A uniform solid sphere of mass `m` and radius `r` is suspended symmetrically by a uniform thin spherical shell of radius `2r` and mass `m`.

Options

A. The gravitational field at a distance of 1.5r from the centre is `(2)/(9)(Gm)/(r^(2))`
B. The gravitational field at a distance of 2.5r from centre is `(8)/(25)(Gm)/(r^(2))`
C. The gravitational field at a distance of 1.5r from centre is zero
D. The gravitational field between the sphere and spherical shell is uniform.

Explanation

To solve the problem, we need to analyze the gravitational fields produced by both the solid sphere and the spherical shell at specific distances from their centers.

### Step-by-Step Solution:

1. **Identify the Setup**:
- We have a solid sphere of mass `m` and radius `r`.
- This sphere is suspended symmetrically inside a thin spherical shell of radius `2r` and mass `m`.

2. **Determine the Gravitational Field at a Distance of 1.5r**:
- The distance of interest, `1.5r`, is located between the center of the solid sphere and the inner surface of the spherical shell.
- According to the properties of gravitational fields:
- The gravitational field inside a uniform spherical shell is zero.
- The gravitational field due to the solid sphere at a distance `d` from its center is given by the formula:
\[ g = -\frac{G \cdot m}{d^2} \]
- Here, `d = 1.5r`. Therefore, we can calculate the gravitational field due to the solid sphere:
\[ g_{\text{solid sphere}} = -\frac{G \cdot m}{(1.5r)^2} = -\frac{G \cdot m}{\frac{9}{4}r^2} = -\frac{4Gm}{9r^2} \]
- The gravitational field due to the spherical shell is zero:
\[ g_{\text{spherical shell}} = 0 \]
- Thus, the total gravitational field at `1.5r` is:
\[ g_{\text{total}} = g_{\text{solid sphere}} + g_{\text{spherical shell}} = -\frac{4Gm}{9r^2} + 0 = -\frac{4Gm}{9r^2} \]

3. **Determine the Gravitational Field at a Distance of 2.5r**:
- The distance of interest, `2.5r`, is outside both the solid sphere and the spherical shell.
- For points outside a uniform spherical shell and a solid sphere, the total gravitational field can be treated as if the entire mass were concentrated at a point at the center.
- The total mass at this point is `2m` (mass of the solid sphere + mass of the shell).
- The distance from the center to the point is `2.5r`.
- The gravitational field at this distance is given by:
\[ g_{\text{total}} = -\frac{G \cdot (2m)}{(2.5r)^2} = -\frac{G \cdot (2m)}{\frac{25}{4}r^2} = -\frac{8Gm}{25r^2} \]

### Final Answers:
- The gravitational field at a distance of `1.5r` is:
\[ g_{\text{total}} = -\frac{4Gm}{9r^2} \]
- The gravitational field at a distance of `2.5r` is:
\[ g_{\text{total}} = -\frac{8Gm}{25r^2} \]