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

### Step 1: Understand the relationship between gravitational acceleration and height
The gravitational acceleration at a height \( h \) above the Earth's surface can be calculated using the formula:
\[ g' = g \left(1 - \frac{2h}{R}\right) \]
where:
- \( g' \) is the gravitational acceleration at height \( h \)
- \( g \) is the gravitational acceleration at the Earth's surface (approximately \( 9.81 \, \text{m/s}^2 \))
- \( R \) is the radius of the Earth (approximately \( 6400 \, \text{km} \))

### Step 2: Substitute the values
Given \( h = 64 \, \text{km} \) and \( R = 6400 \, \text{km} \):
\[ g' = g \left(1 - \frac{2 \times 64}{6400}\right) \]
Calculating the fraction:
\[ \frac{2 \times 64}{6400} = \frac{128}{6400} = \frac{1}{50} = 0.02 \]
Thus:
\[ g' = g \left(1 - 0.02\right) = g \times 0.98 \]

### Step 3: Find the change in gravitational acceleration
The change in gravitational acceleration can be expressed as:
\[ \frac{\Delta g}{g} = -\frac{2h}{R} = -0.02 \]

### Step 4: Use the formula for the time period of a simple pendulum
The time period \( T \) of a simple pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum.

### Step 5: Calculate the change in time period
To find the change in time period, we can use the relationship:
\[ \frac{\Delta T}{T} = \frac{1}{2} \left(\frac{\Delta L}{L} - \frac{\Delta g}{g}\right) \]
Since there is no change in the length of the pendulum (\( \Delta L = 0 \)), this simplifies to:
\[ \frac{\Delta T}{T} = -\frac{1}{2} \left(\frac{\Delta g}{g}\right) \]
Substituting \( \frac{\Delta g}{g} = -0.02 \):
\[ \frac{\Delta T}{T} = -\frac{1}{2} \times (-0.02) = \frac{0.02}{2} = 0.01 \]

### Step 6: Convert to percentage
To express this change as a percentage:
\[ \Delta T = 0.01 \times 100\% = 1\% \]
This indicates that the time period increases by 1%.

### Final Answer
The new time period of the pendulum will increase by 1%.

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