Two concentric spherical shells have masses `M_1,M_2` and radii `R_1,R_2(R_1ltR_2)`. What is the force exerted by this system on a particle of mass `m` if it is placed at a distance `(R_1+R_2)/2` from the center?

Explanation

To solve the problem, we need to determine the gravitational force exerted on a particle of mass `m` placed at a distance `(R_1 + R_2)/2` from the center of two concentric spherical shells with masses `M_1` and `M_2`, and radii `R_1` and `R_2` respectively.

### Step-by-Step Solution:

1. **Identify the Position of the Particle**:
The particle is placed at a distance of `(R_1 + R_2)/2` from the center. Since `R_1 < R_2`, we know that `(R_1 + R_2)/2` will lie between `R_1` and `R_2`.

2. **Determine the Region of Influence**:
The gravitational force exerted by a spherical shell on a point mass located inside it is zero. Since the particle is located at `(R_1 + R_2)/2`, which is greater than `R_1` but less than `R_2`, it is inside the outer shell (mass `M_2`) but outside the inner shell (mass `M_1`).

3. **Calculate the Force Due to the Inner Shell**:
The gravitational force acting on the particle due to the inner shell (mass `M_1`) can be calculated using the formula for gravitational force:
\[ F = \frac{G \cdot M_1 \cdot m}{r^2} \]
where `r` is the distance from the center to the particle, which is `(R_1 + R_2)/2`.

4. **Substitute the Distance into the Formula**:
Substitute `r = (R_1 + R_2)/2` into the gravitational force equation:
\[ F = \frac{G \cdot M_1 \cdot m}{\left(\frac{R_1 + R_2}{2}\right)^2} \]

5. **Simplify the Expression**:
Simplifying the expression gives:
\[ F = \frac{G \cdot M_1 \cdot m}{\frac{(R_1 + R_2)^2}{4}} = \frac{4G \cdot M_1 \cdot m}{(R_1 + R_2)^2} \]

6. **Consider the Outer Shell**:
The gravitational force due to the outer shell (mass `M_2`) is zero because the particle is located inside the outer shell.

7. **Final Result**:
Therefore, the total gravitational force acting on the particle is solely due to the inner shell:
\[ F = \frac{4G \cdot M_1 \cdot m}{(R_1 + R_2)^2} \]