If a smooth tunnel is dug across a diameter of earth and a particle is released from the surface of earth, the particle oscillate simple harmonically along it
Time period of the particle is not equal to

Options

A. `2pisqrt((R )/(g))`
B. `(2pi)/(sqrt(GM))R^(3//2)`
C. 84.6 min
D. None of these

Answer

To solve the problem of determining the time period of a particle oscillating in a smooth tunnel dug across the diameter of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept of Simple Harmonic Motion (SHM)**:
- When a particle is released from the surface of the Earth and falls through a tunnel that passes through the center, it will experience gravitational force which varies with distance from the center. This results in simple harmonic motion.

2. **Formula for Time Period of SHM**:
- The time period \( T \) of a particle in simple harmonic motion can be expressed as:
\[ T = 2\pi \sqrt{\frac{r}{g}} \]
where \( r \) is the radius of the Earth and \( g \) is the acceleration due to gravity at the surface.

3. **Substituting Values**:
- The radius of the Earth \( r \) is approximately \( 6400 \times 10^3 \) meters.
- The acceleration due to gravity \( g \) is approximately \( 9.8 \, \text{m/s}^2 \).

4. **Calculating the Time Period**:
- Substitute the values into the formula:
\[ T = 2\pi \sqrt{\frac{6400 \times 10^3}{9.8}} \]
- Calculate \( \frac{6400 \times 10^3}{9.8} \):
\[ \frac{6400 \times 10^3}{9.8} \approx 653061.2245 \]
- Now calculate \( \sqrt{653061.2245} \):
\[ \sqrt{653061.2245} \approx 808.5 \]
- Finally, calculate \( T \):
\[ T \approx 2\pi \times 808.5 \approx 5086.4 \, \text{seconds} \approx 84.7 \, \text{minutes} \]

5. **Identifying the Incorrect Option**:
- The question asks which time period the particle is **not equal to**. The calculated time period is approximately \( 84.7 \, \text{minutes} \).
- We need to compare this with the given options. If none of the options match this value, that option would be the correct answer.

### Final Answer:
The time period of the particle is not equal to the options provided in the question (assuming none match \( 84.7 \, \text{minutes} \)).