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

Options

A. increase by `1%`
B. decrease by `1%`
C. increase by `2%`
D. decrease `2%`

Answer

To find the new time period of a simple pendulum taken to a height of 64 km above the Earth's surface, we can follow these steps:

### Step 1: Understand the formula for 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.

### Step 2: Determine the value of \( g \) at the height of 64 km
At a height \( h \) above the Earth's surface, the acceleration due to gravity \( g_h \) can be calculated using the formula:
\[ g_h = g \left( \frac{R}{R + h} \right)^2 \]
where:
- \( g \) is the acceleration due to gravity at the Earth's surface (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( R \) is the radius of the Earth (approximately \( 6400 \, \text{km} \)),
- \( h \) is the height above the Earth's surface (64 km).

### Step 3: Substitute the values into the formula
First, convert the height into meters:
\[ h = 64 \, \text{km} = 64000 \, \text{m} \]
Now, calculate \( g_h \):
\[ g_h = g \left( \frac{R}{R + h} \right)^2 = 9.81 \left( \frac{6400}{6400 + 64} \right)^2 \]
Calculate \( R + h \):
\[ R + h = 6400 + 64 = 6464 \, \text{km} = 6464000 \, \text{m} \]
Now substitute:
\[ g_h = 9.81 \left( \frac{6400}{6464} \right)^2 \]

### Step 4: Calculate the ratio
Calculate \( \frac{6400}{6464} \):
\[ \frac{6400}{6464} \approx 0.987 \]
Now square this value:
\[ \left( 0.987 \right)^2 \approx 0.974 \]
Now calculate \( g_h \):
\[ g_h \approx 9.81 \times 0.974 \approx 9.57 \, \text{m/s}^2 \]

### Step 5: Calculate the new time period \( T_h \)
Using the new value of \( g_h \):
\[ T_h = 2\pi \sqrt{\frac{L}{g_h}} \]
The ratio of the new time period to the original time period can be expressed as:
\[ \frac{T_h}{T} = \sqrt{\frac{g}{g_h}} \]

### Step 6: Calculate the percentage change in time period
The percentage change in time period is given by:
\[ \frac{T_h - T}{T} \times 100 = \left( \sqrt{\frac{g}{g_h}} - 1 \right) \times 100 \]
Substituting the values:
\[ \frac{T_h - T}{T} \times 100 \approx \left( \sqrt{\frac{9.81}{9.57}} - 1 \right) \times 100 \]
Calculating \( \sqrt{\frac{9.81}{9.57}} \):
\[ \sqrt{\frac{9.81}{9.57}} \approx 1.012 \]
Now calculate the percentage change:
\[ \approx (1.012 - 1) \times 100 \approx 1.2\% \]

### Conclusion
The new time period will increase by approximately 1.2%.