A particle placed
1.inside a uniform spherical shell of mass M, but not at the center
2.inside a uniform spherical shell of mass M, at the center
3.outside a uniform spherical shell of mass M, a distance r from the center
4.outside a uniform solid sphere of mass M, a distance 2r from the center
If `F_1,F_2,F_3` and `F_4` are gravitational forces acting on the particle in four cases.

Options

A. `F_1 gt F_2 gt F_3 gt F_4`
B. `F_1 = F_2 lt F_3 lt F_4`
C. `F_1=F_2 lt F_4 lt F_3`
D. `F_1 lt F_2 lt F_3 lt F_4`

Explanation

To solve the problem, we will analyze the gravitational forces acting on a particle placed in four different scenarios involving a uniform spherical shell and a solid sphere. We will denote the gravitational forces in each case as \( F_1, F_2, F_3, \) and \( F_4 \).

### Step-by-Step Solution:

1. **Case 1: Inside a Uniform Spherical Shell (Not at the Center)**
- According to the Shell Theorem, the gravitational field inside a uniform spherical shell is zero at any point inside the shell.
- Therefore, the gravitational force \( F_1 \) acting on the particle is:
\[ F_1 = 0 \]

2. **Case 2: Inside a Uniform Spherical Shell (At the Center)**
- Similar to Case 1, since the particle is still inside the shell, the gravitational field remains zero at the center as well.
- Thus, the gravitational force \( F_2 \) acting on the particle is:
\[ F_2 = 0 \]

3. **Case 3: Outside a Uniform Spherical Shell (Distance \( r \) from the Center)**
- When the particle is outside the shell, we can treat the shell as a point mass located at its center. The gravitational force \( F_3 \) can be calculated using Newton's law of gravitation:
\[ F_3 = \frac{G M m}{r^2} \]

4. **Case 4: Outside a Uniform Solid Sphere (Distance \( 2r \) from the Center)**
- Similar to Case 3, for a solid sphere outside its surface, we can also treat it as a point mass at its center. The gravitational force \( F_4 \) is given by:
\[ F_4 = \frac{G M m}{(2r)^2} = \frac{G M m}{4r^2} \]

### Summary of Forces:
- \( F_1 = 0 \)
- \( F_2 = 0 \)
- \( F_3 = \frac{G M m}{r^2} \)
- \( F_4 = \frac{G M m}{4r^2} \)

### Comparison of Forces:
- Since \( F_1 = 0 \) and \( F_2 = 0 \), we have:
\[ F_1 = F_2 < F_4 < F_3 \]
- This means:
\[ 0 < \frac{G M m}{4r^2} < \frac{G M m}{r^2} \]

### Final Conclusion:
The correct order of the gravitational forces is:
\[ F_1 = F_2 < F_4 < F_3 \]