Formula
Variation of g with depth
At depth d below Earth's surface (assuming uniform density): g_d = g (1 - d/R). Decreases with depth, becoming zero at Earth's centre.
-- NCERT Class 11 Physics, Ch. 7, p. 134g Variation Depth
8 MCQs2 revision cards9-step worked example
Source: NCERT GravitationOfficial key: NTA-verifiedLast updated: 26 Sep 2026
g Variation Depth, explained for NEET
Variation of g with depth — the linear decrease most aspirants misapply
When you move below Earth's surface, the behaviour of gravitational acceleration reverses from what happens above: g decreases linearly with depth, not by an inverse-square law.
The derivation assumes Earth is a uniform-density sphere of radius R. At depth d below the surface, only the spherical shell of radius (R − d) contributes gravitationally. By the shell theorem, the mass enclosed is M′ = M(R − d)³/R³. Setting up the gravitational acceleration at distance (R − d) from the centre:
g_d = GM′/(R − d)² = g(1 − d/R)
This is the key formula from NCERT Class 11 Physics, Chapter 7 (page 134). Note two anchor points: at d = 0 (surface), g_d = g; at d = R (centre), g_d = 0.
The depth-vs-altitude confusion. A common NEET trap conflates the depth formula with the altitude formula. Above the surface, g falls as an inverse square: g_h = g(R/(R + h))². Below the surface, g falls linearly: g_d = g(1 − d/R). These are fundamentally different functional forms. A question that gives "a point at distance R/2 from the centre" is asking for d = R/2 (depth formula), not h = R/2 (altitude formula). Mixing them up changes the answer entirely.
Uniform-density assumption. Every NEET problem on this topic assumes uniform density unless explicitly stated otherwise. In reality, Earth's core is denser, so g actually increases slightly before decreasing — but that real-world nuance is outside NEET scope.
Watch out: when a stem says "at a depth equal to half the radius," d = R/2, so g_d = g/2. If it says "at a distance R/2 from the centre," then d = R/2 gives the same result — but only because "distance from centre" = R − d, so R − d = R/2 means d = R/2. Read the reference point (surface vs centre) carefully.
Can you answer these g Variation Depth MCQs?
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
MCQ 1Easy RecallPractice
At what depth below Earth's surface does the acceleration due to gravity become zero? (Assume uniform density.)
Show answer and why every option is right or wrong
Answer: C. From g_d = g(1 − d/R), setting g_d = 0 gives d = R, i.e., at Earth's centre (NCERT Class 11 Physics, Chapter 7, page 134).
Why A is wrong: A is wrong because at d = R/2, g_d = g/2, not zero.
Why B is wrong: B is wrong because depth cannot exceed R (the centre is the deepest point in a sphere of radius R).
Why D is wrong: D is wrong because at the surface d = 0 and g_d = g, the full surface value.
MCQ 2Direct ApplicationPractice
The value of g at a depth of R/4 below Earth's surface is (g = acceleration due to gravity at the surface, R = Earth's radius, uniform density assumed):
Show answer and why every option is right or wrong
Answer: C. g_d = g(1 − d/R) = g(1 − (R/4)/R) = g(1 − 1/4) = 3g/4 (NCERT Class 11 Physics, Chapter 7, page 134).
Why A is wrong: A is wrong because g/4 would require d = 3R/4, not R/4. The formula gives g(1 − d/R), not g(d/R).
Why B is wrong: B is wrong because g/2 corresponds to d = R/2, not d = R/4.
Why D is wrong: D is wrong because g_d equals surface g only when d = 0. At any non-zero depth, g decreases.
MCQ 3Easy RecallPractice
Which of the following correctly describes how g varies with depth inside a uniform-density Earth?
Show answer and why every option is right or wrong
Answer: B. The formula g_d = g(1 − d/R) is linear in d, so g decreases linearly from the surface value to zero at the centre (NCERT Class 11 Physics, Chapter 7, page 134).
Why A is wrong: A is wrong because g decreases, not increases, as you go deeper. Only the enclosed mass shrinks (as the cube of remaining radius), and the net effect is a linear decrease.
Why C is wrong: C is wrong because inverse-square variation applies above the surface (altitude formula g_h = g(R/(R+h))²), not below it.
Why D is wrong: D is wrong because the enclosed mass decreases with depth, so g cannot remain constant.
MCQ 4Direct ApplicationPractice
A body weighs 63 N on Earth's surface. What is its weight at a depth of R/3 below the surface? (Uniform density, R = Earth's radius.)
Show answer and why every option is right or wrong
Answer: A. g_d = g(1 − d/R) = g(1 − 1/3) = 2g/3. Weight = 63 × (2/3) = 42 N (NCERT Class 11 Physics, Chapter 7, page 134).
Why B is wrong: B is wrong because 21 N = 63/3 — this treats weight as proportional to d/R instead of (1 − d/R).
Why C is wrong: C is wrong because 54 N = 63 × (6/7), which doesn't match any correct substitution into the depth formula.
Why D is wrong: D is wrong because weight decreases with depth; it equals the surface value only at d = 0.
MCQ 5Direct ApplicationPractice
At a point inside the Earth at a distance R/2 from the centre (R = Earth's radius, uniform density), the acceleration due to gravity is:
Show answer and why every option is right or wrong
Answer: D. Distance from centre = R − d = R/2, so d = R/2. Then g_d = g(1 − (R/2)/R) = g/2 (NCERT Class 11 Physics, Chapter 7, page 134).
Why A is wrong: A is wrong because g_d = g only at the surface (d = 0). At depth R/2, the enclosed mass is less.
Why B is wrong: B is wrong because g can never exceed the surface value inside a uniform-density sphere.
Why C is wrong: C is wrong because g/4 would come from an inverse-square law applied incorrectly; the depth formula is linear, giving g/2.
MCQ 6Easy RecallPractice
If the depth formula g_d = g(1 − d/R) gives g_d = g/2, what fraction of Earth's radius is the depth d?
Show answer and why every option is right or wrong
Answer: D. g(1 − d/R) = g/2 implies 1 − d/R = 1/2, so d/R = 1/2, meaning d = R/2 (NCERT Class 11 Physics, Chapter 7, page 134).
Why A is wrong: A is wrong because d/R = 1/4 gives g_d = 3g/4, not g/2.
Why B is wrong: B is wrong because d/R = 1/3 gives g_d = 2g/3, not g/2.
Why C is wrong: C is wrong because d/R = 2/3 gives g_d = g/3, not g/2.
MCQ 7CalculationPractice
A mine shaft reaches a depth d below Earth's surface. If the percentage decrease in g at the bottom of the shaft compared to the surface is 0.1%, what is d? (R = 6400 km, uniform density.)
Show answer and why every option is right or wrong
Answer: A. Percentage decrease = (d/R) × 100 = 0.1%, so d/R = 0.001. d = 0.001 × 6400 km = 6.4 km (NCERT Class 11 Physics, Chapter 7, page 134).
Why B is wrong: B is wrong because 3.2 km corresponds to d/R = 0.0005, which gives a 0.05% decrease, not 0.1%.
Why C is wrong: C is wrong because 12.8 km gives d/R = 0.002 and a 0.2% decrease — double the required value.
Why D is wrong: D is wrong because 32 km gives d/R = 0.005 and a 0.5% decrease, five times too large.
MCQ 8CalculationPractice
At what depth below Earth's surface is g the same as at a height R above the surface? (Uniform density. Use exact altitude formula g_h = g(R/(R+h))².)
Show answer and why every option is right or wrong
Answer: B. At height h = R: g_h = g(R/(2R))² = g/4. Set g_d = g(1 − d/R) = g/4, so 1 − d/R = 1/4, giving d = 3R/4 (NCERT Class 11 Physics, Chapter 7, pages 133–134).
Why A is wrong: A is wrong because at d = R/4, g_d = 3g/4 ≠ g/4.
Why C is wrong: C is wrong because at d = R/2, g_d = g/2 ≠ g/4.
Why D is wrong: D is wrong because at d = R, g_d = 0 ≠ g/4. d = R corresponds to Earth's centre where g vanishes.
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g Variation Depth: quick recall before you leave
How do you solve a g Variation Depth question? A worked example
- 1
Given
• Surface weight W = 72 N (exact, problem-defined)• Depth for part (a): d₁ = R/3• Height for part (b): h = R• Earth radius: R (symbolic)
- 2
Required
(a) Weight at depth d₁ = R/3.
(b) Depth d₂ where weight equals weight at height R. - 3
Concept
Inside a uniform-density Earth, g varies linearly with depth: g_d = g(1 − d/R). Above the surface, g varies as inverse square: g_h = g(R/(R+h))². Weight is proportional to local g.
- 4
Formula
• g_d = g(1 − d/R) [NCERT Class 11 Physics, Chapter 7, page 134]• g_h = g(R/(R+h))² [NCERT Class 11 Physics, Chapter 7, page 132]
- 5
Substitution
(a) g_{d₁} = g(1 − (R/3)/R) = g(1 − 1/3) = 2g/3
W_{d₁} = 72 × (2/3)
(b) g_h = g(R/(R+R))² = g(1/2)² = g/4
Set g(1 − d₂/R) = g/4 → 1 − d₂/R = 1/4 → d₂/R = 3/4 - 6
Calculation
(a) W_{d₁} = 72 × 2/3 = 48 N
Note: 72 and the fractions 1/3, 2/3 are exact (problem-defined integers and their ratios), so they do not limit significant figures.
(b) d₂ = 3R/4 - 7
Final answer
(a) Weight at depth R/3 = 48 N
(b) Depth where weight matches the height-R value = 3R/4
The factor 72 and the fractions 1/3, 3/4 are exact problem-defined values and do not contribute to any significant-figure limitation. - 8
Common trap
Confusing the depth and altitude formulas. If you mistakenly apply the inverse-square altitude formula at depth R/3, you'd compute g(R/(R + R/3))² = g(3/4)² = 9g/16, giving a weight of 72 × 9/16 = 40.5 N — wrong. The depth formula is linear, not inverse-square.
- 9
Similar NEET-style question
"A body weighs 200 N on Earth's surface. At what depth below the surface will its weight be 150 N? (Assume uniform density.)"
Approach: 150 = 200(1 − d/R) → d/R = 1/4 → d = R/4.
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What to remember before solving g Variation Depth questions
Which g Variation Depth formulas do you need for NEET?
1 formula — click to collapse
g variation with depth
Inside Earth (uniform density), g decreases linearly with depth, vanishing at centre.
| Symbol | Quantity | SI Unit |
|---|---|---|
| g_d | g at depth | m/s^2 |
| g | surface g | m/s^2 |
| R | Earth radius | m |
| d | depth | m |
Valid when
- Earth treated as uniform density sphere
More in Gravitation: 4 exam traps and mistakes · 7 formulas · 4 question patterns from its other lessons.
g Variation Depth questions from past NEET papers
No question in our NEET 2020–2025 set targets this topic directly.
Sources
Page numbers are the ones printed in the current NCERT textbook (2023 rationalised edition), unless marked pre-2023. The books are free at ncert.nic.in.
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