The binding energy of an object of mass `m` placed on the surface of the earth r is (R = radius of earth, g = acceleration due to gravity)

Options

A. `mg R//2`
B. mg R
C. `(mgR)/(4)`
D. `(mgR)/(8)`

Explanation

To find the binding energy of an object of mass \( m \) placed on the surface of the Earth, we need to understand the concept of gravitational potential energy and how it relates to binding energy.

### Step-by-Step Solution:

1. **Understanding Binding Energy**:
The binding energy of an object is defined as the energy required to remove it from the gravitational influence of the Earth. In this case, we want to calculate how much energy is needed to move the object from the surface of the Earth to a point far away where the gravitational influence is negligible.

2. **Gravitational Potential Energy**:
The gravitational potential energy \( U \) of an object of mass \( m \) at a distance \( r \) from the center of the Earth is given by the formula:
\[ U = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

3. **At the Surface of the Earth**:
At the surface of the Earth, the distance \( r \) is equal to the radius of the Earth \( R \). Therefore, the potential energy at the surface becomes:
\[ U = -\frac{G M m}{R} \]

4. **Energy Required to Escape**:
To escape the gravitational field, the object must gain enough energy to reach a point where the potential energy is zero (far away from Earth). Thus, the binding energy \( BE \) is equal to the negative of the potential energy at the surface:
\[ BE = -U = \frac{G M m}{R} \]

5. **Relating to Acceleration Due to Gravity**:
We know that the acceleration due to gravity \( g \) at the surface of the Earth is given by:
\[ g = \frac{G M}{R^2} \]
Rearranging this gives:
\[ G M = g R^2 \]

6. **Substituting into Binding Energy**:
Now, substituting \( G M \) into the binding energy equation:
\[ BE = \frac{g R^2 m}{R} = g R m \]

### Final Answer:
Thus, the binding energy of an object of mass \( m \) placed on the surface of the Earth is:
\[ BE = g R m \]