Gravity’s Half: Calculating the Altitude Where Earth’s Gravity Somewhat Diminishes

Have you ever pondered at what height the gravity you feel on Earth’s surface is halved? In this exploration, we will delve into the physics behind gravitational acceleration and discover the height at which this occurs.

Understanding Gravitational Acceleration

The fundamental concept here is gravitational acceleration g. The gravitational acceleration at the Earth’s surface is approximately 9.81 m/s2. This can be derived from Newton’s Universal Law of Gravitation, expressed as:

Fg G M m / R2

Where:

Fg is the gravitational force, G is the gravitational constant, M is the mass of the Earth, m is the mass of the object, R is the radius of the Earth.

When we consider a small mass m on the Earth’s surface, Fg simplifies to:

mg G M m / R2

From this, we can derive:

g GM / R2

At the Earth’s surface, substituting known values:

g GM / (6371×103)2

Calculating the Altitude

Now, let's find the height where the gravitational acceleration is exactly half of its value at the Earth’s surface. The equation to solve is:

Canceling out g0 from both sides:

Taking the reciprocal:

2 1 frac{h}{R}^2

sqrt{2} 1 frac{h}{R}

This gives:

h Rsqrt{2} - 1

Simplifying, with R 6371×103 meters:

h ≈ (6371000 m) × 1.414 - 1 ≈ 2640000 m ≈ 2640 km

Therefore, the altitude where the gravitational acceleration is half of its surface value is approximately 2640 kilometers above the Earth's surface.

Verification with Another Method

Another method to find this height is by considering that gravitational acceleration follows an inverse square law. Therefore, to halve the gravitational acceleration, the distance must be increased by a factor of √2. Consequently, the altitude is half the Earth’s radius √2 times the radius:

h √2 × 6371 - 6371 ≈ 2640 km

This approach aligns with the previous calculation, confirming our result.

Conclusion

The height at which Earth's gravity reduces to half of its strength at the surface is a fascinating concept. Understanding this height and the principles behind it can lead to a deeper appreciation of celestial mechanics and the natural world around us.

Key Takeaways: Gravitational acceleration is modeled using Newton's Universal Law of Gravitation. The inverse square law explains why gravity diminishes as distance from the center of the Earth increases. The altitude required to halve the gravitational acceleration is approximately 2640 kilometers.

Engage with these concepts further to explore more intricate aspects of physics and astronomy!