The gravitational potential energy is maximum at :

Options

A. infinity
B. the earth's surface
C. The centre of the earth
D. Twice the radius of the earth

Answer

To solve the question regarding where the gravitational potential energy is maximum, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Gravitational Potential Energy (U)**:
The gravitational potential energy (U) between two masses (M and m) is given by the formula:
\[ U = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of one object, \( m \) is the mass of the other object, and \( r \) is the distance between the centers of the two masses.

2. **Analyzing the Formula**:
The formula shows that gravitational potential energy is negative due to the negative sign in front of the fraction. This indicates that the potential energy is always less than zero when the masses are separated by a finite distance.

3. **Finding the Maximum Value**:
The maximum value of U occurs when it approaches zero. Since U is negative for all finite distances (r > 0), the only way for U to be maximum (i.e., least negative) is when \( r \) approaches infinity. As \( r \) increases, the value of \( U \) approaches zero.

4. **Conclusion**:
Therefore, the gravitational potential energy is maximum (or least negative) when the distance \( r \) between the two masses is infinite. Thus, the answer to the question is:
\[ \text{The gravitational potential energy is maximum at } r = \infty. \]

### Final Answer:
The gravitational potential energy is maximum at infinity.

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