1. What is Acceleration Due to Gravity?
When you drop an object, it does not fall at a constant speed. Instead, it becomes faster and faster as it moves downward. This increase in speed is caused by gravity, and the rate at which the object accelerates is called the acceleration due to gravity, represented by \( g \).
On the surface of the Earth, the average value of \( g \) is:
\( g = 9.8\; \text{m/s}^2 \)
This means every second, a freely falling object increases its speed by about 9.8 m/s (ignoring air resistance).
1.1. Why Do All Objects Fall at the Same Rate?
In a vacuum, where there is no air resistance, all objects fall with the same acceleration—regardless of mass. A feather and a stone dropped in a vacuum fall together because gravity accelerates all masses equally.
1.1.1. Galileo’s Insight
Galileo showed that objects fall with the same acceleration if air resistance is removed. This was a major step in understanding gravity scientifically.
2. Formula for Acceleration Due to Gravity
The value of \( g \) near the Earth's surface is derived from Newton’s law of gravitation. Using:
\( F = G\dfrac{Mm}{R^2} \)
and Newton’s second law \( F = ma \), we get:
\( g = \dfrac{GM}{R^2} \)
Where:
- \( G \) = gravitational constant
- \( M \) = mass of Earth
- \( R \) = radius of Earth
2.1. What This Formula Means
The acceleration due to gravity depends only on Earth’s mass and radius—not on the mass of the falling object. This is why all bodies accelerate equally under gravity.
3. Variation of g with Height
The value of \( g \) decreases as you move away from the Earth's surface. At a height \( h \) above the surface, gravity becomes weaker.
The relation is approximately:
\( g_h = g \left(1 - \dfrac{2h}{R}\right) \)
This decrease is small for everyday heights but significant for satellites and space travel.
3.1. Examples
- At the top of a mountain, \( g \) is slightly smaller than at sea level.
- In space, far above Earth, \( g \) becomes much weaker.
4. Variation of g with Depth
As you go below the Earth’s surface, the value of \( g \) decreases. At depth \( d \):
\( g_d = g \left(1 - \dfrac{d}{R}\right) \)
At the centre of the Earth, \( g \) becomes zero because the mass pulls equally in all directions.
4.1. Why g Decreases Inside Earth
When you go deeper, less mass lies below you. Since gravitational force depends on the mass under your feet, gravity gets weaker.
5. Variation of g with Latitude
The Earth is not a perfect sphere—it is slightly flattened at the poles and bulged at the equator. This shape affects gravitational acceleration.
- At poles: Earth’s radius is smaller → \( g \) is higher
- At equator: Earth’s radius is larger → \( g \) is lower
Additionally, Earth's rotation slightly reduces \( g \) at the equator due to centrifugal effect.
5.1. Combined Effect
Both shape and rotation make gravity vary with location. This is why your weight is slightly higher at the poles than at the equator.
6. Why Acceleration Due to Gravity is Important
The value of \( g \) is essential for understanding:
- How objects fall
- Motion of projectiles
- The weight of objects
- Satellite orbits
- The behaviour of tides and oceans
In the next topic, we will study how objects move when gravity is the only force acting—this is the motion of objects under gravity.