The value of g at depth h is two third the value that on the earth's surface. The value of h in terms of radius of earth R is

Options

A. `(2R)/(3)`
B. `(R )/(3)`
C. `(R )/(6)`
D. `(R )/(2)`

Answer

To find the value of depth \( h \) in terms of the radius of the Earth \( R \) when the acceleration due to gravity at depth \( h \) is two-thirds of the value at the Earth's surface, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The acceleration due to gravity at a depth \( h \) below the Earth's surface is given by the formula:
\[ g' = g \left(1 - \frac{h}{R}\right) \]
where:
- \( g' \) is the acceleration due to gravity at depth \( h \),
- \( g \) is the acceleration due to gravity at the Earth's surface,
- \( R \) is the radius of the Earth.

2. **Setting Up the Equation**:
According to the problem, the value of \( g' \) at depth \( h \) is two-thirds of the value of \( g \) at the surface:
\[ g' = \frac{2}{3} g \]
Substituting this into the formula gives:
\[ \frac{2}{3} g = g \left(1 - \frac{h}{R}\right) \]

3. **Cancelling \( g \)**:
Since \( g \) is common on both sides (assuming \( g \neq 0 \)), we can cancel it out:
\[ \frac{2}{3} = 1 - \frac{h}{R} \]

4. **Rearranging the Equation**:
Rearranging the equation to isolate \( \frac{h}{R} \):
\[ \frac{h}{R} = 1 - \frac{2}{3} \]

5. **Calculating \( \frac{h}{R} \)**:
Simplifying the right side:
\[ \frac{h}{R} = \frac{1}{3} \]

6. **Finding \( h \)**:
Multiplying both sides by \( R \) to find \( h \):
\[ h = \frac{R}{3} \]

### Final Answer:
Thus, the value of \( h \) in terms of the radius of the Earth \( R \) is:
\[ h = \frac{R}{3} \]