Variation of g with Depth || Variation of acceleration due to gravity with depth

Summary

This video explains how the gravitational force (g) varies inside and on the surface of the Earth. The instructor demonstrates the concept by considering a point beneath the Earth's surface, analyzing the changes in mass and distance that affect gravitational acceleration. Using formulas for mass, volume, and density, the video shows that gravity decreases both when moving above the surface and deeper into the Earth. A special case highlights that at the Earth's center, gravitational acceleration becomes zero. The explanation is detailed, step-by-step, and illustrated with calculations, making the concepts of internal and surface gravity variations clear and easy to understand.

Takeaways

• 🌍 The video explains how acceleration due to gravity (g) changes when moving below the surface of the Earth.
• 📘 Earth is considered as a sphere with mass M and radius R for deriving the variation of gravity.
• 🕳️ A point inside the Earth at a depth d is taken to study the change in gravitational acceleration.
• 📏 The distance of the internal point from the Earth’s center becomes (R - d).
• ⚖️ Inside the Earth, only the mass enclosed within the radius (R - d) contributes to gravity at that point.
• 🧮 The formula for gravity at the Earth’s surface is given as g = GM / R².
• 🔬 Density is used in the derivation with the relation: Mass = Density × Volume.
• 🌐 The Earth’s volume is represented using the sphere formula: (4/3)πR³.
• 📉 By substituting the reduced mass and reduced radius, the derived equation becomes g(d) = g(1 - d/R).
• ⬇️ The derivation shows that gravity decreases linearly as depth inside the Earth increases.
• 🚀 Similar to increasing height above Earth, moving deeper inside Earth also reduces the value of g.
• ⭐ The maximum value of acceleration due to gravity exists at the Earth’s surface.
• 🧲 At the center of the Earth, the depth becomes equal to the radius (d = R).
• ❌ Substituting d = R into the equation gives g = 0 at the Earth’s center.
• 🎯 The final conclusion is that acceleration due to gravity continuously decreases from the surface toward the center of the Earth.

Q & A

What is the main topic discussed in the video?

The video discusses the variation of gravitational acceleration (g) without a solution, focusing on how it changes when moving above or below the Earth's surface.

How does the gravitational acceleration change when moving below the Earth's surface?

When moving below the Earth's surface, the gravitational acceleration decreases because only the mass up to the distance from the center contributes to gravity at that point.

What is the significance of point B in the explanation?

Point B represents a point inside a dug hole below the Earth's surface. The script calculates the effective mass and distance contributing to gravity at point B.

Why is the mass considered different when calculating gravity below the surface?

Because only the portion of Earth's mass located at a radius less than or equal to the distance from the center affects gravity at that point, the effective mass is reduced compared to the full Earth's mass.

What formula is used to calculate gravitational acceleration inside the Earth?

The simplified formula g = G * M / R^2 is adapted by replacing M with the mass enclosed within radius r and R with the distance from the center, giving g = G * M_enclosed / r^2.

Does the density of the Earth change with depth in this explanation?

No, the density is assumed constant (average density), so only the volume changes when calculating the mass enclosed at a particular depth.

What happens to gravitational acceleration at the center of the Earth?

At the center of the Earth, the gravitational acceleration becomes zero because the net gravitational pull from all directions cancels out.

How does the volume of Earth considered for gravity calculation change with depth?

The volume considered decreases as we go deeper because only the spherical portion up to that depth contributes to the gravitational force at that point.

What analogy is made with the 'high case' scenario in the video?

The 'high case' scenario refers to points above the Earth's surface where gravitational acceleration decreases with increasing altitude, similar to how it decreases below the surface but for opposite reasons.

Why is it important to consider only part of the Earth's mass below a certain depth?

It is important because the gravitational force at a point inside the Earth depends only on the mass located closer to the center than that point. Mass outside that radius does not contribute, according to the shell theorem.

What key concept explains why gravity decreases both above and below the surface?

Gravity decreases above the surface because the distance from the Earth's center increases, and below the surface because the effective mass contributing to gravity decreases with depth.