The magnitude of 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)/(4r)`
C. `(u)/(2r)`
D. `(4r)/(u)`

Explanation

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

### Step-by-Step Solution:

1. **Understand the Gravitational Potential Energy (U)**:
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:
\[ U = -\frac{GMm}{r} \]
Here, \(G\) is the gravitational constant.

2. **Relate U to the Given Value**:
According to the problem, the magnitude of gravitational potential energy at a distance \(r\) is given as \(u\). Thus, we can write:
\[ u = -\frac{GMm}{r} \]

3. **Find the Gravitational Acceleration (g) at Distance \(2r\)**:
The weight \(W\) of the body is given by:
\[ W = mg \]
To find \(g\) at a distance \(2r\), we use the formula for gravitational acceleration:
\[ g = \frac{GM}{r^2} \]
At a distance \(2r\), the gravitational acceleration becomes:
\[ g_{2r} = \frac{GM}{(2r)^2} = \frac{GM}{4r^2} \]

4. **Substitute g into the Weight Formula**:
Now, substituting \(g_{2r}\) into the weight formula:
\[ W = m \cdot g_{2r} = m \cdot \frac{GM}{4r^2} \]

5. **Relate GM/r to u**:
From the earlier step, we know:
\[ \frac{GM}{r} = -u \quad \text{(from the potential energy formula)} \]
Therefore, we can express \(GM\) as:
\[ GM = -\frac{u \cdot r}{m} \]

6. **Substitute GM into the Weight Equation**:
Now substituting \(GM\) into the weight equation:
\[ W = m \cdot \frac{-\frac{u \cdot r}{m}}{4r^2} \]
Simplifying this gives:
\[ W = -\frac{u}{4r} \]

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