A spherical shell has inner radius `R_1`, outer radius `R_2`, and mass M, distributed uniformly throughout the shell. The magnitude of the gravitational force exerted on the shell by a point mass particle of mass m, located at a distance d from the center, inside the inner radius, is:

Options

A. 0
B. `GMm//R_1^2`
C. `GMm//d^2`
D. `GMm//(R_2^2-d^2)`

Explanation

To solve the problem, we need to determine the gravitational force exerted on a point mass \( m \) located inside a spherical shell with inner radius \( R_1 \), outer radius \( R_2 \), and mass \( M \) uniformly distributed throughout the shell.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a spherical shell with inner radius \( R_1 \) and outer radius \( R_2 \).
- A point mass \( m \) is located at a distance \( d \) from the center of the shell, where \( d < R_1 \) (inside the shell).

2. **Applying Gauss's Law for Gravitation**:
- According to Gauss's Law for gravitation, the gravitational field \( E_g \) at a distance \( r \) from the center of a spherical shell is determined by the mass enclosed within that radius.
- For a Gaussian surface inside the shell (where \( r < R_1 \)), the mass enclosed is zero because all the mass \( M \) is distributed in the shell itself.

3. **Calculating the Gravitational Field**:
- Since there is no mass enclosed within the Gaussian surface, we have:
\[ E_g = 0 \]
- This means that the gravitational field inside the shell is zero.

4. **Calculating the Gravitational Force**:
- The gravitational force \( F \) on the mass \( m \) can be calculated using the formula:
\[ F = m \cdot E_g \]
- Substituting \( E_g = 0 \):
\[ F = m \cdot 0 = 0 \]

5. **Conclusion**:
- The magnitude of the gravitational force exerted on the point mass \( m \) located inside the inner radius of the shell is:
\[ F = 0 \]

### Final Answer:
The gravitational force exerted on the mass \( m \) by the spherical shell is \( 0 \).

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