Binding energy of moon and earth is :-

Options

A. `(GM_(e)M_(m))/r_(em)`
B. `(GM_(e)M_(m))/(2r_(em))`
C. `-(GM_(e)M_(m))/r_(em)`
D. `-(GM_(e)M_(m))/(2r_(em))`

Answer

To find the binding energy of the Moon and Earth, we will follow these steps:

### Step 1: Understand the Concept of Binding Energy
Binding energy is defined as the minimum energy required to remove an object from the gravitational influence of another body. In this case, we want to calculate the binding energy of the Moon in relation to the Earth.

### Step 2: Write the Formula for Gravitational Potential Energy
The gravitational potential energy (U) between two masses (M1 and M2) separated by a distance (r) is given by the formula:
\[ U = -\frac{G \cdot M_1 \cdot M_2}{r} \]
where \(G\) is the gravitational constant.

### Step 3: Identify the Masses and Distance
Let:
- \(M_E\) = mass of the Earth
- \(M_M\) = mass of the Moon
- \(R_{EM}\) = distance between the Earth and the Moon

### Step 4: Calculate the Gravitational Potential Energy
Using the formula from Step 2, the gravitational potential energy of the Earth-Moon system can be expressed as:
\[ U = -\frac{G \cdot M_E \cdot M_M}{R_{EM}} \]

### Step 5: Calculate the Total Energy of the System
The total mechanical energy (E) of the system is the sum of kinetic energy (K) and potential energy (U). However, for the binding energy, we focus on the potential energy:
\[ E = U = -\frac{G \cdot M_E \cdot M_M}{2R_{EM}} \]
This is because the kinetic energy at this distance can be considered as half of the potential energy in a bound system.

### Step 6: Define Binding Energy
The binding energy (B.E) is defined as the negative of the total energy:
\[ B.E = -E = \frac{G \cdot M_E \cdot M_M}{2R_{EM}} \]

### Final Result
Thus, the binding energy of the Moon and Earth is given by:
\[ B.E = \frac{G \cdot M_E \cdot M_M}{2R_{EM}} \]