A mass is taken from surface to a height h. The change in potential energy in this process is equal to the change in potential energy if it is now taken from that point to infinity. What is the value of h?

Options

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

Answer

To solve the problem, we need to find the height \( h \) such that the change in potential energy when a mass is taken from the surface of the Earth to height \( h \) is equal to the change in potential energy when it is taken from height \( h \) to infinity.

### Step-by-Step Solution:

1. **Understanding Potential Energy**:
The gravitational potential energy \( U \) at a distance \( r \) from the center of the Earth is given by:
\[ U = -\frac{GMm}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( m \) is the mass of the object.

2. **Potential Energy at the Surface**:
At the surface of the Earth (radius \( R \)):
\[ U_{\text{surface}} = -\frac{GMm}{R} \]

3. **Potential Energy at Height \( h \)**:
When the mass is raised to a height \( h \), the distance from the center of the Earth becomes \( R + h \):
\[ U_h = -\frac{GMm}{R + h} \]

4. **Change in Potential Energy from Surface to Height \( h \)**:
The change in potential energy when moving from the surface to height \( h \) is:
\[ \Delta U_1 = U_h - U_{\text{surface}} = -\frac{GMm}{R + h} + \frac{GMm}{R} \]
Simplifying this, we get:
\[ \Delta U_1 = GMm \left( \frac{1}{R} - \frac{1}{R + h} \right) \]

5. **Potential Energy at Infinity**:
At infinity, the potential energy is zero. The change in potential energy when moving from height \( h \) to infinity is:
\[ \Delta U_2 = U_{\infty} - U_h = 0 - \left(-\frac{GMm}{R + h}\right) = \frac{GMm}{R + h} \]

6. **Setting the Changes Equal**:
According to the problem, these two changes in potential energy are equal:
\[ \Delta U_1 = \Delta U_2 \]
Thus:
\[ GMm \left( \frac{1}{R} - \frac{1}{R + h} \right) = \frac{GMm}{R + h} \]

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

8. **Combining Terms**:
Rearranging gives:
\[ \frac{1}{R} = \frac{2}{R + h} \]

9. **Cross-Multiplying**:
Cross-multiplying leads to:
\[ R + h = 2R \]

10. **Solving for \( h \)**:
Rearranging gives:
\[ h = 2R - R = R \]

### Final Answer:
Thus, the value of \( h \) is:
\[ h = R \]