Question

If you were to triple the size of the Earth (R = 3R⊕) and double the mass of the Earth (M = 2M⊕), how much would it change the gravity on the Earth (Fg = XFg⊕)?

Answer 1

To determine how gravity would change if Earth's size and mass were altered, Newton's law of universal gravitation is used. If Earth's radius were tripled and its mass were doubled, the gravitational force on the surface would be (2/9) of the current Earth's gravity.

Step-by-step explanation:

The student is asking how gravity would change if Earth's size and mass were altered. Specifically, they want to know the effect on the gravitational force (Fg) if Earth's radius (R) is tripled and Earth's mass (M) is doubled. Using Newton's law of universal gravitation, which states that Fg is directly proportional to the product of the two masses (M1 and M2) and inversely proportional to the square of the distance between the centers of the two masses (R), we can calculate the change in gravity.

The gravitational force is given by Fg = G(M1M2/R²), where G is the gravitational constant. If we increase the Earth's radius to 3R₂⊕ and the mass to 2M₂⊕, we alter the gravitational force equation to Fg' = G(2M₂⊕)/(3R₂⊕)². Simplifying, we find that Fg' = G(2M₂⊕)/(9R₂⊕²) = (2/9)*G(M₂⊕/R₂⊕²), which means the new gravitational force is (2/9) times the original Earth's gravitational force.

Answer 2

Decreased by a factor of 4.5

Step-by-step explanation:

"We have Newton formula for attraction force between 2 objects with mass and a distance between them:

`F_G = G(M_1M_2)/(R^2)`

where
`G =6.67408 × 10^(-11) m^3/kgs^2` is the gravitational constant on Earth.
`M_1, M_2` are the masses of the object and Earth itself. and R distance between, or the Earth radius.

So when R is tripled and mass is doubled, we have the following ratio of the new gravity over the old ones:

`(F_G)/(f_g) = (G(M_1M_2)/(R^2))/(G(M_1m_2)/(r^2))`

`(F_G)/(f_g) = ((M_2)/(R^2))/((m_2)/(r^2))`

`(F_G)/(f_g) = (M_2)/(R^2)(r^2)/(m_2)`

`(F_G)/(f_g) = (M_2)/(m_2)((r)/(R))^2`

Since
`M_2 = 2m_2` and
`r = R/3`

`(F_G)/(f_g) = (2)/(3^2) = 2/9 = 1/4.5`

So gravity would have been decreased by a factor of 4.5