Understanding the Concepts
* Newton's Law of Universal Gravitation: The force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
* Acceleration due to Gravity (g): The acceleration experienced by an object due to the Earth's gravitational pull. It depends on the Earth's mass (M) and the distance (r) from the object to the Earth's center.
Calculations
1. Start with the Formula:
The acceleration due to gravity (g) at a distance (r) from the Earth's center is given by:
g = GM/r²
where:
* G is the gravitational constant (6.674 × 10⁻¹¹ N m²/kg²)
* M is the mass of the Earth (5.972 × 10²⁴ kg)
2. Find the Acceleration at the Earth's Surface (g₀):
Let the radius of the Earth be R. At the Earth's surface (r = R):
g₀ = GM/R²
3. Set up the Equation for Half the Surface Gravity:
We want to find the height (h) above the surface where the acceleration due to gravity (g) is half of g₀:
g = g₀/2
GM/(R + h)² = (GM/R²)/2
4. Solve for h:
* Simplify the equation: (R + h)² = 2R²
* Take the square root of both sides: R + h = √2R
* Solve for h: h = √2R - R = R(√2 - 1)
5. Substitute Earth's Radius:
The Earth's radius (R) is approximately 6371 km.
h ≈ 6371 km * (√2 - 1) ≈ 2639 km
Therefore, the height above the Earth's surface where the acceleration due to gravity is half its value on the surface is approximately 2639 kilometers.
