A body of mass `m` is kept at a small height `h` above the ground. If the radius of the earth is R and its mass is M, the potential energy of the body and earth system (with `h=infty` being the reference position) is

Options

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

Explanation

To find the potential energy of the body and Earth system when the body is at a height \( h \) above the ground, we can follow these steps:

### Step 1: Understand the reference point for potential energy
The problem states that the reference position for potential energy is at \( h = \infty \). This means that the potential energy is considered to be zero when the body is infinitely far away from the Earth.

### Step 2: Determine the potential energy at the surface of the Earth
The gravitational potential energy \( U \) of a mass \( m \) at the surface of the Earth (radius \( R \)) is given by the formula:
\[ U = -\frac{GMm}{R} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

### Step 3: Calculate the potential energy at height \( h \)
When the body is raised to a height \( h \) above the surface of the Earth, the potential energy of the body-Earth system can be expressed as:
\[ U_h = -\frac{GMm}{R + h} \]

### Step 4: Use the approximation for small height \( h \)
Since \( h \) is small compared to \( R \) (the radius of the Earth), we can use the binomial approximation:
\[ \frac{1}{R + h} \approx \frac{1}{R} - \frac{h}{R^2} \]
Thus, the potential energy at height \( h \) can be approximated as:
\[ U_h \approx -\frac{GMm}{R} + \frac{GMmh}{R^2} \]

### Step 5: Combine the potential energy contributions
The total potential energy of the body and the Earth system at height \( h \) is:
\[ U_h = -\frac{GMm}{R} + mgh \]
where \( mgh \) is the potential energy due to the height \( h \).

### Final Expression
Therefore, the potential energy of the body and Earth system at height \( h \) is:
\[ U_h = -\frac{GMm}{R} + mgh \]