Acceleration Due to Gravity (Earth’s Gravitational Acceleration)

The Earth exerts a gravitational force on all objects near its surface, pulling them toward its centre.  This force is responsible for the free fall of objects and keeps celestial bodies like the Moon and satellites in orbit.

• According to Newton’s Second Law of Motion, a force acting on an object results in acceleration. When this force is due to Earth’s gravity, the resulting acceleration is called acceleration due to gravity (g).
• Since the gravitational force is always directed toward the Earth's centre, the acceleration due to gravity also acts in the same direction—vertically downward. This acceleration is independent of an object's mass, meaning all objects experience the same gravitational pull in the absence of air resistance.
• The value of g on Earth's surface is approximately 9.8 m/s², but it varies with location, altitude, and depth due to the Earth's shape and mass distribution. This fundamental force plays a crucial role in planetary motion, satellite dynamics, and everyday physical phenomena.

Using Newton’s Universal Law of Gravitation, the force exerted by the Earth on an object of mass m at a distance r from its centre is:

`(F)=(GMm)/(r^2)`

According to Newton’s Second Law:

F=mg

Equating both equations:

`(mg)=(GMm)/(r^2)`

Cancelling m from both sides:

`(g)=(GM)/(r^2)`

For an object on the surface of the Earth, r = R (radius of Earth), so:

`(g)=(GM)/(R^2)`

Substituting values:

• G = 6.67×10⁻¹¹N⋅m²/kg²
• M = 6×10²⁴kg
• R = 6.4×10⁶m

`"g" = (6.67 xx 10^-11 xx 6 xx 10^24)/(6.4 xx 10^6)^2`

Thus, g is nearly 9.8 m/s² on Earth’s surface.

• The value of g depends only on the Earth’s mass and radius.
• It is the same for all objects, regardless of their properties.

A. Change Along the Surface of the Earth:

The Earth is not a perfect sphere; it is flattened at the poles and bulging at the equator. The radius at the equator is larger than at the poles, so g is slightly lower at the equator and higher at the poles.

• g at the poles = 9.832 m/s²
• g at the equator = 9.78 m/s²

B. Change with Height:

As we move above Earth's surface, the distance (r) increases, so g decreases. However, for small heights (like aeroplanes at 10 km), the change in g is negligible. For satellites at very high altitudes, g decreases significantly.

C. Change with Depth:

If we move inside the Earth, the distance (r) decreases, but the mass (M) also decreases because only the inner portion of the Earth contributes to gravity. As a result, g decreases inside the Earth. At the Earth’s centre, g becomes zero because the gravitational pull is equal in all directions.

Gravitational Acceleration on Other Celestial Bodies:

• Different planets and moons have different masses and radii, so g varies across celestial bodies.
• On the Moon, g is 1/6th of Earth's gravity. This means a person can jump 6 times higher on the Moon compared to Earth using the same force.