Question

The centre of mass of an extended body on the surface of the earth and its centre of gravity

• A. can never be at the same point
• B. are always at the same point for any size of the body
• C. centre of mass coincides with centre of gravity of a body if the size of the body is negligible as compared to the size (or radius) of the earth
• D. are always at the same point only for spherical bodies

Answer 1

The Correct Option is C

The center of mass and the center of gravity are related concepts but have key differences:

Center of Mass:

• The center of mass of an extended body is the point where the entire mass of the body can be considered to be concentrated.
• It is the average position of all the individual particles' masses that make up the body.
• The center of mass is determined based on the distribution of mass within the object.

Center of Gravity:

• The center of gravity is the point where the force of gravity can be considered to act on the body.
• In a uniform gravitational field, the center of gravity coincides with the center of mass.
• In non-uniform gravitational fields, the center of gravity may differ slightly from the center of mass.

In summary: The center of mass is related to mass distribution, while the center of gravity is related to the distribution of the gravitational force. In most practical situations where gravity is uniform, the two points coincide.

Answer 2

Let's define the Center of Mass (CM) and Center of Gravity (CG) for an extended body. 

• Center of Mass (CM): The center of mass is a point representing the mean position of the matter in a body or system. It is the weighted average of the positions of the particles, weighted by their mass. Its position \( \vec{r}_{CM} \) is given by: \[ \vec{r}_{CM} = \frac{\sum m_i \vec{r}_i}{\sum m_i} \] The location of the CM depends only on the distribution of mass within the body.
• Center of Gravity (CG): The center of gravity is the point where the net gravitational force (weight) on the body effectively acts. Its position \( \vec{r}_{CG} \) is given by the weighted average of the positions of the particles, weighted by their weight \( m_i \vec{g}_i \): \[ \vec{r}_{CG} = \frac{\sum (m_i \vec{g}_i) \times \vec{r}_i}{\sum m_i \vec{g}_i} \] The location of the CG depends on both the distribution of mass and the gravitational field \( \vec{g} \).

The CM and CG coincide if and only if the gravitational field \( \vec{g} \) is uniform across the entire extent of the body. If \( \vec{g} \) is constant (same magnitude and direction for all particles \( i \)), then \( \vec{g}_i = \vec{g} \) for all \( i \), and the formula for CG simplifies: \[ \vec{r}_{CG} = \frac{\sum m_i \vec{g} \times \vec{r}_i}{\sum m_i \vec{g}} = \frac{\vec{g} \sum m_i \vec{r}_i}{\vec{g} \sum m_i} = \frac{\sum m_i \vec{r}_i}{\sum m_i} = \vec{r}_{CM} \]

Now consider an extended body on the surface of the Earth. The Earth's gravitational field is not perfectly uniform. The magnitude of \( \vec{g} \) decreases with altitude (distance from the Earth's center), and its direction always points towards the Earth's center. For a large extended body (e.g., a very tall building or a mountain), the gravitational field at the top of the body will be slightly weaker and point in a slightly different direction compared to the field at the bottom.

However, if the size of the body is negligible compared to the radius of the Earth (which is approximately 6400 km), the variation in the gravitational field \( \vec{g} \) across the body is extremely small and can be ignored. In this common scenario, we can approximate the gravitational field as uniform over the body.

Therefore:

• For bodies whose dimensions are small compared to the Earth's radius, the gravitational field is effectively uniform over the body, and thus CM coincides with CG.
• For very large bodies (e.g., comparable in size to Earth's features), the variation in \( \vec{g} \) is significant, and the CM and CG will generally be at different points (typically, CG is slightly lower than CM because the lower parts experience stronger gravity).

Evaluating the options:

1. Can never be at the same point: Incorrect. They coincide in a uniform field.
2. Are always at the same point for any size of the body: Incorrect. For very large bodies, they differ.
3. Centre of mass coincides with centre of gravity of a body if the size of the body is negligible as compared to the size (or radius) of the earth: Correct. This is the condition under which the gravitational field can be considered uniform over the body.
4. Are always at the same point only for spherical bodies: Incorrect. Coincidence depends on the uniformity of the field, not the shape, although for a large sphere, CM and CG would both be at the geometric center due to symmetry, but the *reason* they coincide relies on the assumption of uniformity which breaks down if the sphere is large enough relative to Earth. For small bodies, they coincide regardless of shape.