If the radius of the earth is 6400km, the depth at which the acceleration due to gravity will be 5% less than the value on the surface of the earth is

Options

A. 320 km
B. 2.0 km
C. 480 km
D. 500 km

Answer

To find the depth at which the acceleration due to gravity is 5% less than its value on the surface of the Earth, we can follow these steps:

### Step 1: Define the known values
- The radius of the Earth (R) = 6400 km
- The acceleration due to gravity on the surface (g) = 9.8 m/s² (approximately, but we can use a relative value for simplicity)
- The value of gravity at depth (g_d) = g - 5% of g = 0.95g

### Step 2: Set up the equation for gravity at depth
The formula for the acceleration due to gravity at a depth (d) below the surface is given by:
\[ g_d = g \left(1 - \frac{d}{R}\right) \]

### Step 3: Substitute the known values into the equation
We know that:
\[ g_d = 0.95g \]
So we can set up the equation:
\[ 0.95g = g \left(1 - \frac{d}{R}\right) \]

### Step 4: Simplify the equation
Dividing both sides by g (assuming g ≠ 0):
\[ 0.95 = 1 - \frac{d}{R} \]

### Step 5: Solve for d/R
Rearranging the equation gives:
\[ \frac{d}{R} = 1 - 0.95 \]
\[ \frac{d}{R} = 0.05 \]

### Step 6: Solve for d
Now, multiply both sides by R to find d:
\[ d = R \times 0.05 \]
Substituting R = 6400 km:
\[ d = 6400 \times 0.05 \]
\[ d = 320 \text{ km} \]

### Final Answer
The depth at which the acceleration due to gravity will be 5% less than the value on the surface of the Earth is **320 km**.

---