Different gravitational and inertial mass

As has been discussed already, different objects would fall at different speeds. Why? Wikipedia led me to an interesting derivation.

Consider Newton's second law: $$F=m_{i}a$$ Now consider his law of universal gravitation: $$F=G\frac{m_{g1}{m}_{g2}}{r^2}$$ Now, because the force here is due to gravity, we set the equations equal: $$m_{i}a=G\frac{m_{g1}{m}_{g2}}{r^2}$$ But because $G\frac{m_{g2}}{r^2}=g$, $$m_{i}a=m_{g1}g$$ and, thus, the actual acceleration due to gravity (denoted here as $g_{actual}$) is $$g_{actual}=\frac{m_{g1}}{m_i}g$$ So, the greater the gravitational mass, or the lesser the inertial mass, the greater the acceleration.

Okay, that was boring, but I figured you might want some rationale behind the ramifications. You probably did this yourself, but someone reading this might not have, so I decided to stick it in here.

An object with larger gravitational mass would undergo a greater acceleration. Strange but true; it's an artifact of the equation. So heavy objects would fall a lot faster than lighter objects - if they had the same gravitational mass. I'll take you up on what you ended your question with:

Not really in astronomy or in how the world came to exist under these circumstances. (Though feel free to post that for future readers if you feel so inclined).

and post an answer related to astronomy, simply because I love it. I'm going to base it partly off this answer.

Planets orbit due to the force of gravity; this manifests itself as centripetal force. The relevant equation here is $$F_c=\frac{m_i{v}^2}{r}$$ We set that equal to $$F=G\frac{m_{g1}{m}_{g2}}{r^2}$$ and write this as $$\frac{m_i{v}^2}{r}=G\frac{m_{g1}{m}_{g2}}{r^2}$$ We can cancel out an $r$ and make it $$m_i{v}^2=G\frac{m_{g1}{m}_{g2}}{r}$$ We solve for velocity and find that $$v=\sqrt{G\frac{m_{1g}{m}_{2g}}{m_{i}r}}$$ Thus, the speed of planets will depend on their gravitational and inertial masses. It could also mean that Trojan asteroids might not be at stable positions (although I'm not positive about this; I could be wrong).

Given that angular velocity $\omega$ is equal to $\frac{v}{r}$, the spin of the Sun's initial protoplanetary disk could be impacted. It might be unstable, with different objects moving at different speeds. I would think that this would make the early solar system fairly chaotic; planetary formation would be a lot different. Perhaps Earth would not have formed - the original planetesimals might never have coalesced into planets.

I've obviously strayed far from the area where you wanted an answer, but I wanted to explore some of the interesting consequences when it comes to astronomy. Feel free to disregard this answer if you want to.

posted over 10 years ago

HDE 226868‭

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