as the known physical constants of the Earth.
Further Reading
For a deeper understanding of gravitational acceleration and its applications, refer to the following articles:
Gravity and Gravitation Constants Applications of Gravitational Force Earth's Physical Constants