Acceleration Due to Gravity
Introduction
When objects fall towards the Earth, their velocity keeps increasing. This means they are accelerated. This acceleration is due to Earth’s gravitational force and is called acceleration due to gravity.
Acceleration Due to Gravity
Acceleration due to gravity is the acceleration produced in a body when it falls freely under gravity.
It is denoted by g⃗and its direction is towards the center of the Earth.
Derivation of Formula
From Newton’s Second Law of Motion:
F⃗ = m × g⃗
From Universal Law of Gravitation (vector form):
F⃗ = − (G M m / R²) r̂
Here, r̂ is the unit vector directed away from the center of the Earth. The negative sign shows that the force is directed towards the Earth.
Equating both equations:
m × g⃗ = − (G M m / R²) r̂
Canceling m:
g⃗ = − (G M / R²) r̂
Calculation of g
Substituting values:
- G = 6.67 × 10-11 N m²/kg²
- M = 6 × 1024 kg
- R = 6.4 × 106 m
|g⃗| = (6.67 × 10-11 × 6 × 1024) / (6.4 × 106)²
|g⃗| ≈ 9.8 m/s²
Therefore,
g⃗ = 9.8 m/s² (−r̂)
Explanation
The vector g⃗ always points towards the center of the Earth. Its magnitude depends on Earth’s mass and radius, but it does not depend on the mass of the falling body.
Examples from Daily Life
- Fruits fall towards the ground.
- Rain falls towards the Earth.
- A stone accelerates downward when dropped.
Illustration
The acceleration g⃗ acts towards the center of the Earth, showing its direction clearly.
Important Points
- g⃗ = − (G M / R²) r̂
- |g⃗| = 9.8 m/s² near Earth surface
- g⃗ is always directed towards Earth’s center
- r̂ is unit vector directed outward
Conclusion
Acceleration due to gravity is a vector quantity. Its magnitude is 9.8 m/s² and its direction is always towards the Earth, which is represented using the unit vector r̂.
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