# Barbells and gravity

A giant set of bar bells floating in space (like two identical sized planets connected by a long rod) would have a centre of mass midway between the two on the connecting rod. But surely it would have two centres of gravity, one at each end? If you were standing on one of the "bells" or planets, and threw a rock in the air, it wouldn't fly to the middle of the rod, surely? And if I'm correct, then say, a big wobbly jelly shaped planet would also have multiple points of gravity. We have to except a sphere on which the centre of mass and gravity are the same.

My interest in dark matter was brought about by a friend who explained that the observed mass was calculated with reference to the centres of galaxies - but in what sense can a galaxy have a centre if the above confusions come into play? Isn't a galaxy like a set of interconnected barbells? Is there really a "centre" for gravitational calcuations?

• Related: physics.stackexchange.com/q/50107 . In this case you can't talk about centres of gravity. Jul 8, 2013 at 9:49
• It would be courteous if the questionning answer you deleted were edited into this initial question, so that the additional answer I took the time to write would actually answer some question. Jul 8, 2013 at 15:00
• Tim, you seem to have lost contact with the posting account. You can have accounts merged, but if you continue to use cookie-based (unregistered) access this will probably happen again and again. Perhaps you should consider registering if you intend to use the site regularly. Jul 8, 2013 at 15:37
• @user26796 Tim, that is why your edit was not taken immediately, as it normally is for original author. The system did not recognize you as the author. Not easy to understand at first ... and that is not the end :-) Fair question, and thanks for explaining. I was wondering though I suspected the origin of the problem. Jul 8, 2013 at 16:39

If you are close to a 'bell' you will be pulled towards it, assuming it's mass is far greater than that of the bar and that the other 'bell' is not far far greater.

The centre of mass of a system of objects is separate from where the force of gravity pulls you towards.

For instance, the centre of mass of the Earth-moon system lies somewhere in space between the two bodies far above the Earth's surface about halfway between the surface and the centre of the Earth (see comments). However, inhabitants of Earth, like us experience gravity pulling them towards the centre of Earth, not towards the centre of mass of the Earth-moon system. One of the consequences of having it the other way round would be that regions directly beneath the moon would feel four times heavier than normal and those on the opposite side of the Earth would feel ~half as heavy.

Another example is the Earth-Sun system where the centre of mass is very close to the Sun. We don't fall off the Earth towards the Sun during the day.

To take this to an extreme, the centre of mass of the galaxy (changed from universe for simplicity) is certainly not the direction we feel gravity pulling us.

The centre of mass is simply the average position of all the mass in the system. The strength of gravity follows an inverse square law (let's stick to Newton for simplicity) so bodies you're closest to (such as the massive 'bell') will be the dominating source.

• Earth-moon mass ratio is 1.23% (wikipedia). Moon distance is about 400,000km. That places the common center of mass at about 3,900 km from the center of mass of earth which has a radius of about 6,300 km. Hence it is underground, not above the surface. Also, I am not sure there is such a thing as the center of mass of the universe, but others will be more qualified than I to comment on this. Jul 8, 2013 at 9:23
• Fair enough I should have looked it up but the point is still valid. I could change it to the Earth-Sun or Earth-Mars or Jupiter-Saturn centre of mass to make the same point which is that the gravitational force does not act towards the centre of mass of the system.
– ejrb
Jul 8, 2013 at 9:30
• "We don't fall off the Earth towards the Sun during the day". But we do fall towards the sun, day and night. It is not off the Earth because Earth is falling with us. Actually, Earth gravity helps a tiny bit to keep us on board, though we would tend to fall slightly faster during the day (and slower at night). That is what causes tides (the solar part of it). I have not done the actual computation but I would suspect centrifugal force from Earth rotation to have more expelling effect than tidal force from the sun. Jul 8, 2013 at 12:28
• I'm trying to avoid overcomplicating the answer. Obviously the centre of mass the us-Earth-Moon system is constantly falling towards, and is orbiting around the centre of mass of the us-Earth-Moon-Sun system but that does not mean we do not keep our feet firmly on Earth at all times. The proximity of Earth is by far the most dominating factor in our local gravitational field, not the location of the centre of mass of as many arbitrary bodies as we want to include.
– ejrb
Jul 8, 2013 at 12:54

The center of mass can be used as representing a whole system only when the interactions considered are the same (up to approximations) for all points of the system, or average linearly with respect to distance, like the computation of the center of mass itself.

This is not so for gravitation, because of the inverse square law of attraction. The center of gravitation of earth is 6,300 km from us, and it is 400,000 km for the moon. So the moon is 63 times further away, and its gravitational effect on objects on the surface of Earth is then 4,000 times weaker for the same amount of mass. It does have an influence, seen as tides. But we are in no way atracted towards the center of mass of the moon-earth system.

Note that I talked of center of gravitation of earth, because, if I remember correctly, a homogenous sphere, or spherical shell, attracts towards its center. You were quite right on this.

To answer your next question (which you should delete and edit into the first): (sorry if this is a bit messy, the question, from a very new user, has been evolving a bit too much)

The center of gravity could be defined as the point with respect to which the torque of external gravitational forces applied to a rigid system averages to 0 (I do not know whether that is actually done). It is the same as the center of mass if the system is in a uniform gravity field. Actually the expression was created for objects in a uniform gravitaty field. It allows you to compute trajectories, independently ot rotations. It is often loosely used in place of center of mass.

Except for spheres, it is not the point of gravitational attraction of a system. As we have just shown, such a point does not exists in general. If you consider Earth-Moon as a single objet, you are attracted to the Moon when close to it, and to the Earth when close to it. In general the gravity vectors do not converge through a unique point.

I think you should just forget center of gravity in non uniform gravity fields. Regarding dark matter, I believe astrophisicists look at observable mass, and analyze motion with respect to the observed mass. Given that observed motion does not follow the usual laws, they infere that it could be because there is some invisible mass that contributes to the gravity field. It could be that the invisible mass does not have the same center of mass (visible and invisible mass attract each other, but they could be rotating around a common center of mass). But it also has an effect on the speed of rotation of visible bodies. So, if a proper distribution of such an invisible mass can explain observed motions, that is a simple explanation for it and thus it has a good chance of being correct. I think that current Big Bang theories consider that dark matter collapsed before visible matter did (but do not trust me).

Added later (question evolving): As for the galactic center, even ignoring the fact that there is usually a very massive black hole there, it is the center of mass. When two free bodies are satellised around each other (Earth and Moon for example) there revolve around their common center of mass. But this does not contradict the fact that a third body can get more attraction from one than from the other if it is closer to the former. If big enough, it can also perturb the system. Indeed the planet Neptune was discovered (actually predicted) by Leverrier because of the perturbations caused to the orbit of Uranus.

Rigid bodies can also revolve around their center of mass, in a uniform gravity field, and your barbell might well do that. The galaxy itself can be seen as a body revolving around it center of mass, though this is a bit more complex since it is not a rigid body (I am reaching the limits of my competence).

I guess (literally) a simple way to analyse the galaxy is to approximate it as a unique body with a given shape (ellipsoid, disk, ...) and a given density distribution. Then you can analyse gravity within that shape (and outside too if you are a fan of intergalactic travel). This will give you good approximations, as long as you stay away from large bodies that locally strongly change the gravitational field, as the end masses of your barbell.

Note also that from far away, compared to the barbell size, it does not make much difference what point in the barbell is considered as center of gravitational attraction.

Last remark : you can actually find asteroïds that have pretty much the shape of your barbell. They do not have a strong gravity field, but computing trajectories is not easy, as it rotates also. It is somewhere on the net, probably Nasa.

• Babou - I have submitted the dark matter question in a slightly different form for inestertion to my original question - it was not deleted by myself. Jul 8, 2013 at 15:16
• Well ... I thought it might be that. System is a bit hard to understand at first. Your question is to me a reasonable one. You cannot comment the answers, but you can comment your own question, and you can edit it. You should use answers only when you are actually giving the answer to a question. Whoever deleted should probably have done the editing or ask you to do it. Thanks for your reply anyway. Jul 8, 2013 at 16:28