A simple pendulum has time period `T_1`. The point of suspension is now moved upward according to the relation `y = K t^2, (K = 1 m//s^2)` where (y) is the vertical displacement. The time period now becomes `T_2`. The ratio of `(T_1^2)/(T_2^2)` is `(g = 10 m//s^2)`.

Options

A. `5//6`
B. `6//5`
C. 1
D. `4//5`

Explanation

To solve the problem, we need to find the ratio of the squares of the time periods \( T_1^2 \) and \( T_2^2 \) of a simple pendulum when the point of suspension is moved upward according to the relation \( y = K t^2 \), where \( K = 1 \, \text{m/s}^2 \).

### Step-by-Step Solution:

1. **Understanding the Time Period of a Simple Pendulum:**
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 and \( g \) is the acceleration due to gravity.

2. **Identifying Initial Conditions:**
Initially, the time period is \( T_1 \) with the acceleration due to gravity \( g_1 = 10 \, \text{m/s}^2 \). Thus, we can express \( T_1 \) as:
\[ T_1 = 2\pi \sqrt{\frac{L}{10}} \]

3. **Analyzing the Movement of the Suspension Point:**
The point of suspension is moved upward according to the relation \( y = K t^2 \). Given \( K = 1 \, \text{m/s}^2 \), the upward acceleration \( a \) of the suspension point is:
\[ a = \frac{d^2y}{dt^2} = 2 \, \text{m/s}^2 \]

4. **Calculating the Effective Gravity:**
When the point of suspension accelerates upward, the effective acceleration due to gravity \( g_2 \) experienced by the pendulum becomes:
\[ g_2 = g + a = 10 + 2 = 12 \, \text{m/s}^2 \]

5. **Finding the New Time Period:**
The new time period \( T_2 \) can be expressed as:
\[ T_2 = 2\pi \sqrt{\frac{L}{g_2}} = 2\pi \sqrt{\frac{L}{12}} \]

6. **Calculating the Ratio of Time Periods:**
To find the ratio of the squares of the time periods, we calculate:
\[ \frac{T_1^2}{T_2^2} = \frac{\left(2\pi \sqrt{\frac{L}{10}}\right)^2}{\left(2\pi \sqrt{\frac{L}{12}}\right)^2} = \frac{\frac{L}{10}}{\frac{L}{12}} = \frac{12}{10} = \frac{6}{5} \]

Thus, the final answer is:
\[ \frac{T_1^2}{T_2^2} = \frac{6}{5} \]