Different gravitational and inertial mass
This is going to take some imagination and assumptions (and knowledge about physics).
Imagine gravitational mass and inertial mass would be different. In classical mechanics it seems a coincidence that they are the same in our universe.
Gravitational mass is how much objects attract other things and are attracted by gravity (passive and active gravitational mass is still the same, $m_{g1} = m_{g2}$). Newton's law of universal gravitation: $\vec{F} = G \frac{m_{g1} m_{g2}}{r^2}\hat{r}$
Inertial mass is how much objects resist change in velocity. This is the mass from Newton's second law, $\vec{F} = m_i \vec{a}$.
Furthermore, forget anything you might know about subatomic physics or non-classical mechanics. Each atom has a gravitational mass and an inertial mass (isotopes have little effect).
Assume that somehow the universe has evolved somewhat similarly with a planet like Earth existing. Both masses are still positive.
Examples:
- Objects would not fall at the same speed even in vacuum. A half-iron, half-carbon item would rotate while falling to have the fast-falling part point down (or it might break).
- If iron would have low inertial mass and high gravitational mass, then an iron vehicle would stay on the road more firmly but would be easy to turn or accelerate.
- If copper would have high inertial mass and low gravitational mass, then a cannonball made up copper could travel spectacular distances and still have a big impact. EDIT: that is given a fixed initial velocity; more realistically it would have a fixed initial energy, in which case the trajectory is the same but the impact is higher.
I am interested in what happens on Earth at technology levels up to around 1900 (so no relativity). 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). Specifically, the scenario is for a computer RPG, but no need to focus on that.
This post was sourced from https://worldbuilding.stackexchange.com/q/3396. It is licensed under CC BY-SA 3.0.
1 answer
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: $$F=m_ia$$ 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: $$m_ia=G\frac{m_{g1}m_{g2}}{r^2}$$ But because $G\frac{m_{g2}}{r^2}=g$, $$m_ia=m_{g1}g$$ 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 $$F_c=\frac{m_iv^2}{r}$$ We set that equal to $$F=G\frac{m_{g1}m_{g2}}{r^2}$$ and write this as $$\frac{m_iv^2}{r}=G\frac{m_{g1}m_{g2}}{r^2}$$ We can cancel out an $r$ and make it $$m_iv^2=G\frac{m_{g1}m_{g2}}{r}$$ We solve for velocity and find that $$v=\sqrt{G\frac{m_{1g}m_{2g}}{m_ir}}$$ 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.
0 comment threads
0 comment threads