Centre of mass of the earth-moon system lies

Options

A. on the surface of the earth
B. on the surface of the moon
C. with in the earth
D. at the midpoint of the line joinining their centres

Answer

To determine the center of mass of the Earth-Moon system, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Center of Mass**:
The center of mass (COM) of a system of particles is the point where the total mass of the system can be considered to be concentrated. For two bodies, the position of the center of mass can be calculated using the formula:
\[ R_{cm} = \frac{m_1 r_1 + m_2 r_2}{m_1 + m_2} \]
where \( m_1 \) and \( m_2 \) are the masses of the two bodies, and \( r_1 \) and \( r_2 \) are their respective positions.

2. **Assign Values**:
Let:
- \( m_1 = m_E \) (mass of the Earth)
- \( m_2 = m_M \) (mass of the Moon)
- Assume the position of the Earth \( r_1 = 0 \) (taking the center of the Earth as the origin)
- The distance between the Earth and the Moon is \( D \), so \( r_2 = D \).

3. **Plug Values into the Formula**:
Substitute the values into the center of mass formula:
\[ R_{cm} = \frac{m_E \cdot 0 + m_M \cdot D}{m_E + m_M} = \frac{m_M \cdot D}{m_E + m_M} \]

4. **Analyze the Masses**:
Since the mass of the Earth \( m_E \) is much greater than the mass of the Moon \( m_M \) (i.e., \( m_E \gg m_M \)), the term \( m_E + m_M \) can be approximated as \( m_E \).

5. **Simplify the Expression**:
Therefore, we can simplify the expression for the center of mass:
\[ R_{cm} \approx \frac{m_M \cdot D}{m_E} \]

6. **Interpret the Result**:
Since \( m_E \) is significantly larger than \( m_M \), the value of \( R_{cm} \) will be a very small fraction of \( D \). This means that the center of mass is very close to the center of the Earth.

7. **Conclusion**:
Thus, the center of mass of the Earth-Moon system lies within the Earth.

### Final Answer:
The center of mass of the Earth-Moon system lies within the Earth.

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