Gravitational potential energy of body of mass `m` at a height of `h` above the surface of earth (M = mass of earth , R = radius of earth) is

Options

A. `(GMm)/(h)`
B. `(GMm)/((R+h))`
C. `(-GM)/((R+h))`
D. `-(GMm)/((R+h))`

Explanation

To find the gravitational potential energy (U) of a body of mass \( m \) at a height \( h \) above the surface of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let \( M \) be the mass of the Earth.
- Let \( R \) be the radius of the Earth.
- Let \( h \) be the height above the Earth's surface.
- The distance from the center of the Earth to the body at height \( h \) is \( r = R + h \).

2. **Understand Gravitational Force**:
- The gravitational force \( F \) acting on the body of mass \( m \) at a distance \( r \) from the center of the Earth is given by:
\[ F = -\frac{G M m}{r^2} \]
where \( G \) is the universal gravitational constant.

3. **Work Done Against Gravitational Force**:
- The work done \( W \) to move the mass \( m \) from a point at infinity (where potential energy is defined as zero) to a height \( h \) above the surface of the Earth can be expressed as:
\[ W = \int_{\infty}^{R + h} F \, dr \]
- Since \( F \) is negative (attractive force), we can write:
\[ W = -\int_{\infty}^{R + h} \frac{G M m}{r^2} \, dr \]

4. **Evaluate the Integral**:
- The integral can be evaluated as follows:
\[ W = -\left[-\frac{G M m}{r}\right]_{\infty}^{R + h} = -\left(0 - \frac{G M m}{R + h}\right) = \frac{G M m}{R + h} \]

5. **Gravitational Potential Energy**:
- The gravitational potential energy \( U \) at height \( h \) is equal to the work done against the gravitational force:
\[ U = -W = -\frac{G M m}{R + h} \]

### Final Answer:
The gravitational potential energy of a body of mass \( m \) at a height \( h \) above the surface of the Earth is:
\[ U = -\frac{G M m}{R + h} \]