The change in the gravitational potential energy when a body of a mass `m` is raised to a height `nR` above the surface of the earth is (here `R` is the radius of the earth)

Options

A. `(mgR)/n`
B. `mmgR`
C. `(n/(n-1))mgR`
D. `(n/(n+1))mgR`

Explanation

To find the change in gravitational potential energy when a body of mass \( m \) is raised to a height \( nR \) above the surface of the Earth, we can follow these steps:

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

### Step 2: Calculate the gravitational potential energy at the surface of the Earth
At the surface of the Earth, the distance \( d \) is equal to the radius of the Earth \( R \). Therefore, the potential energy at the surface \( U_{\text{surface}} \) is:
\[ U_{\text{surface}} = -\frac{G M m}{R} \]

### Step 3: Calculate the gravitational potential energy at height \( nR \)
When the body is raised to a height \( nR \), the distance from the center of the Earth becomes \( R + nR = (1 + n)R \). Thus, the potential energy at this height \( U_{\text{height}} \) is:
\[ U_{\text{height}} = -\frac{G M m}{(1+n)R} \]

### Step 4: Find the change in gravitational potential energy
The change in gravitational potential energy \( \Delta U \) when the body is raised from the surface to height \( nR \) is given by:
\[ \Delta U = U_{\text{height}} - U_{\text{surface}} \]
Substituting the values we calculated:
\[ \Delta U = \left(-\frac{G M m}{(1+n)R}\right) - \left(-\frac{G M m}{R}\right) \]
This simplifies to:
\[ \Delta U = -\frac{G M m}{(1+n)R} + \frac{G M m}{R} \]
\[ \Delta U = G M m \left(\frac{1}{R} - \frac{1}{(1+n)R}\right) \]
\[ \Delta U = G M m \left(\frac{(1+n) - 1}{(1+n)R}\right) \]
\[ \Delta U = G M m \left(\frac{n}{(1+n)R}\right) \]

### Step 5: Express in terms of acceleration due to gravity \( g \)
We know that the gravitational acceleration \( g \) at the surface of the Earth is given by:
\[ g = \frac{G M}{R^2} \]
Thus, we can express \( G M \) as:
\[ G M = g R^2 \]
Substituting this back into our equation for \( \Delta U \):
\[ \Delta U = \frac{g R^2 m n}{(1+n)R} \]
\[ \Delta U = \frac{g R m n}{1+n} \]

### Final Answer
The change in gravitational potential energy when a body of mass \( m \) is raised to a height \( nR \) above the surface of the Earth is:
\[ \Delta U = \frac{g R m n}{1+n} \]