A flat horizontal board moves up and down under SHM vertically with amplitude A. The shortest permissible time period of the vibration such that an object placed on the board may not lose contact with the board is

Options

A. `2pi sqrt((g)/(A))`
B. `2pi sqrt((A)/(g))`
C. `2pi sqrt((2A)/(g))`
D. `(pi)/(2) sqrt((A)/(g))`

Answer

To solve the problem, we need to determine the shortest permissible time period of the vibration of a flat horizontal board moving up and down under Simple Harmonic Motion (SHM) such that an object placed on the board does not lose contact with it.

### Step-by-Step Solution:

1. **Understanding the Forces**:
- The board is moving vertically with an amplitude \( A \).
- An object placed on the board experiences a normal force \( F_n \) due to the board's upward motion and gravitational force \( mg \) acting downward.

2. **Condition for Maintaining Contact**:
- For the object to remain in contact with the board, the normal force must be equal to or greater than the weight of the object. If the upward acceleration of the board exceeds the gravitational acceleration, the object will lose contact.
- Mathematically, this can be expressed as:
\[ F_n \geq mg \]

3. **Acceleration in SHM**:
- The maximum upward acceleration \( a_{max} \) of the board in SHM is given by:
\[ a_{max} = \omega^2 A \]
- Here, \( \omega \) is the angular frequency of the SHM.

4. **Setting Up the Equation**:
- For the object to just maintain contact, we set the maximum acceleration equal to the gravitational acceleration \( g \):
\[ \omega^2 A = g \]

5. **Solving for Angular Frequency**:
- Rearranging the equation gives:
\[ \omega^2 = \frac{g}{A} \]
- Taking the square root:
\[ \omega = \sqrt{\frac{g}{A}} \]

6. **Finding the Time Period**:
- The time period \( T \) of SHM is related to the angular frequency by the formula:
\[ T = \frac{2\pi}{\omega} \]
- Substituting for \( \omega \):
\[ T = \frac{2\pi}{\sqrt{\frac{g}{A}}} \]
- This simplifies to:
\[ T = 2\pi \sqrt{\frac{A}{g}} \]

7. **Conclusion**:
- The shortest permissible time period of the vibration such that the object placed on the board may not lose contact with the board is:
\[ T = 2\pi \sqrt{\frac{A}{g}} \]