The gravitational potential energy of a body at a distance r from the center of the earth is U. The force at that point is :

Options

A. `U/r^(2)`
B. `U/r`
C. `Ur`
D. `Ur^(2)`

Answer

To solve the problem, we need to find the gravitational force acting on a body at a distance \( r \) from the center of the Earth, given that the gravitational potential energy at that distance is \( U \).

### Step-by-Step Solution:

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

2. **Relate Force to Potential Energy**:
The gravitational force \( F \) can be derived from the gravitational potential energy. The force is related to the potential energy by the negative gradient of potential energy:
\[ F = -\frac{dU}{dr} \]

3. **Differentiate the Potential Energy**:
To find the force, we differentiate \( U \) with respect to \( r \):
\[ F = -\frac{d}{dr}\left(-\frac{GMm}{r}\right) \]
This simplifies to:
\[ F = \frac{GMm}{r^2} \]

4. **Express Force in Terms of Potential Energy**:
We can express \( F \) in terms of \( U \). From the expression for \( U \):
\[ U = -\frac{GMm}{r} \]
Rearranging gives:
\[ GMm = -Ur \]
Substituting this back into the force equation gives:
\[ F = \frac{-Ur}{r^2} = -\frac{U}{r} \]

5. **Final Expression for Force**:
Thus, we find that the gravitational force \( F \) at a distance \( r \) from the center of the Earth is:
\[ F = -\frac{U}{r} \]

### Conclusion:
The force at that point is given by:
\[ F = -\frac{U}{r} \]