A thin spherical shell of mass `M` and radius `R` has a small hole. A particle of mass `m` released at its mouth. Then

Options

A. the particle will execute simple harmonic motion inside the shell
B. the particle will oscillate inside the shell, but the oscillation are not harmonic
C. the particle will not oscillate, but the speed of the particle will go on increasing
D. None of the above

Answer

To solve the problem, we need to analyze the situation of a particle of mass \( m \) being released from the mouth of a thin spherical shell of mass \( M \) and radius \( R \).

### Step-by-Step Solution:

1. **Understanding the Gravitational Field Inside a Spherical Shell**:
According to the shell theorem, a uniform spherical shell exerts no net gravitational force on any object located inside it. This means that if a particle is placed anywhere inside the shell, it will not experience any gravitational attraction from the mass of the shell.

2. **Position of the Particle**:
When the particle of mass \( m \) is released at the mouth of the hole in the shell, it is technically still inside the shell. Therefore, it is not influenced by the gravitational field of the shell.

3. **Effect of the Gravitational Field**:
Since the net gravitational field inside the shell is zero, the particle will not experience any gravitational pull towards the shell. Consequently, it will not oscillate or move in a harmonic manner. Instead, it will simply remain at rest or move freely without any acceleration.

4. **Conclusion**:
As a result, the particle will not oscillate, and there will be no increase in speed due to gravitational attraction. Therefore, the correct conclusion is that the particle will not be influenced by the gravitational field of the shell.

5. **Selecting the Correct Option**:
From the options provided:
- Option 1: The particle executes simple harmonic motion inside the shell. (Incorrect)
- Option 2: The particle oscillates inside the shell but the oscillations are not harmonic. (Incorrect)
- Option 3: The particle does not oscillate, but the speed of the particle will go on increasing. (Incorrect)
- Option 4: None of the above. (Correct)

Thus, the correct answer is **Option 4: None of the above**.