If radius of the earth is 6347 km, then what will be difference between acceleration of free falls and acceleration due to gravity near the earth's surface?

Options

A. A. 0.34
B. B. 0.034
C. C. 0.0034
D. D. 0.24

Explanation

To solve the problem of finding the difference between the acceleration of free fall and the acceleration due to gravity near the Earth's surface, we can follow these steps:

### Step 1: Understand the Definitions
- The acceleration due to gravity (g) at the Earth's surface is approximately \(9.8 \, \text{m/s}^2\).
- The acceleration of free fall (G_free_fall) can be expressed as:
\[ G_{\text{free fall}} = g - \omega^2 r \]
where \( \omega \) is the angular velocity of the Earth and \( r \) is the radius of the Earth.

### Step 2: Calculate the Angular Velocity (ω)
- The angular velocity \( \omega \) can be calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
where \( T \) is the time period of rotation of the Earth (one day = 24 hours).

### Step 3: Convert Time Period to Seconds
- Convert the time period \( T \) from hours to seconds:
\[ T = 24 \, \text{hours} \times 60 \, \text{minutes/hour} \times 60 \, \text{seconds/minute} = 86400 \, \text{seconds} \]

### Step 4: Substitute Values to Find ω
- Now substitute \( T \) into the formula for \( \omega \):
\[ \omega = \frac{2\pi}{86400} \, \text{rad/s} \]

### Step 5: Calculate ω²
- Calculate \( \omega^2 \):
\[ \omega^2 = \left(\frac{2\pi}{86400}\right)^2 \]

### Step 6: Substitute Radius of the Earth
- The radius of the Earth \( r \) is given as \( 6347 \, \text{km} = 6347000 \, \text{m} \).

### Step 7: Calculate the Difference
- Substitute \( \omega^2 \) and \( r \) into the equation for the difference:
\[ \text{Difference} = g - G_{\text{free fall}} = \omega^2 r \]

### Step 8: Final Calculation
- Calculate the difference:
\[ \text{Difference} = \left(\frac{2\pi}{86400}\right)^2 \times 6347000 \]
- After performing the calculations, you will find that the difference is approximately \( 0.03401 \, \text{m/s}^2 \).

### Conclusion
- The difference between the acceleration of free fall and the acceleration due to gravity near the Earth's surface is approximately \( 0.034 \, \text{m/s}^2 \).