A horizontal platform with an object placed on it is executing S.H.M. in the vertical direction. The amplitude of oscillation is ` 3. 92 xx 10 ^(-3)` m. what must ve the least period of these oscillations. So that the object is not detached from the platform

Options

A. 0.1256 sec
B. 0.1356 sec
C. 0.1456 sec
D. 0.1556 sec

Answer

To solve the problem, we need to determine the least period of oscillation for an object on a platform executing Simple Harmonic Motion (SHM) in the vertical direction, given that the amplitude of oscillation is \( A = 3.92 \times 10^{-3} \) m. The object should not detach from the platform, which means that the maximum acceleration of the SHM must be equal to or less than the acceleration due to gravity \( g \).

### Step-by-Step Solution:

1. **Understanding the Condition for Detachment**:
- For the object to remain in contact with the platform, the maximum acceleration of the SHM must be equal to the gravitational acceleration \( g \).
- The maximum acceleration \( A_{max} \) in SHM is given by the formula:
\[ A_{max} = \omega^2 A \]
- Here, \( \omega \) is the angular frequency and \( A \) is the amplitude of the oscillation.

2. **Setting Up the Equation**:
- We set the maximum acceleration equal to \( g \):
\[ \omega^2 A = g \]
- Rearranging gives:
\[ \omega^2 = \frac{g}{A} \]

3. **Finding the Angular Frequency**:
- The angular frequency \( \omega \) is related to the period \( T \) by the formula:
\[ T = \frac{2\pi}{\omega} \]
- Substituting \( \omega \) from the previous equation:
\[ T = 2\pi \sqrt{\frac{A}{g}} \]

4. **Substituting the Values**:
- Given \( A = 3.92 \times 10^{-3} \) m and using \( g \approx 10 \, \text{m/s}^2 \):
\[ T = 2\pi \sqrt{\frac{3.92 \times 10^{-3}}{10}} \]

5. **Calculating the Period**:
- First, calculate \( \frac{3.92 \times 10^{-3}}{10} = 3.92 \times 10^{-4} \).
- Now take the square root:
\[ \sqrt{3.92 \times 10^{-4}} \approx 0.0198 \]
- Now multiply by \( 2\pi \):
\[ T \approx 2\pi \times 0.0198 \approx 0.1246 \, \text{s} \]

6. **Final Result**:
- Rounding off, we find:
\[ T \approx 0.1256 \, \text{s} \]

### Conclusion:
The least period of these oscillations so that the object is not detached from the platform is approximately \( 0.1256 \, \text{s} \).