The gravitational potential energy of a body at a distance `r` from the centre of earth is `U`. Its weight at a distance `2r` from the centre of earth is

Options

A. `(U)/(r)`
B. `(U)/(2r)`
C. `(U)/(4r)`
D. `(U)/(sqrt2r)`

Explanation

To solve the problem, we need to determine the weight of a body at a distance of \(2r\) from the center of the Earth, given that its gravitational potential energy at a distance \(r\) from the center of the Earth is \(U\).

### Step-by-step Solution:

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

2. **Weight of the Body**:
The weight \(W\) of the body is given by:
\[ W = m \cdot g \]
where \(g\) is the acceleration due to gravity at that distance.

3. **Finding \(g\) at Distance \(2r\)**:
The acceleration due to gravity \(g\) at a distance \(r\) from the center of the Earth is:
\[ g = \frac{G M}{r^2} \]
At a distance \(2r\), the acceleration due to gravity becomes:
\[ g_{2r} = \frac{G M}{(2r)^2} = \frac{G M}{4r^2} \]

4. **Substituting \(g_{2r}\) into the Weight Formula**:
Now, substituting \(g_{2r}\) into the weight formula:
\[ W_{2r} = m \cdot g_{2r} = m \cdot \frac{G M}{4r^2} \]

5. **Relating Weight to Gravitational Potential Energy**:
We know from the potential energy formula that:
\[ U = -\frac{G M m}{r} \]
Rearranging this gives:
\[ G M = -\frac{U \cdot r}{m} \]
Now substituting \(G M\) into the weight formula:
\[ W_{2r} = m \cdot \frac{-\frac{U \cdot r}{m}}{4r^2} \]

6. **Simplifying the Expression**:
The mass \(m\) cancels out:
\[ W_{2r} = -\frac{U \cdot r}{4r^2} = -\frac{U}{4r} \]

7. **Final Result**:
Thus, the weight of the body at a distance \(2r\) from the center of the Earth is:
\[ W_{2r} = -\frac{U}{4r} \]