Electron bound to proton by gravity

[Juli]

Homework Statement

Suppose an electron were bound to a proton not by the electrical force, but by gravity. What would be the radius and energy of Bohr's first orbit?

Relevant Equations

$F_G = \frac{G\cdot m_1 m_2}{r^2}$
$F_C = m_e r \omega^2$

Hello everyone,
I have the problem above. I chose to put $F_G = F_Z$ to solve it and end up with a radius $r = 1.04\cdot 10^{-7}$m.
Solutions on the internet choose to put the gravitational force equal to the centrifugal force and obviously end up with a completely different solution. I can kind of understand both ways, but for me my way is the solution to the above statement, and to put $F_G$ equal to the coulomb force would just show how big the radius has to be to equal the Coulomb force. Which of course is valid because this is how the electron is bound to the proton. But in the problem we think about, how it is, when the Coulomb force is not there.
Which way would you think is correct to solve what is asked?

 

[Meow12]

Why do you think the gravitational force between the proton and electron has to equal the electrical force?

In a normal hydrogen atom, the sum  of the gravitational and electrical forces between the proton and electron provides the centripetal force on the electron. You can neglect the gravitational force as it is  much  weaker than the electric force.

But if there is no electric force, the centripetal force on the electron has to be provided entirely by the gravitational force.

Does that make sense?

 

[Juli]

Hello, thank you for your answer. I was just about to delete this post, since I got the right solution by using my method. I was wondering anyway, that if the Coulomb force was set equal to the centrifugal force and the gravitational force was set equal to the centrifugal force, that the Coulomb and the gravitational force had to be equal. And that is what I got now. I think I made som e major mistakes in the way I calculated my first solutions.

 

[Steve4Physics]

I was just about to delete this post, since I got the right solution by using my method.

What answers (radius and energy) did you get? Hopefully the radius was very (and I mean very!) large.

I was wondering anyway, that if the Coulomb force was set equal to the centrifugal force and the gravitational force was set equal to the centrifugal force, that the Coulomb and the gravitational force had to be equal. And that is what I got now. I think I made som e major mistakes in the way I calculated my first solutions.

You mean centripetal, not centrifugal.

You shouldn't set the Coulomb and gravitational forces equal here; it's wrong. The 1st Bohr orbit is determined by ensuring that the electron's angular momentum is the required value.