A horizontal platform with an object placed on it is executing SHM in the vertical direction . The amplitude of oscillation is 2.5 cm what must be the least period of these oscillations so that the object is not detached ?

Options

A. `pis`
B. `pi/5s`
C. `pi/10s`
D. `pi/15s`

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, such that the object does not detach from the platform.

### Step-by-Step Solution:

1. **Understand the Forces Involved**:
- When the platform oscillates upwards, the object experiences a force due to the upward acceleration of the platform.
- When the platform oscillates downwards, the object experiences gravitational force acting downwards.

2. **Condition for Detachment**:
- The object will not detach from the platform if the maximum upward acceleration of the platform is equal to the acceleration due to gravity (g). This means:
\[ a_{max} = g \]

3. **Maximum Acceleration in SHM**:
- The maximum acceleration (a_max) in SHM can be expressed as:
\[ a_{max} = \omega^2 A \]
where \( \omega \) is the angular frequency and \( A \) is the amplitude of oscillation.

4. **Setting Up the Equation**:
- From the condition for detachment, we have:
\[ \omega^2 A = g \]
- Rearranging gives:
\[ \omega^2 = \frac{g}{A} \]

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

6. **Substituting Values**:
- Given \( A = 2.5 \, \text{cm} = 0.025 \, \text{m} \) (convert cm to m) and \( g \approx 9.81 \, \text{m/s}^2 \):
\[ T = 2\pi \sqrt{\frac{0.025}{9.81}} \]

7. **Calculating the Value**:
- First, calculate the fraction:
\[ \frac{0.025}{9.81} \approx 0.00255 \]
- Now, take the square root:
\[ \sqrt{0.00255} \approx 0.0505 \]
- Finally, calculate \( T \):
\[ T \approx 2\pi \times 0.0505 \approx 0.317 \, \text{s} \]

### Final Answer:
The least period of oscillation \( T \) such that the object does not detach is approximately \( 0.317 \, \text{s} \).