Effect on Acceleration Due to Gravity (g) at Depth d | Gravitation Class 11 Physics Notes, Important Questions & Answers, MCQS, Quiz

📚 Buy Complete Study Material for JEE, NEET, CBSE Class 11

Get high-quality study material at Anand Technical Publishers for JEE, NEET, and CBSE Class 11 exams.

🔍 Understanding the Variation of g with Depth

The acceleration due to gravity (g) is the force that attracts objects toward the Earth’s center. However, its value changes depending on depth beneath the Earth’s surface.

Acceleration Due to Gravity on the Surface of Earth

Earth is assumed to be a uniform solid sphere with a mean density (ρ). We know that,

Density = mass/volume

Then, ρ = M/[4/3 πR³]

M = ρ × [4/3 πR³]

We know that g = GM/R².

On substituting the values of M, we get,

g = 4/3 [πρRG]

At any distance ‘r’ from the centre of the earth,

g = 4/3 [πρRG]

Variation of Acceleration Due to Gravity (g) at Depth d

Consider a test mass (m) taken to a distance (d) below the earth’s surface, the acceleration due to gravity at that point (g_d) is obtained by taking the value of g in terms of density (ρ) because density of earth is uniform at all places of interior of earth.

On the surface of the earth, the value of g is given by

g = 4/3 × πρRG

At a distance (d) below the earth’s surface, the acceleration due to gravity is given by

g_d = 4/3 × πρ × (R – d) G

On dividing the above equations, we get,

g_d /g = (R – d)/R

Therefore,

g_d = g (1 – d/R)

Therefore, g_d < g

And this acceleration, (g_d), decreases as we move towards the center of the earth, which is experienced only when we move very deep towards the center of the earth. Hence, the value of g changes with height and depth both.

• When the depth d = 0, the value of g on the surface of the earth g_d = g.
• When the depth d = R, the value of g at the centre of the earth g_d = 0.

❓ Question-Answer Format

Q1: What is the formula for acceleration due to gravity at depth d?

A: The value of acceleration due to gravity at depth d is given by:

g_d = g (1 – d/R)

where:

• g = acceleration due to gravity on the surface of earth
• g_d​ = acceleration due to gravity at depth d
• R = radius of the Earth
• d = depth below the Earth’s surface

Q2: How does gravity change with depth?

A: As depth increases, the effective acceleration due to gravity decreases linearly. This happens because at greater depths, part of the Earth’s mass no longer contributes to the gravitational attraction.

Q3: Why does g decrease inside the Earth?

A: Below the surface, only the mass enclosed within a sphere of radius (R – d) exerts a gravitational force. Since the effective mass decreases, gravity reduces accordingly.

Q4: What is the acceleration due to gravity at the Earth’s center?

A: At the Earth’s center (d = R),

g_c = g(1−R/R)=0

Thus, gravity is zero at the center of the Earth.

📖 Multiple-Choice Questions (MCQs)

Q1: What happens to g when we go deeper inside the Earth?

A) Increases
B) Decreases
C) Remains Constant
D) Becomes Zero

✅ Answer: B) Decreases
📌 Explanation: As depth d increases, the effective mass contributing to gravitational attraction reduces, causing g to decrease.

Q2: At what depth does g become half of its surface value?

A) d=R/2
B) d=R/4
C) d=R/3
D) d=2R/3

✅ Answer: A) d=R/2
📌 Explanation: Using the formula, setting g_d = g/2, we solve for d:

g/2 = g(1−d/R)

Solving for d, we get d = R/2.

❓ Frequently Asked Questions (FAQs)

Q1: Does g increase or decrease with altitude?

A: g decreases with altitude because the distance from the Earth’s center increases, reducing gravitational attraction.

Q2: What is the value of g on the Moon?

A: The acceleration due to gravity on the Moon is 1/6th of that on Earth, approximately 1.63 m/s².

Q3: Why is g maximum at the poles?

A: The Earth is an oblate spheroid, meaning its radius is smaller at the poles than at the equator. Since g is inversely proportional to the square of the radius, it is maximum at the poles.

Q4: Can gravity ever become negative?

A: No, gravity is always an attractive force. However, in the case of repulsive effects (like dark energy), it behaves differently in cosmological contexts.

Test Your Knowledge

📚 Buy Complete Study Material

📖 Get JEE, NEET, and CBSE Class 11 study material from Anand Technical Publishers to boost your exam preparation.

🏫 Published By:

✍ Proprietor: NIRMAL ANAND Educations
✍ Written By: Neeraj Anand
🏢 Published By: Anand Technical Publishers
📞 Contact: +91-9463138669
📧 Email: anandclasses1996@gmail.com