In the earth-moon system, if T1 and T2 are period of revolution of earth and moon respectively about the centre of mass of the system then

a) T1>T2

b) T1=T2

c) T1

d) Insufficient data.

To tackle the question regarding the periods of revolution of the Earth and the Moon about the center of mass of the Earth-Moon system, we need to consider the dynamics of how these two celestial bodies interact with each other. The correct answer is a) T1 > T2, meaning the period of revolution of the Earth (T1) is greater than that of the Moon (T2).

Understanding the Earth-Moon System

The Earth and Moon both orbit around a common center of mass, known as the barycenter. This point is located inside the Earth, but not at its center, due to the significant difference in mass between the two bodies. The Earth is much more massive than the Moon, which influences their orbital characteristics.

Mass and Orbital Periods

The relationship between the masses of two bodies and their orbital periods can be understood through Kepler's laws of planetary motion and Newton's law of gravitation. According to these principles, the period of revolution of a body is related to its distance from the center of mass and its mass. The more massive body (in this case, the Earth) will have a longer orbital period compared to the less massive body (the Moon).

• Mass of Earth: Approximately 5.97 x 10²⁴ kg
• Mass of Moon: Approximately 7.35 x 10²² kg

Given that the Earth is about 81 times more massive than the Moon, it exerts a stronger gravitational pull, which means it will take longer to complete its orbit around the barycenter compared to the Moon.

Calculating the Periods

To further illustrate this, we can use the formula derived from Kepler's Third Law, which states that the square of the orbital period (T) is proportional to the cube of the semi-major axis (a) of the orbit:

T² ∝ a³

In our case, since the Earth is much more massive, its distance from the barycenter is smaller than that of the Moon. This means that the Moon, being further away, has a shorter orbital period. Thus, we can conclude:

• T1 (Earth's period) > T2 (Moon's period)

Conclusion

In summary, the Earth takes longer to complete its orbit around the barycenter of the Earth-Moon system than the Moon does. Therefore, the correct answer to the question is a) T1 > T2. This relationship highlights the fascinating dynamics of celestial mechanics and how mass influences orbital behavior.