Center of Mass

[Heinsbergrelatz]

Say that we have a piece of tissue paper, how do we find the center of mass ? step by step experimentally???

[imatfaal]

Could you not just hang it from each of the four corners, photograph from multiple angles each time with every shot including a view of definitive vertical plumb line coinciding with point of attachment, on each print consider a plane which includes both the point of view of the camera and vertical plumb line, the centre of gravity is the point of intersection of the planes. It is worth bearing in mind that the centre of mass may well lie outside the body of the tissue paper if it is at all curved. Might be really quite tedious to do in practice - but would be greatly aided by a pattern on the tissue paper that would allow you to make eye judgments

[michel123456]

Why tissue paper?

 

i mean ; the centre of mass is a geometric feature, if you change the shape of the object (it easily happens with tissue paper) then the centre of mass will change location.

For example if you wrap the tissue in donut shape, the centre of mass will not be on the tissue.

[Przemyslaw.Gruchala]

Or dual-star system- center of mass won't be in any of star, but somewhere between them.

[elfmotat]

If the shape of the tissue paper is changing in time then the center of mass will also change in time. If you know the mass distribution as a function of time [math]\rho (\mathbf{r},t)[/math] then you can calculate the center of mass as:

 

[math]\mathbf{R}(t)=\frac{\int_V \rho (\mathbf{r},t)~\mathbf{r}~\mathrm{dV}}{\int_V \rho (\mathbf{r},t)~\mathrm{dV}}[/math]

[SamBridge]

I can't remember how to do it exactly but you have to use some kind of average value integral based on it's dimensional properties of how they change, I think elf is pretty close if not right on it. Normally its the integral of a to b and the integrand of which is divided by b-a and you somehow us the coordinate of x=average value and f(average value), but I don't know how you find the exact coordinate.

Edited February 8, 2013 by SamBridge

[Heinsbergrelatz]

Ok thank you for all the replies. Interesting, however i want to assume that the tissue will be let flat in air (even though its not realistically possible given the equipments i have). Now i have another question, how would i drop the tissue paper such that it will fall with equal velocity at all 4 ends? (given that the tissue [paper is held flat on air)

[SamBridge]

Ok thank you for all the replies. Interesting, however i want to assume that the tissue will be let flat in air (even though its not realistically possible given the equipments i have). Now i have another question, how would i drop the tissue paper such that it will fall with equal velocity at all 4 ends? (given that the tissue [paper is held flat on air)

The tissue paper will already have its ends falling at equal velocity otherwise the ends would tear off, at least in an ideal situation but there's air resistance of course and you'd need some kind of computer to model how the complex folds in air would warp it as it fell. If you were on the moon however it would just automatically fall flat with all the parts falling at equal acceleration.

Edited February 8, 2013 by SamBridge

[michel123456]

Ok thank you for all the replies. Interesting, however i want to assume that the tissue will be let flat in air (even though its not realistically possible given the equipments i have). Now i have another question, how would i drop the tissue paper such that it will fall with equal velocity at all 4 ends? (given that the tissue [paper is held flat on air)

I suspect like a Chef would do with his sauce: you hold the tissue flat on a pan (if you have a large pan) as high as possible and then you draw quickly the pan down, vertically and quicker than gravity. The tissue should fall evenly, not flat, but with all 4 sides at the same velocity.