G Variation Depth

8 MCQs2 revision cards9-step worked example

Source: NCERT GravitationPYQ coverage: NEET 2021, 2022, 2023, 2024, 2025Official key: NTA-verifiedLast reviewed: May 2026

Lesson

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 7). 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.

Practice 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.)

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):

MCQ 3Easy RecallPractice

Which of the following correctly describes how g varies with depth inside a uniform-density Earth?

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.)

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:

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?

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.)

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.)

Quick recall before you leave

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 7] - g_h = g(R/(R+h))² [NCERT Class 11 Physics, Chapter 7, page 6]
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. ---

Before solving, remember these

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. 7

Formulas

8 formulas — click to collapse

Escape velocity from a body's surface

v_{e} = \frac{2 GM}{R} = 2 g R

Minimum speed for an object to escape gravity to infinity from radius R. Earth: ~11.2 km/s.

Symbol	Quantity	SI Unit
v_e	escape velocity	m/s
M	planet mass	kg
R	planet radius	m
g	surface gravity	m/s^2

Valid when

Launched from surface
No air drag
Body treated as point/sphere

Gravitational potential energy (point masses)

U = - \frac{GM m}{r}

PE of two-body system; negative because gravity is attractive (work to separate them is positive).

Symbol	Quantity	SI Unit
U	grav PE	J
M, m	two masses	kg
r	separation	m

Valid when

Reference U=0 at r=infinity
Point or spherical masses

g variation with altitude

g_{h} = g (\frac{R}{R + h})^{2} \approx g (1 - 2 h / R)

Gravitational acceleration decreases with altitude above Earth's surface.

Symbol	Quantity	SI Unit
g_h	g at height h	m/s^2
g	surface g	m/s^2
R	Earth radius	m
h	altitude	m

Valid when

Static observer at altitude
Earth treated as uniform sphere

g variation with depth

g_{d} = g (1 - d / R)

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

Kepler's third law

T^{2} = \frac{4 π ^{2}}{GM} a^{3}

Square of orbital period proportional to cube of semi-major axis. Holds for elliptic orbits about a central mass.

Symbol	Quantity	SI Unit
T	orbital period	s
a	semi-major axis	m
M	central mass	kg

Valid when

Two-body system with central mass M >> orbiting mass
Bound orbit

Orbital velocity for circular orbit

v = \frac{GM}{R + h}

Speed of circular orbit at altitude h above body of mass M, radius R.

Symbol	Quantity	SI Unit
v	orbital speed	m/s
M	central mass	kg
R+h	orbit radius	m

Valid when

Circular orbit
M >> orbiting mass

Satellite total mechanical energy

E = - \frac{GM m}{2 ( R + h )}

Total energy = KE + PE = -KE (virial). Always negative for bound orbit; E -> 0 at infinity.

Symbol	Quantity	SI Unit
E	total energy	J
M, m	central mass and satellite mass	kg
R+h	orbit radius	m

Valid when

Circular orbit
Bound (E < 0)

Newton's law of gravitation

F = \frac{G m _{1} m _{2}}{r ^{2}}

Attractive force between any two masses. Inverse-square central force.

Symbol	Quantity	SI Unit
F	force	N
G	grav constant = 6.674e-11	N*m^2/kg^2
m1, m2	masses	kg
r	centre-to-centre distance	m

Valid when

Point masses or spherically symmetric distributions
r > sum of body radii (else use shell theorem)

Exam Traps & Common Mistakes

These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.

4 items — click to collapse

Trap

g at altitude: linear vs inverse-square

Category: Similar Terms

Student treats g(h) as linear in h. Actual: g(R/(R+h))² (inverse-square).

When it triggers

Question asks for g at significant altitude (e.g. R/2 above surface).

How to avoid

Use g_h = g (R/(R+h))². Linear approximation g(1 - 2h/R) only valid for h << R.

Trap

Kepler's third law: linear vs 3/2 power

Category: Similar Terms

Student treats T proportional to a (linear) instead of a^(3/2).

When it triggers

Question gives change in semi-major axis and asks for new period.

How to avoid

T² ∝ a³, so T ∝ a^(3/2). Doubling a multiplies T by 2^(3/2) ≈ 2.83.

Mistake

Treats g(h) as linear in h for significant heights.

Root cause: formula misuse

Correction

Use g_h = g(R/(R+h))² (inverse-square). Linear approximation g(1-2h/R) is only valid for h << R. For h = R/2, the exact formula gives g_h = (2/3)² g = 4g/9, not g(1-1) = 0.

Mistake