Find the depth at which g becomes 4% less than value at surface

Options

A. 0.2 R
B. 0.04 R
C. 0.3 R
D. 0.03 R

Answer

To find the depth at which the acceleration due to gravity (g) becomes 4% less than its value at the surface, we can follow these steps:

### Step 1: Define the surface value of g
At the surface of the Earth, we can assume the value of g to be 100% or simply 100 (as a reference point).

### Step 2: Calculate the reduced value of g
Since we want to find the depth where g is 4% less than its surface value, we can calculate:
\[ g_{depth} = g_{surface} - 0.04 \times g_{surface} = 100 - 4 = 96 \]

### Step 3: Use the formula for g at depth
The formula for the acceleration due to gravity at a depth \(d\) below the surface is given by:
\[ g_d = g_{surface} \left(1 - \frac{d}{R}\right) \]
where \(R\) is the radius of the Earth.

### Step 4: Set up the equation
Substituting the known values into the equation:
\[ 96 = 100 \left(1 - \frac{d}{R}\right) \]

### Step 5: Simplify the equation
Rearranging the equation gives:
\[ \frac{d}{R} = 1 - \frac{96}{100} = 0.04 \]

### Step 6: Solve for d
Now, we can express \(d\) in terms of \(R\):
\[ d = 0.04R \]

### Step 7: Express d in terms of R
To find the depth where g is 4% less than its surface value, we can write:
\[ d = \frac{4R}{100} = \frac{R}{25} \]

### Final Result
Thus, the depth \(d\) at which g becomes 4% less than its value at the surface is:
\[ d = \frac{R}{25} \]