Summary of "Value of g on surface of earth class 12 nbf || Variation of g with altitude || By Atif Ahmed"

Summary of the Video: “Value of g on surface of earth class 12 nbf || Variation of g with altitude || By Atif Ahmed”

Main Ideas and Concepts

1. Introduction to Gravitational Acceleration (g)

Gravitational acceleration, denoted by g, is the acceleration experienced by a body due to Earth’s gravitational force.
It represents the rate at which the velocity of a freely falling object increases (approximately 9.8 m/s² near Earth’s surface).
The direction of g is always towards the center of the Earth.
According to Newton’s laws, acceleration occurs when a net force acts on a body; here, the net force is the gravitational force.

2. Definition of Gravitational Acceleration

Gravitational acceleration is the acceleration experienced by a body when it falls freely under Earth’s gravity.
Example: Near Earth’s surface, a freely falling object’s velocity increases by 9.8 m/s every second.

3. Derivation of the Mathematical Formula for g on Earth’s Surface

Newton’s Law of Gravitation: [ F = \frac{G M_e M_o}{r^2} ] where:
- ( G ) = gravitational constant
- ( M_e ) = mass of Earth
- ( M_o ) = mass of the object
- ( r ) = distance between centers of Earth and object
Since the object’s radius is negligible compared to Earth’s radius, ( r \approx R_e ) (Earth’s radius).
Newton’s second law states: [ F = M_o g ]
Equating and simplifying: [ M_o g = \frac{G M_e M_o}{R_e^2} \implies g = \frac{G M_e}{R_e^2} ]
Using known values:
- ( R_e = 6.4 \times 10^6 \, m )
- ( M_e = 6 \times 10^{24} \, kg )
- ( G = 6.673 \times 10^{-11} \, N \cdot m^2/kg^2 )
Calculated value: [ g \approx 9.8 \, m/s^2 ]

4. Concept of Free Fall and Independence from Object’s Mass

The value of g is independent of the mass of the falling object.
Both heavy and light objects fall with the same acceleration in the absence of air resistance.
Air resistance affects falling speed differently (e.g., paper vs. book), but when minimized (e.g., placing paper on a book), both fall together.
This illustrates free fall: acceleration due to gravity is constant for all masses.

5. Dependency of g on Planetary Properties

The value of g depends on the mass and radius of the celestial body (planet or moon).
For example, the Moon’s gravitational acceleration is about one-sixth that of Earth’s due to its smaller mass and radius.
Weight ( W = mg ) changes accordingly on different planets because g changes.
This explains why a person weighs less on the Moon than on Earth.

6. Variation of g with Altitude (Height above Earth’s Surface)

As height ( h ) above Earth’s surface increases, the value of ( g ) decreases.
Reason: Gravitational force (and acceleration) is inversely proportional to the square of the distance from Earth’s center.
At height ( h ): [ g_h = \frac{G M_e}{(R_e + h)^2} ]
Since ( (R_e + h)^2 > R_e^2 ), [ g_h < g ]
Thus, gravitational acceleration and weight decrease with altitude.

7. Mathematical Explanation of Decrease in g with Height

Comparing ( g ) at surface and ( g_h ) at height ( h ): [ g_h = g \times \left(\frac{R_e^2}{(R_e + h)^2}\right) ]
Since the fraction ( \frac{R_e^2}{(R_e + h)^2} < 1 ), ( g_h ) is less than ( g ).
This confirms the decrease in gravitational acceleration with altitude.

Methodology / Step-by-Step Instructions

To Calculate the Value of g on Earth’s Surface:

Use Newton’s law of gravitation: [ F = \frac{G M_e M_o}{r^2} ]
Approximate the distance: [ r \approx R_e ]
Apply Newton’s second law: [ F = M_o g ]
Equate and solve for ( g ): [ g = \frac{G M_e}{R_e^2} ]
Substitute known values for ( G ), ( M_e ), and ( R_e ) to compute ( g ).

To Find the Value of g at a Height ( h ) Above Earth’s Surface:

Use the modified distance: [ r = R_e + h ]
Apply the formula: [ g_h = \frac{G M_e}{(R_e + h)^2} ]
Express ( g_h ) in terms of ( g ): [ g_h = g \times \left(\frac{R_e^2}{(R_e + h)^2}\right) ]
Understand that since the denominator increases, ( g_h ) decreases.

Conceptual Understanding of Free Fall:

Gravitational acceleration is constant for all masses.
Air resistance affects falling speed but can be minimized in experiments.

Understanding Weight Variation on Different Celestial Bodies:

Weight depends on local gravitational acceleration: [ W = m \times g_{planet} ]
Gravitational acceleration varies based on the planet’s mass and radius.

Key Takeaways

Gravitational acceleration ( g ) is approximately 9.8 m/s² near Earth’s surface.
( g ) depends only on Earth’s mass and radius, not on the mass of the falling object.
Free fall acceleration is the same for all objects regardless of mass (ignoring air resistance).
( g ) decreases with altitude as one moves away from Earth’s surface.
Weight changes on different planets because their ( g ) values differ.
Newton’s laws provide the mathematical basis for these phenomena.

Speaker / Source

Atif Ahmed — Physics teacher and presenter of the video.

This summary captures the core physics concepts, mathematical derivations, and conceptual clarifications presented by Atif Ahmed in the video on gravitational acceleration and its variation with altitude.