At what depth below the surface does the acceleration due to gravity becomes 70% of its value in the surface of earth ?

Answer

To find the depth below the surface of the Earth at which the acceleration due to gravity becomes 70% of its value at the surface, we can follow these steps:

### Step 1: Understand the relationship between gravity at depth and at the surface
The acceleration due to gravity at a depth \( D \) below the surface of the Earth is given by the formula:
\[ g_D = g \left(1 - \frac{D}{R}\right) \]
where:
- \( g_D \) is the acceleration due to gravity at depth \( D \),
- \( g \) is the acceleration due to gravity at the surface,
- \( R \) is the radius of the Earth.

### Step 2: Set up the equation for 70% of surface gravity
We need to find the depth \( D \) where \( g_D \) is 70% of \( g \):
\[ g_D = 0.7g \]
Substituting this into the formula gives:
\[ 0.7g = g \left(1 - \frac{D}{R}\right) \]

### Step 3: Simplify the equation
We can cancel \( g \) from both sides (assuming \( g \neq 0 \)):
\[ 0.7 = 1 - \frac{D}{R} \]

### Step 4: Solve for \( D \)
Rearranging the equation gives:
\[ \frac{D}{R} = 1 - 0.7 = 0.3 \]
Now, multiplying both sides by \( R \):
\[ D = 0.3R \]

### Step 5: Substitute the radius of the Earth
The average radius of the Earth \( R \) is approximately 6400 km. Thus:
\[ D = 0.3 \times 6400 \text{ km} \]

### Step 6: Calculate the depth
Calculating this gives:
\[ D = 1920 \text{ km} \]

### Final Answer
The depth below the surface at which the acceleration due to gravity becomes 70% of its value at the surface of the Earth is **1920 km**.

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