Mark the correct statement/s-:

Options

A. Gravitational potential at curvature centre of a thin hemispherical shell of radius R and mass M is equal to `(GM)/(R)`
B. Gravitational field strength at point lying on the axis of a thin, uniform circular ring of radius R and mass M is equal to `(GMx)/((R^(2)+x^(2))^(3//2))` where x is distance of that point from centre of the ring.
C. Newton's law of gravitation for gravitational force between two bodies is applicable only when bodies have spherically symmetric distribution of mass
D. None of these

Explanation

To solve the question, we need to evaluate the statements regarding gravitational potential and gravitational field strength. Let's analyze each statement step by step.

### Step 1: Evaluating the first statement
**Statement:** Gravitational potential at the center of a thin hemispherical shell of radius r and mass m is equal to \( \frac{gm}{r} \).

**Solution:**
- The gravitational potential \( V \) at a point due to a mass \( m \) is given by the formula:
\[ V = -\frac{Gm}{r} \]
where \( G \) is the universal gravitational constant and \( r \) is the distance from the mass to the point where the potential is being calculated.
- For a thin hemispherical shell, every point on the shell is at a distance \( r \) from the center.
- Therefore, the gravitational potential at the center of the hemispherical shell is:
\[ V = -\frac{Gm}{r} \]
- Since the statement claims that the potential is \( \frac{gm}{r} \) (without the negative sign), this statement is **false**.

### Step 2: Evaluating the second statement
**Statement:** Gravitational field strength at a point on the axis of a thin uniform circular ring of radius r and mass m is given by a specific formula, where r is the distance of that point from the center of the ring.

**Solution:**
- The gravitational field \( E \) at a point along the axis of a circular ring can be derived using symmetry.
- Consider a ring of radius \( r \) and mass \( m \). The gravitational field at a point along the axis at distance \( x \) from the center of the ring can be calculated by considering the contributions from each mass element on the ring.
- Due to symmetry, the horizontal components of the gravitational field from opposite mass elements cancel out, while the vertical components add up.
- The resultant gravitational field strength \( E \) at a distance \( x \) from the center of the ring is given by:
\[ E = \frac{Gm}{r^2 + x^2} \cdot \frac{x}{\sqrt{r^2 + x^2}} \]
- This expression shows that the gravitational field strength at that point is indeed dependent on the distance \( x \) from the center of the ring and the radius \( r \) of the ring.
- Thus, this statement is **true**.

### Step 3: Evaluating the third statement
**Statement:** Newton's law of gravitation for gravitational force between two bodies is applicable only when two bodies are spherically symmetrically distributed mass.

**Solution:**
- Newton's law of gravitation states that every point mass attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
- This law is valid for all mass distributions, not just for spherically symmetric bodies. It can be applied to point masses, spherical bodies, and even non-spherical distributions using the principle of superposition.
- Therefore, this statement is **false**.

### Final Evaluation of Statements
- **Statement 1:** False
- **Statement 2:** True
- **Statement 3:** False

### Conclusion
The correct statements are:
- Only Statement 2 is true.