if the mass of the earth is twice as much as at present then the time period of moon revolving at the same distance at present will be

To understand how the mass of the Earth affects the Moon's orbital period, we can refer to Kepler's Third Law of Planetary Motion. This law states that the square of the orbital period of a planet (or moon) is directly proportional to the cube of the semi-major axis of its orbit. However, when considering the gravitational influence of a celestial body, we can also use Newton's law of gravitation to derive the relationship between mass and orbital period.

Understanding Orbital Mechanics

The gravitational force between two bodies, like the Earth and the Moon, can be described by the formula:

F = G * (m1 * m2) / r²

Where:

• F is the gravitational force between the two masses.
• G is the gravitational constant.
• m1 is the mass of the Earth.
• m2 is the mass of the Moon.
• r is the distance between the centers of the two bodies.

Applying Kepler's Third Law

According to Kepler's Third Law, the relationship can be expressed as:

T² = (4π² / G * m) * r³

Here, T is the orbital period, m is the mass of the Earth, and r is the distance from the Earth to the Moon. If we double the mass of the Earth (let's call the new mass m'), the equation becomes:

T'² = (4π² / G * m') * r³

Calculating the New Orbital Period

Since we are doubling the mass of the Earth, we can substitute m' = 2m into the equation:

T'² = (4π² / G * 2m) * r³

This simplifies to:

T'² = (2 * 4π² / G * m) * r³

Now, we can relate this back to the original period T:

T² = (4π² / G * m) * r³

From this, we can see that:

T'² = 2 * T²

Finding the New Period

To find the new period T', we take the square root of both sides:

T' = √(2) * T

This means that if the mass of the Earth were to double, the new orbital period of the Moon would be approximately 1.414 times the current period. Since the Moon currently takes about 27.3 days to orbit the Earth, the new period would be:

T' ≈ 1.414 * 27.3 days ≈ 38.6 days

Final Thoughts

In summary, if the mass of the Earth were to double, the time period of the Moon revolving at the same distance would increase to about 38.6 days. This illustrates the fascinating interplay between mass and gravitational forces in celestial mechanics, demonstrating how changes in one variable can significantly impact the dynamics of an entire system.