If the mass of Earth were increased by a factor of 2.00 , by how many times would the force between Earth and the Sun increase?

Answer:

If the mass of Earth were increased by a factor of 2.00, the force between Earth and the Sun would increase by a factor of 2.00 as well.

Explanation:

The gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The formula for gravitational force (F) is given by:

F = (G * m1 * m2) / r^2

Where:

F = gravitational force

G = gravitational constant (a constant value)

m1 = mass of the first object

m2 = mass of the second object

r = distance between the centers of the two objects

If the mass of Earth is increased by a factor of 2.00 (meaning it becomes 2 times its original mass), we can denote the original mass of Earth as "M" and the increased mass as "2M."

Now, let's calculate how the force between Earth and the Sun changes:

Original force (F1) = (G * M * Msun) / r^2

New force (F2) = (G * 2M * Msun) / r^2

Now, we can find the ratio of the new force to the original force:

F2 / F1 = [(G * 2M * Msun) / r^2] / [(G * M * Msun) / r^2]

Notice that the gravitational constant (G), distance (r), and the mass of the Sun (Msun) are the same in both equations and cancel out when calculating the ratio. Therefore, you're left with:

F2 / F1 = (2M) / M = 2

So, if the mass of Earth were increased by a factor of 2.00, the force between Earth and the Sun would increase by a factor of 2.00 as well.