Gm1m2 / d2 = kq1q2 / d2What is the value of Q to maintain the present orbit?

[Littlemin5]

Homework Statement

Supppose that electrical attraction, rather than gravity were responsible for holding the moon in orbit around the earth. If equal and opposite charges Q were placed on the Earth and the Moon, What should be the value of Q to maintain the present orbit?

Treat the Earth and Moon as point particles

Homework Equations

Mass of earth: 5.97x10²⁴kg
Mass of moon: 7.35x10²²kg
radius of orbit: 3.84x10⁸m

Fg=Gm₁m₂ / d²
Fe= kq₁q₂ / d²

The Attempt at a Solution

Well I know to start by finding the force of gravity, but from there i am unsure about how to do this problem. Could someone please help?

 

[davieddy]

If the orbit remains the same, then so does the acceleration.
Since the masses are the same, so are the forces: Fg=Fq

 

[thomate1]

I would like to clarify that the formula provided, Gm1m2 / d2 = kq1q2 / d2, is not accurate in this scenario. The formula for gravitational force, Fg = Gm1m2 / d2, is based on the masses of two objects and the distance between them. The formula for electrical force, Fe = kq1q2 / d2, is based on the charges of two objects and the distance between them. In this scenario, we are assuming that the moon is held in orbit by electrical attraction between the Earth and the Moon. Therefore, the formula for electrical force should be used.

To find the value of Q that would maintain the present orbit, we can set the electrical force equal to the gravitational force, since they are both acting on the moon and keeping it in orbit. We can rearrange the equation to solve for Q:

Fe = Fg

kq1q2 / d2 = Gm1m2 / d2

kq1q2 = Gm1m2

Q = √(Gm1m2 / k)

Substituting the given values, we get:

Q = √((6.67x10^-11 Nm^2/kg^2)(5.97x10^24 kg)(7.35x10^22 kg) / (9x10^9 Nm^2/C^2))

Q = 5.95x10^14 C

Therefore, to maintain the present orbit, the Earth and the Moon would need to have equal and opposite charges of approximately 5.95x10^14 Coulombs. However, it is important to note that this scenario is not possible in reality as the Earth and Moon do not have significant electric charge. This is just a hypothetical scenario to understand the concept of electrical attraction in relation to orbital motion.