A simple pendulum has a time period $T_1$, when it is on earth's surface and $T_2$, when taken to a height R above the earth's surface. Where R is the radius of the earth. The value of $T_2/T_1$ will be :

Question

Accepted Answer

To determine the ratio $ T_2/T_1 $ for a simple pendulum when taken to a height equal to the radius of the Earth, we need to consider how the gravitational acceleration affects the time period of a pendulum.

### Explanation for Option 3:

1. **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. **Gravitational Acceleration at Different Heights:**
   On the surface of the Earth, the gravitational acceleration is $ g $. At a height $ R $ (where $ R $ is the radius of the Earth), the gravitational acceleration $ g' $ is given by:
   $$
   g' = \frac{g}{(1 + \frac{R}{R})^2} = \frac{g}{4}
   $$
   This is because the gravitational force decreases with the square of the distance from the center of the Earth.

3. **Time Period at Height $ R $:**
   At height $ R $, the time period $ T_2 $ becomes:
   $$
   T_2 = 2\pi \sqrt{\frac{L}{g'}}
   $$
   Substituting $ g' = \frac{g}{4} $:
   $$
   T_2 = 2\pi \sqrt{\frac{L}{\frac{g}{4}}} = 2\pi \sqrt{\frac{4L}{g}} = 2 \times 2\pi \sqrt{\frac{L}{g}} = 2T_1
   $$

4. **Ratio $ T_2/T_1 $:**
   Therefore, the ratio of the time periods is:
   $$
   \frac{T_2}{T_1} = \frac{2T_1}{T_1} = 2
   $$

Thus, the correct answer is Option 3, which states that $ T_2/T_1 = 2 $.

### Explanation for Why Other Options are Incorrect:

- **Option 1 (1):** This would imply that the time period does not change with height, which is incorrect because the gravitational acceleration decreases with height, affecting the time period.

- **Option 2 ($\sqrt{2}$):** This suggests a different relationship between $ T_2 $ and $ T_1 $ that does not align with the derived formula for gravitational acceleration at height $ R $.

- **Option 4 (4):** This would imply a much larger increase in the time period than what is calculated. The correct calculation shows that the time period doubles, not quadruples.

In conclusion, the correct answer is Option 3, as the time period of the pendulum doubles when taken to a height equal to the Earth's radius.