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 a body of mass `m` kept at a small height `h` above the ground, with the radius of the Earth being `R` and its mass being `M`, 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 height \( h \) above the surface of the Earth can be expressed as:
\[ U = -\frac{G M m}{R + h} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the body,
- \( R \) is the radius of the Earth,
- \( h \) is the height above the surface of the Earth.

### Step 2: Simplify the formula for small height \( h \)
Since \( h \) is small compared to \( R \) (i.e., \( h \ll R \)), we can approximate \( R + h \) as \( R \):
\[ U \approx -\frac{G M m}{R + h} \approx -\frac{G M m}{R} \]

### Step 3: Use binomial expansion for further simplification
To be more precise, we can use the binomial expansion for \( (1 + \frac{h}{R})^{-1} \):
\[ \frac{1}{R + h} = \frac{1}{R(1 + \frac{h}{R})} \approx \frac{1}{R} \left(1 - \frac{h}{R}\right) \]
Thus, we can write:
\[ U \approx -\frac{G M m}{R} \left(1 - \frac{h}{R}\right) = -\frac{G M m}{R} + \frac{G M m h}{R^2} \]

### Step 4: Identify the gravitational acceleration at the surface
The gravitational acceleration \( g \) at the surface of the Earth is given by:
\[ g = \frac{G M}{R^2} \]
Substituting this into our equation, we have:
\[ U \approx -\frac{G M m}{R} + mgh \]
where \( gh = \frac{G M m h}{R^2} \).

### Step 5: Final expression for potential energy
Thus, the potential energy of the body-Earth system at height \( h \) is:
\[ U \approx -\frac{G M m}{R} + mgh \]

### Conclusion
The potential energy of the body and Earth system, taking \( h \) as small compared to \( R \), can be approximated as:
\[ U \approx -\frac{G M m}{R} + mgh \]