The gravitational potential energy at a body of mass `m` at a distance `r` from the centre of the earth is U. What is the weight of the body at this distance ?

Options

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

Explanation

To find the weight of a body of mass \( m \) at a distance \( r \) from the center of the Earth, we can use the relationship between gravitational potential energy and weight. Here’s the step-by-step solution:

### Step 1: Understand the relationship between gravitational potential energy and weight
The gravitational potential energy \( U \) of a body of mass \( m \) at a distance \( r \) from the center of the Earth can be expressed as:
\[ U = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth. However, for this problem, we will use the expression provided in the video transcript.

### Step 2: Use the expression for gravitational potential energy
According to the video transcript, the gravitational potential energy can also be expressed as:
\[ U = m g r \]
where \( g \) is the acceleration due to gravity at distance \( r \) from the center of the Earth.

### Step 3: Relate weight to gravitational potential energy
The weight \( W \) of the body is given by:
\[ W = m g \]
From the expression for gravitational potential energy, we can rearrange it to find \( g \):
\[ g = \frac{U}{m r} \]

### Step 4: Substitute \( g \) into the weight equation
Now, substituting \( g \) back into the weight equation:
\[ W = m g = m \left(\frac{U}{m r}\right) \]

### Step 5: Simplify the expression
This simplifies to:
\[ W = \frac{U}{r} \]

### Final Answer
Thus, the weight of the body at a distance \( r \) from the center of the Earth is:
\[ W = \frac{U}{r} \]