The work done in slowly lifting a body from earth's surface to a height R (radius of earth) is equal to two times the work done in lifting the same body from earth's surface to a height h. Here h is equal to

Options

A. `(R)/(4)`
B. `(R)/(3)`
C. `(R)/(6)`
D. `(R)/(2)`

Answer

To solve the problem, we need to find the height \( h \) such that the work done in lifting a body from the Earth's surface to a height \( R \) (the radius of the Earth) is equal to two times the work done in lifting the same body to a height \( h \).

### Step-by-Step Solution:

1. **Understanding Work Done in Lifting**:
The work done in lifting a mass \( m \) to a height \( h \) in a gravitational field is equal to the change in gravitational potential energy. The formula for gravitational potential energy at a height \( h \) is given by:
\[ U = -\frac{GMm}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the distance from the center of the Earth.

2. **Work Done to Height \( R \)**:
When lifting the body to a height \( R \):
\[ W_1 = U_{\text{final}} - U_{\text{initial}} = \left(-\frac{GMm}{R}\right) - \left(-\frac{GMm}{R}\right) = \frac{GMm}{R} - \frac{GMm}{2R} \]
Simplifying this gives:
\[ W_1 = \frac{GMm}{2R} \]

3. **Work Done to Height \( h \)**:
Now, for lifting the body to height \( h \):
\[ W_2 = U_{\text{final}} - U_{\text{initial}} = \left(-\frac{GMm}{R+h}\right) - \left(-\frac{GMm}{R}\right) = \frac{GMm}{R} - \frac{GMm}{R+h} \]
Simplifying this gives:
\[ W_2 = GMm \left(\frac{1}{R} - \frac{1}{R+h}\right) = GMm \left(\frac{(R+h) - R}{R(R+h)}\right) = \frac{GMmh}{R(R+h)} \]

4. **Setting Up the Equation**:
According to the problem, the work done to lift to height \( R \) is equal to twice the work done to lift to height \( h \):
\[ W_1 = 2W_2 \]
Substituting the expressions for \( W_1 \) and \( W_2 \):
\[ \frac{GMm}{2R} = 2 \left(\frac{GMmh}{R(R+h)}\right) \]

5. **Canceling Common Terms**:
We can cancel \( GMm \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{1}{2R} = \frac{2h}{R(R+h)} \]

6. **Cross-Multiplying**:
Cross-multiplying gives:
\[ R(R+h) = 4Rh \]
This simplifies to:
\[ R^2 + Rh = 4Rh \]
Rearranging gives:
\[ R^2 = 3Rh \]

7. **Solving for \( h \)**:
Dividing both sides by \( R \) (assuming \( R \neq 0 \)):
\[ h = \frac{R}{3} \]

### Final Answer:
Thus, the height \( h \) is:
\[ h = \frac{R}{3} \]