What would be the value of acceleration due to gravity at a point 5 km below the earth's surface ? `(R_(E) = 6400 km, g_(E) = 9.8 "ms"^(-2))`

Options

A. `9.6 " ms"^(-2)`
B. `9.79 " ms"^(-2)`
C. `9.89 " ms"^(-2)`
D. `10 " ms"^(-2)`

Explanation

To find the value of acceleration due to gravity at a point 5 km below the Earth's surface, we can use the formula for gravitational acceleration at a depth \( h \):

\[ g_d = g_e \left(1 - \frac{d}{R}\right) \]

where:
- \( g_d \) is the acceleration due to gravity at depth \( d \),
- \( g_e \) is the acceleration due to gravity at the Earth's surface (given as \( 9.8 \, \text{m/s}^2 \)),
- \( d \) is the depth below the Earth's surface (5 km in this case),
- \( R \) is the radius of the Earth (given as \( 6400 \, \text{km} \)).

### Step 1: Convert the depth and radius to the same units
Since the depth \( d \) is given in kilometers, we need to convert it to meters:
\[ d = 5 \, \text{km} = 5000 \, \text{m} \]
And the radius of the Earth:
\[ R = 6400 \, \text{km} = 6400000 \, \text{m} \]

### Step 2: Substitute the values into the formula
Now we can substitute \( g_e \), \( d \), and \( R \) into the formula:
\[ g_d = 9.8 \left(1 - \frac{5000}{6400000}\right) \]

### Step 3: Calculate the fraction
Calculate the fraction:
\[ \frac{5000}{6400000} = 0.00078125 \]

### Step 4: Substitute back into the equation
Now substitute this value back into the equation:
\[ g_d = 9.8 \left(1 - 0.00078125\right) \]
\[ g_d = 9.8 \left(0.99921875\right) \]

### Step 5: Calculate \( g_d \)
Now, calculate \( g_d \):
\[ g_d \approx 9.8 \times 0.99921875 \approx 9.79 \, \text{m/s}^2 \]

### Final Answer
Thus, the value of acceleration due to gravity at a point 5 km below the Earth's surface is approximately:
\[ g_d \approx 9.79 \, \text{m/s}^2 \]

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