At What Altitude Does Earth's Gravitational Acceleration Drop to 9.71 m/s2?

Earth's gravitational acceleration, commonly denoted as g, is approximately 9.81 m/s2 at sea level. However, this value varies depending on the altitude above the Earth's surface due to the inverse square law governing gravitational force. This article explores the altitude at which Earth's gravitational acceleration drops to 9.71 m/s2 and explains the mathematical derivation behind this.

The Gravitational Force and Its Dependence on Altitude

Gravitational force can be described by Newton's law of universal gravitation:F (G*m1*m2) / r2. Here, F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers. For our purposes, m1 represents the Earth's mass, and m2 represents the mass of a point object at a certain altitude h. The gravitational acceleration g at a distance r from the center of the Earth can be expressed as:

[ g frac{G * M}{r^2} ]

Where:

G Universal gravitational constant (6.674 x 10-11 m3kg-1s-2) M Mass of Earth (5.972 x 1024 kg) r Distance from the center of the Earth

Deriving the Altitude for 9.71 m/s2

We need to find the altitude h at which the gravitational acceleration g is 9.71 m/s2. Let's start by rearranging the formula for gravitational acceleration to solve for h: [ 9.71 frac{6.674 times 10^{-11} times 5.972 times 10^{24}}{(6.371 times 10^6 h)^2} ]

Simplifying, we have:

[ (6.371 times 10^6 h)^2 frac{6.674 times 5.972 times 10^{13}}{9.71} approx 4.102 times 10^{13} ]

Taking the square root of both sides, we get:

[ 6.371 times 10^6 h approx sqrt{4.102 times 10^{13}} approx 6.405 times 10^6 ]

Solving for h yields:

[ h approx 6.405 times 10^6 - 6.371 times 10^6 approx 34 text{ km} ]

Understanding the Result

The result indicates that the acceleration due to gravity will be 9.71 m/s2 at an altitude of approximately 34 kilometers above Earth's surface. This finding aligns closely with the earlier calculations, demonstrating how the inverse square law affects gravitational acceleration at different altitudes.

Conclusion

Understanding the gravitational force and its relationship with altitude is crucial for various fields, including aerospace engineering, geophysics, and satellite communications. By utilizing the principles of gravitational physics, we can accurately predict and calculate the gravitational acceleration at different points in space, ensuring the safety and success of various scientific and technological endeavors.

FAQ

Q: Why is gravitational acceleration at sea level different from 9.71 m/s2?

A: Gravitational acceleration varies with altitude because the force of gravity decreases as you move farther from the center of the Earth. At sea level, the effect of the Earth's curvature and the surrounding atmosphere contribute to the observed value of 9.81 m/s2. As altitude increases, the gravitational pull decreases, resulting in a lower value.

Q: How accurate are these calculations?

A: These calculations are highly accurate for practical purposes, though they may vary slightly due to factors such as the Earth's slightly ellipsoidal shape and the distribution of mass within the Earth. For most applications, these approximations are sufficient.