A simple pendulum has time period `2s`. The point of suspension is now moved upward accoding to relation `y = (6t - 3.75t^(2))m` where `t` is in second and y is the vertical displacement in upward direction. The new time period of simple pendulum will be

Options

A. `2s`
B. `1s`
C. `4s`
D. None of these

Explanation

To solve the problem, we need to determine the new time period of the simple pendulum after the point of suspension is moved upward according to the given relation.

### Step-by-Step Solution:

1. **Identify the Initial Time Period**:
The initial time period \( T \) of the pendulum is given as \( 2 \, \text{s} \).

2. **Understand the Relation of Vertical Displacement**:
The vertical displacement of the point of suspension is given by the equation:
\[ y = 6t - 3.75t^2 \]
where \( y \) is in meters and \( t \) is in seconds.

3. **Calculate the Acceleration**:
To find the effective acceleration due to gravity, we need to differentiate the displacement function \( y \) twice with respect to time \( t \).

- First, find the velocity \( v \):
\[ v = \frac{dy}{dt} = \frac{d}{dt}(6t - 3.75t^2) = 6 - 7.5t \]

- Next, find the acceleration \( a \):
\[ a = \frac{d^2y}{dt^2} = \frac{d}{dt}(6 - 7.5t) = -7.5 \, \text{m/s}^2 \]

4. **Determine the Effective Gravity**:
The effective acceleration due to gravity \( g' \) acting on the pendulum is the actual gravitational acceleration \( g \) minus the upward acceleration \( a \):
\[ g' = g - |a| = 10 \, \text{m/s}^2 - 7.5 \, \text{m/s}^2 = 2.5 \, \text{m/s}^2 \]

5. **Relate the New Time Period to the Old Time Period**:
The time period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum.

For the new time period \( T' \) with effective gravity \( g' \):
\[ T' = 2\pi \sqrt{\frac{L}{g'}} = 2\pi \sqrt{\frac{L}{2.5}} = 2\pi \sqrt{\frac{L}{\frac{g}{4}}} = 2\pi \sqrt{\frac{4L}{g}} = 2 \times 2\pi \sqrt{\frac{L}{g}} = 2T \]

6. **Calculate the New Time Period**:
Since the initial time period \( T \) is \( 2 \, \text{s} \):
\[ T' = 2 \times 2 = 4 \, \text{s} \]

### Conclusion:
The new time period of the simple pendulum after the upward displacement is \( 4 \, \text{s} \).