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. Ue
D. Ur

Explanation

To find the gravitational force at a distance \( r \) from the center of the Earth when the gravitational potential energy \( U \) is given, we can follow these steps:

### Step 1: Write the expression for gravitational potential energy
The gravitational potential energy \( U \) of a body of mass \( m \) at a distance \( r \) from the center of the Earth (with mass \( M \)) is given by the formula:
\[ U = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant.

### Step 2: Write the expression for gravitational force
The gravitational force \( F \) between the Earth and the body is given by Newton's law of gravitation:
\[ F = \frac{G M m}{r^2} \]

### Step 3: Relate potential energy to force
We can relate the gravitational potential energy \( U \) to the gravitational force \( F \) using the relationship:
\[ F = -\frac{dU}{dr} \]
This means that the force can be derived from the potential energy by taking the derivative of \( U \) with respect to \( r \).

### Step 4: Differentiate the potential energy
Differentiating \( U \) with respect to \( r \):
\[ \frac{dU}{dr} = \frac{d}{dr}\left(-\frac{G M m}{r}\right) = \frac{G M m}{r^2} \]
Thus, the force \( F \) can be expressed as:
\[ F = -\frac{dU}{dr} = -\left(\frac{G M m}{r^2}\right) \]

### Step 5: Substitute the expression for \( U \)
From the expression for \( U \):
\[ U = -\frac{G M m}{r} \]
We can rearrange this to express \( G M m \) in terms of \( U \):
\[ G M m = -U r \]
Now substituting this into the force equation:
\[ F = -\left(-\frac{U r}{r^2}\right) = \frac{U}{r} \]

### Step 6: Conclusion
The magnitude of the gravitational force at a distance \( r \) from the center of the Earth is:
\[ F = -\frac{U}{r} \]
Since we are interested in the magnitude, we can drop the negative sign:
\[ F = \frac{-U}{r} \]

### Final Answer
Thus, the force at that point is given by:
\[ F = -\frac{U}{r} \]

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