Does centre of mass and centre of gravity body always coincide?

Explanation

To determine whether the center of mass and the center of gravity of a body always coincide, we can analyze the definitions and conditions under which these concepts are applied.

### Step-by-Step Solution:

1. **Definitions**:
- **Center of Mass (CM)**: The point at which the mass of a body can be considered to be concentrated. It is determined by the distribution of mass in the body.
- **Center of Gravity (CG)**: The point at which the weight of a body can be considered to act. It is influenced by the gravitational field acting on the body.

2. **Uniform Gravitational Field**:
- In a uniform gravitational field (where the gravitational acceleration \( g \) is constant throughout the body), the center of gravity coincides with the center of mass. This is because the gravitational force acts equally on all parts of the body.

3. **Non-Uniform Gravitational Field**:
- In cases where the gravitational field varies from one point to another (which can occur for larger objects or when the object is in a non-uniform field), the center of gravity may not coincide with the center of mass. This is because different parts of the object experience different gravitational forces, leading to a shift in the effective point of weight application.

4. **Conclusion**:
- Therefore, the center of mass and center of gravity do not always coincide. They coincide only in a uniform gravitational field and for small objects. For larger objects or in a non-uniform gravitational field, they can be different.

### Summary:
- **Center of Mass and Center of Gravity Coincidence**: They coincide in a uniform gravitational field and for small objects. They do not coincide in a non-uniform gravitational field or for larger objects.