What will be the acceleration due to gravity at a distance of 3200 km below the surface of the earth ? (Take `R_(e)=6400` km)

Explanation

To find the acceleration due to gravity at a distance of 3200 km below the surface of the Earth, we can use the formula for gravitational acceleration at a depth \( d \):

\[ g' = g \left(1 - \frac{d}{R}\right) \]

Where:
- \( g' \) is the acceleration due to gravity at depth \( d \).
- \( g \) is the acceleration due to gravity at the surface of the Earth (approximately \( 9.8 \, \text{m/s}^2 \)).
- \( d \) is the depth below the surface (in this case, \( 3200 \, \text{km} \)).
- \( R \) is the radius of the Earth (given as \( 6400 \, \text{km} \)).

### Step 1: Identify the values
- \( g = 9.8 \, \text{m/s}^2 \)
- \( d = 3200 \, \text{km} \)
- \( R = 6400 \, \text{km} \)

### Step 2: Substitute the values into the formula
First, we need to convert the depth and radius into the same units. Since \( g \) is in \( \text{m/s}^2 \), we should convert kilometers to meters:
- \( d = 3200 \, \text{km} = 3200 \times 1000 = 3,200,000 \, \text{m} \)
- \( R = 6400 \, \text{km} = 6400 \times 1000 = 6,400,000 \, \text{m} \)

Now we can substitute these values into the formula:

\[ g' = 9.8 \left(1 - \frac{3,200,000}{6,400,000}\right) \]

### Step 3: Calculate the fraction
Calculate \( \frac{d}{R} \):

\[ \frac{d}{R} = \frac{3,200,000}{6,400,000} = \frac{1}{2} \]

### Step 4: Substitute back into the equation
Now substitute this back into the equation for \( g' \):

\[ g' = 9.8 \left(1 - \frac{1}{2}\right) = 9.8 \left(\frac{1}{2}\right) \]

### Step 5: Calculate \( g' \)
Now calculate \( g' \):

\[ g' = 9.8 \times 0.5 = 4.9 \, \text{m/s}^2 \]

### Final Answer
The acceleration due to gravity at a distance of 3200 km below the surface of the Earth is \( 4.9 \, \text{m/s}^2 \).

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