Study Notes on Acceleration Due to Gravity-Below and Above the Earth's Surface-Class 11 Physics

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Introduction:

The concept of acceleration due to gravity is central to understanding gravitational interactions in Class 11 Physics. It explains the force exerted by Earth on objects, pulling them toward their center. The value of acceleration due to gravity, represented by (g), is not constant but varies depending on altitude (above the Earth's surface) and depth (below the Earth's surface). These variations are crucial for understanding phenomena such as free-fall motion, satellite dynamics, and geophysical studies.

Acceleration Due to Gravity on the Earth's Surface:

At the Earth's surface, the acceleration due to gravity is approximately (9.8m/s^2). The formula for (g) at the surface is derived using Newton's Law of Gravitation and is expressed as:

g = GM / R²

Where:

• G is the gravitational constant (6.674 × 10⁻¹¹ Nm²/kg²),

• M is the mass of the Earth,

• R is the radius of the Earth.

Variation of Gravity Above the Earth's Surface:

When an object is at a height (h) above the Earth's surface, the effective distance from the Earth's center becomes (R + h). The value of (g) decreases with altitude and is given by:

gₕ = GM / (R + h)²

Simplifying for small heights (h ≪ R) using the binomial approximation:

gₕ ≈ g (1 - 2h / R)

Key Points:

• Gravity decreases as the square of the distance from the Earth's center.

• Satellites and other objects at higher altitudes experience reduced gravitational pull.

Variation of Gravity Below the Earth's Surface:

When an object moves inside the Earth's surface, only the mass enclosed within the radius (r) contributes to gravitational force. Assuming the Earth is spherical and has uniform density (ρ):

gᵣ = G Mᵣ / r²

Where Mᵣ is the mass enclosed, given by:

Mᵣ = ρ ⋅ 4/3 π r³

Substituting Mᵣ into the formula:

gᵣ = g ⋅ r / R

Key Points:

• Gravity decreases linearly with depth.

• At the Earth's center (r = 0), g = 0.

Graphical Representation:

1. Above the Surface: Exponential decay as h increases.

2. Below the Surface: Linear decrease with r.

Factors Affecting Acceleration Due to Gravity:

1. Altitude: Decreases with height above the surface.

2. Depth: Decreases linearly with depth below the surface.

3. Shape of Earth: Due to Earth's oblateness, g is slightly higher at the poles than at the equator.

4. Rotation of Earth: Causes a reduction in effective g due to centrifugal force.

Conclusion:

Understanding how gravity changes above and below the Earth's surface is essential for comprehending many physical and astronomical phenomena. These variations impact the motion of objects, the behavior of fluids, and even technological advancements like satellite launches. Grasping the principles behind g helps students build a strong foundation in classical mechanics and gravitational studies.