The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhe

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Q. The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration due to gravity

(a) will be directed towards the centre but not the same everywhere.

(b) will have the same value everywhere but not directed towards the centre.

(d) cannot be zero at any point.

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Concept understanding — Variation Of Gravity

Variation of Gravity: Why Your Weight Changes Even When You Don't

Imagine you step on a weighing scale at sea level in Mumbai, then carry that same scale to the top of Mount Everest. The scale would show a smaller number — you'd weigh less. But you haven't lost any mass. What changed?

The force pulling you down — gravity — is not constant everywhere on Earth. It varies. That's what we mean by variation of gravity.

The Core Idea

Gravity is the force with which the Earth pulls objects toward its centre. The strength of this pull depends on two things: the mass of the Earth and your distance from its centre. Since the Earth is not a perfect sphere and it spins, that distance and the effective pull change from place to place.

The acceleration due to gravity, denoted by $g$ , is approximately $9.8 \, \text{m/s}^2$ at sea level. But that's an average. The actual value can be slightly higher or lower depending on where you are.

Why Does Gravity Vary? Three Main Reasons

1. Altitude (Height Above Sea Level)

This is the most intuitive one. As you go higher, you move farther from the Earth's centre. Gravity follows an inverse-square law: double the distance, and the force becomes one-fourth.

The formula for $g$ at a height $h$ above the Earth's surface (where $R$ is Earth's radius, about 6400 km) is:

g_h = \frac{GM}{(R+h)^2}

For small heights compared to $R$ , we can approximate:

g_h \approx g \left(1 - \frac{2h}{R}\right)

Note

This means for every kilometre you go up, $g$ decreases by roughly $0.003 \, \text{m/s}^2$ . That's why at the top of a tall mountain, you weigh about 0.5% less than at sea level.

2. Depth (Going Underground)

What happens if you go down a mine or into the Earth's crust? Intuition might say gravity increases because you're closer to the centre. But the opposite happens.

Inside the Earth, the mass above you pulls upward, partially cancelling the pull from below. For a uniform Earth, only the mass inside the sphere of radius $r$ (your distance from the centre) contributes to gravity at that point.

g_d = \frac{GM'}{r^2}

Where $M'$ is the mass of the sphere of radius $r$ . If Earth had uniform density $\rho$ , then $M' = \frac{4}{3}\pi r^3 \rho$ , giving:

g_d = \frac{4}{3}\pi G \rho r

This means gravity decreases linearly as you go deeper. At the centre of the Earth, $g = 0$ — you'd be weightless, pulled equally in all directions.

Watch out

This linear decrease assumes uniform density. The real Earth has a dense iron core, so the actual variation is more complicated — gravity actually increases slightly as you go down through the crust before eventually decreasing.

3. Rotation of the Earth (Latitude Effect)

The Earth spins once every 24 hours. This rotation creates a centrifugal force that acts outward, away from the axis of rotation. This force effectively reduces the weight you feel.

The effect is strongest at the equator (where the rotational speed is highest, about 1670 km/h) and zero at the poles (where you're on the axis of rotation).

The effective $g$ at latitude $\phi$ is:

g_{\text{eff}} = g - \omega^2 R \cos^2 \phi

Where $\omega$ is Earth's angular speed ( $7.3 \times 10^{-5} \, \text{rad/s}$ ) and $R$ is Earth's radius.

At the equator ( $\phi = 0^\circ$ ), the reduction is about $0.034 \, \text{m/s}^2$ — roughly 0.35% of $g$ .

Tip

| Location | Approximate $g$ (m/s²) | Why? |

|----------|------------------------|------|

| Equator (sea level) | 9.78 | Fastest rotation + bulging equator |

| 45° latitude | 9.81 | Intermediate |

| North Pole | 9.83 | No rotation effect + closer to centre |

4. Shape of the Earth (Oblateness)

The Earth is not a perfect sphere. Because of its rotation, it bulges at the equator and flattens at the poles. The equatorial radius is about 21 km larger than the polar radius.

This means:

At the poles, you're closer to the Earth's centre → stronger gravity
At the equator, you're farther from the centre → weaker gravity

This shape effect combines with the rotation effect to give the latitude variation shown in the table above.

The Complete Picture

Putting it all together, the variation of gravity with latitude $\phi$ and height $h$ is given by:

g(\phi, h) = g_0 \left(1 - \frac{2h}{R}\right) \left(1 - \frac{\omega^2 R}{g_0} \cos^2 \phi\right)

Where $g_0 \approx 9.806 \, \text{m/s}^2$ is the standard value at 45° latitude at sea level.

Important

The variation of gravity is small — typically less than 0.5% across the Earth's surface. But it matters for precise measurements, satellite orbits, and even for defining the kilogram (since a spring scale calibrated in Mumbai would read differently in London).

A Quick Summary

Factor	Effect on $g$	Why
Going up (altitude)	Decreases	Farther from Earth's centre
Going down (depth)	Decreases (linearly for uniform Earth)	Less mass below you
Moving to equator	Decreases	Rotation + bulge
Moving to poles	Increases	No rotation + closer to centre

The key takeaway: gravity is not a fixed number. It's a local property that depends on where you are on (or inside) the Earth. The $9.8 \, \text{m/s}^2$ you memorise is just a convenient average — the real story is richer, and now you know why.

"Variation Of Gravity derivation" and "Variation Of Gravity numerical problems" are two of the most common searches tied to this topic, and Variation Of Gravity is a core, NCERT-aligned topic from the Gravitation portion of the Class 11 Physics curriculum, and it is tested regularly in CBSE board exams as well as in JEE Main and NEET. Pairing this explanation with NCERT Physics textbook practice and previous years' questions is the surest way to lock the concept in before an exam.

Earth is still described as an approximate sphere, so gravity at any surface point continues to act essentially along the line to the centre — but with density no longer uniform, the amount of mass gathered near different points of the surface differs, so the strength of gravity is no longer the same everywhere. This matches option (A): directed towards the centre, but not the same magnitude everywhere.

Why direction stays (essentially) radial

For a sphere, gravity is a sum of pulls from every bit of interior mass. As long as the overall shape stays spherical, the dominant contribution to the field at any surface point is still directed toward the geometric centre — this is exact for a uniform sphere (and for any purely radial density variation, layer by layer, by the shell theorem), and remains the leading-order behaviour even for departures from that idealisation. The question is explicit that Earth remains an approximate sphere — so we are not dealing with a wildly lopsided shape, just a non-uniform density inside a still roughly spherical body.

Why the magnitude is no longer uniform

If every point on the surface had the same distance from the centre, the same total mass, and the same distribution of that mass, gravity's magnitude would be identical everywhere. But once the interior density is allowed to vary — some regions denser, some lighter — the amount and distribution of mass pulling on different surface points is no longer identical from point to point. So $g$ can be slightly stronger over a denser region and slightly weaker over a lighter one, even though the direction stays essentially centre-ward.

Checking against the other options

(B) requires the same magnitude everywhere but not toward the centre — the opposite pairing of what a near-spherical, non-uniform Earth actually gives.
(C) requires both same magnitude and exactly centre-directed — only guaranteed for a perfectly uniform (or purely radially-symmetric) density, which the question explicitly rules out.
(D) claims gravity can never be zero anywhere on the surface — this isn't what the non-uniform-density condition implies, and it isn't the case being tested here.

✓Final answer

Option (A) — the acceleration due to gravity will be directed towards the centre, but its magnitude will not be the same everywhere.

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Q.The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration due to gravity

Why direction stays (essentially) radial

Why the magnitude is no longer uniform

Checking against the other options

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