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. (a)`F_1 gt F_2 gt F_3 gt F_4`
B. (b)`F_1 = F_2 lt F_3 lt F_4`
C. (c)`F_1=F_2 lt F_4 lt F_3`
D. (d)`F_1 lt F_2 lt F_3 lt F_4`

Explanation

To solve the problem, we need to analyze the gravitational forces acting on a particle in four different scenarios involving a uniform spherical shell and a uniform solid sphere. Let's break down each case step by step.

### Step 1: Analyze the first case (F1)
**Case 1:** A particle placed inside a uniform spherical shell of mass M, but not at the center.

According to the shell theorem, the gravitational force acting on a particle located inside a uniform spherical shell is zero, regardless of its position within the shell.

**Conclusion:**
\[ F_1 = 0 \]

### Step 2: Analyze the second case (F2)
**Case 2:** A particle placed inside a uniform spherical shell of mass M, at the center.

Again, by the shell theorem, the gravitational force acting on a particle at the center of a uniform spherical shell is also zero.

**Conclusion:**
\[ F_2 = 0 \]

### Step 3: Analyze the third case (F3)
**Case 3:** A particle placed outside a uniform spherical shell of mass M, at a distance r from the center.

For a particle outside a spherical shell, the gravitational force can be calculated using Newton's law of gravitation. The force is given by:
\[ F_3 = \frac{G M m}{r^2} \]
where G is the gravitational constant, M is the mass of the shell, m is the mass of the particle, and r is the distance from the center of the shell.

### Step 4: Analyze the fourth case (F4)
**Case 4:** A particle placed outside a uniform solid sphere of mass M, at a distance 2r from the center.

Using the same law of gravitation, the force acting on the particle at a distance 2r is:
\[ F_4 = \frac{G M m}{(2r)^2} = \frac{G M m}{4r^2} \]

### Step 5: Compare the forces
Now we can compare the forces \( F_3 \) and \( F_4 \):
- From the equations derived:
- \( F_3 = \frac{G M m}{r^2} \)
- \( F_4 = \frac{G M m}{4r^2} \)

Since \( F_3 \) is greater than \( F_4 \) because:
\[ F_3 = 4 \cdot F_4 \]

### Final Conclusion
Now we can summarize the results:
- \( F_1 = 0 \)
- \( F_2 = 0 \)
- \( F_3 > F_4 \)

Thus, we can conclude:
\[ F_1 = F_2 < F_4 < F_3 \]

### Answer:
The correct option is that \( F_1 = F_2 \) and \( F_4 < F_3 \).

---