A simple pendulum is taken to 64 km above the earth's surface. Its new time period will

Options

A. Increases by 1%
B. Decrease by 1%
C. Increases by 2%
D. Decrease by 2%

Answer

To solve the problem of finding the new time period of a simple pendulum when taken 64 km above the Earth's surface, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Time Period Formula**:
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 at the location of the pendulum.

2. **Determine the Effective Gravity at the New Height**:
When the pendulum is taken to a height \(h = 64 \text{ km}\) above the Earth's surface, the effective gravitational acceleration \(g'\) can be calculated using the formula:
\[ g' = \frac{g}{(1 + \frac{h}{R})^2} \]
where \(R\) is the radius of the Earth (approximately \(6400 \text{ km}\)) and \(g\) is the acceleration due to gravity at the Earth's surface (approximately \(9.81 \text{ m/s}^2\)).

3. **Calculate the Height Ratio**:
Convert the height into the same units as the radius of the Earth:
\[ h = 64 \text{ km} = 64000 \text{ m} \]
\[ R = 6400 \text{ km} = 6400000 \text{ m} \]
Now calculate the ratio:
\[ \frac{h}{R} = \frac{64000}{6400000} = \frac{1}{100} \]

4. **Substitute into the Gravity Formula**:
Substitute this ratio into the formula for \(g'\):
\[ g' = \frac{g}{(1 + \frac{1}{100})^2} = \frac{g}{(1.01)^2} = \frac{g}{1.0201} \]

5. **Calculate the New Time Period**:
The new time period \(T'\) at the height can be expressed as:
\[ T' = 2\pi \sqrt{\frac{L}{g'}} \]
Substituting \(g'\):
\[ T' = 2\pi \sqrt{\frac{L}{\frac{g}{1.0201}}} = 2\pi \sqrt{\frac{L \cdot 1.0201}{g}} = 2\pi \sqrt{\frac{L}{g}} \cdot \sqrt{1.0201} \]
Therefore:
\[ T' = T \cdot \sqrt{1.0201} \]

6. **Calculate the Percentage Change in Time Period**:
The percentage change in the time period can be calculated as:
\[ \text{Percentage Change} = \left(\frac{T' - T}{T}\right) \times 100\% \]
Substituting \(T'\):
\[ \text{Percentage Change} = \left(\sqrt{1.0201} - 1\right) \times 100\% \]
Approximating \(\sqrt{1.0201} \approx 1.01\):
\[ \text{Percentage Change} \approx (1.01 - 1) \times 100\% = 1\% \]

### Final Result:
The new time period will increase by approximately **1%**.