Introduction:
The concept of acceleration due to gravity is fundamental in understanding the effects of Earth's gravitational force on objects. Represented by gg, it defines the rate at which an object accelerates when falling freely under the influence of gravity alone. Its value on the Earth's surface is approximately 9.8m/s^2, but this value changes depending on the object's position above or below the Earth's surface. These variations are crucial for various fields, such as satellite dynamics, mining, and geophysics. This document explains how gg varies with height and depth, supported by mathematical derivations and practical insights.
Acceleration Due to Gravity: Below and Above the Earth's Surface
1. Acceleration Due to Gravity
The acceleration experienced by an object due to the gravitational pull of the Earth is called the acceleration due to gravity (g). On the Earth's surface, g is approximately 9.8 m/s².
2. Variation of g Above and Below the Earth's Surface
The value of g varies with the position of the object relative to the Earth's surface.
2.1. g Above the Earth's Surface
When an object is at a height h above the Earth's surface:
- The distance from the center of the Earth = R + h, where R is the radius of the Earth.
- The acceleration due to gravity decreases as we move away from the Earth’s surface.
Formula:
gₕ = g (R / (R + h))²
Key Points:
1. Direct Relation: gₕ ∝ 1 / (R + h)².
2. Near the Surface: If h ≪ R, then gₕ ≈ g (1 - 2h/R). This is a linear approximation for small heights.
2.2. g Below the Earth's Surface
When an object is at a depth d below the Earth's surface:
- The effective mass contributing to the gravitational pull reduces, as the outer shell of the Earth at a distance greater than R-d does not contribute (by the Shell Theorem).
- The value of g decreases linearly as we go deeper.
Formula:
gₑ = g (1 - d/R)
Key Points:
1. Linear Decrease: gₑ ∝ (1 - d/R).
2. At the center of the Earth (d = R), gₑ = 0, as there is no net gravitational force.
3. Explanation Using Gravitational Force
The gravitational force is given by:
F = G (m₁ m₂) / r²
Where:
- G is the universal gravitational constant.
- m₁ and m₂ are the masses of the two objects.
- r is the distance between their centers.
The acceleration due to gravity (g) is derived from the gravitational force:
g = G M / R²
Read Also: Notes on Gravitational Potential Energy Class 11 Physics
4. Summary of Key Formulas
Position
Expression
Remarks
On the surface (h = 0)
g = G M / R²
Maximum g value.
At height h above surface
gₕ = g (R / (R + h))²
g decreases with height.
At depth d below surface
gₑ = g (1 - d/R)
g decreases linearly with depth.
5. Graphical Representation
1. Above the Surface: gₕ decreases non-linearly and approaches zero as h → ∞.
2. Below the Surface: gₑ decreases linearly and becomes zero at the Earth's center.
6. Applications
Satellite orbits: Use the formula for gₕ to calculate gravitational forces at higher altitudes.
- Mining: Estimate gₑ to understand forces at depths within the Earth.
Conclusion:
Acceleration due to gravity (gg) is not a constant but varies based on an object’s position relative to the Earth's surface. It decreases with height above the Earth due to the inverse square law and diminishes linearly with depth below the surface due to the reduction in effective gravitational mass. Understanding these variations is essential in fields like astrophysics, engineering, and geophysics, as it provides insight into the Earth's structure and influences the design of space and underground technologies. Mastering these concepts builds a strong foundation for exploring advanced topics in gravitation and planetary motion.