Q.The centre of mass of an extended body on the surface of the earth and its centre of gravity (Note: more than one of the given options may be correct.)

Concept: Variation of Gravity

The centre of mass is the point where the entire mass of a body can be considered concentrated for translational motion; it depends only on mass distribution. The centre of gravity is the point where the resultant gravitational force acts; it depends on the local gravitational field.

Near Earth's surface, gravity varies with position—both in magnitude (inversely with distance from Earth's centre) and direction (radially inward). For an extended body, different parts experience slightly different gravitational forces.

Key reasoning:

1. If gravity were perfectly uniform, the two centres would coincide. But Earth's gravity decreases with altitude and points toward Earth's centre, so the gravitational field is non-uniform.

2. For small objects (sizes $\ll$ Earth's radius $\approx 6400$ km), the variation in $g$ across the body is negligible. The centres of mass and gravity are then practically coincident. Objects of size ~100 m satisfy this condition.

3. For very large objects, the lower parts experience stronger gravity than the upper parts, shifting the centre of gravity slightly below the centre of mass. They are distinct but close. …

Why centre of mass and centre of gravity differ

The centre of mass is purely geometric: it is the weighted average position of all the mass elements in a body, independent of any external field. If you have mass elements $m_i$ at positions $\vec{r}_i$, then

The centre of gravity, on the other hand, is the point where the total gravitational force (weight) can be considered to act. It is the weighted average of positions using weight as the weighting factor:

where $g_i$ is the local acceleration due to gravity at position $\vec{r}_i$.

If gravity is uniform—the same magnitude and direction everywhere in the body—then $g_i = g$ factors out and the two definitions become identical. But Earth's gravity is not perfectly uniform: it decreases with altitude (or increases with depth, up to a point), and its direction is always radial toward Earth's centre. For an extended body these variations matter.

Examining each option

1. Option (A): "are always at the same point for any size of the body"

This would require gravity to be uniform across the entire body, regardless of size. On Earth's surface, $g$ varies with height as

where $R_E \approx 6400\,\text{km}$. For a tall building (say 100 m), the variation is about $\frac{2 \times 100}{6.4 \times 10^6} \approx 3 \times 10^{-5}$, or 0.003%. This is negligible, so CM and CG nearly coincide.

But for a very large object—imagine a body stretching hundreds of kilometres vertically—the top experiences noticeably weaker gravity than the bottom. The centre of gravity shifts downward (toward the stronger field) relative to the centre of mass.

Option (A) is false.

2. Option (B): "are always at the same point only for spherical bodies"

Spherical symmetry of the body does not guarantee uniform gravity within the body. A large sphere on Earth's surface still has its top farther from Earth's centre than its bottom, so $g$ varies across it. The statement confuses the symmetry of the object with the uniformity of the external field.

Option (B) is false.

3. Option (C): "can never be at the same point"

In the limit of a point mass, or for any body small enough that $g$ is effectively constant across it, CM and CG coincide to any measurable precision. So "never" is too strong.

Option (C) is false.

4. Option (D): "is close to each other for objects, say of sizes less than 100 m"

For a 100 m tall object, the fractional change in $g$ from bottom to top is

This tiny variation means the centre of gravity is displaced from the centre of mass by a distance of order $h \times 10^{-5} \sim 1\,\text{mm}$—utterly negligible for any practical purpose. The two points are indeed "close."

Option (D) is correct.

5. Option (E): "both can change if the object is taken deep inside the earth"

The centre of mass is an intrinsic property of the body's mass distribution; moving the body does not change it (assuming the body is rigid). …

Step 1: centre of mass is purely geometric; centre of gravity is weighted by local g. Step 2: they coincide exactly only in a uniform field. Step 3: for objects of modest size near Earth's surface, g varies by ~1e-5 fractionally -- CM and CG nearly coincide but not exactly for any size -- (d) true, (a)/(c) false. Step 4: spherical symmetry is irreleva …