Interlacing pages of books - Need help with understanding

[Atham]

Hello all,

I am working on a science project based upon paper, and I have chosen the affect of friction between two interlaced books. The attachment below is something I have found on the internet, as part of my research.

I don't understand why the R_av = 0.5 * (mass₁ + mass₂) *9.81. Why is the total mass of the books halved? Could you please explain this?

Thanks in advance,

A.T.

EDIT: I am not sure if this is the correct section. If it isn't the correct section, please do let me know and I will delete this post and put it in the appropriate place.

 

[Doc Al]

I don't understand why the R_av = 0.5 * (mass₁ + mass₂) *9.81. Why is the total mass of the books halved? Could you please explain this?

They want to estimate the normal force on any given sheet. The bottom sheets support the full weight of both books, while the top sheets have zero normal force. Multiplying the full weight by half gives the average normal force on the sheets.

 

[Atham]

So let's say I have only 4 pages, each with the mass of 2grams.
========== 0 grams of force
------------- 2 grams of force
========== 4 grams of force
------------- 6 grams of force

The top one will not have any force acting upon it. The second sheet from the top will have 2 grams of force acting upon it. The third would have 4 grams acting upon it. And the fourth would have 6 grams acting upon it.
So the average acting force here would be (6+4+2+0)/4 = 3 grams of force, or would it be (6+4+2)/3 = 4?
Because the average force by the 0.5*(mass1+mass2) would equal to 0.5(2*2+2*2) = 0.5(8) = 4grams of force, which would make the latter method ((6+4+2)/3 ) correct, but I would not be counting one of the forces.

EDIT: It is only 3 forces that are being averaged because there are only 3 forces acting upon the bottom page, right?

 

[Doc Al]

If there are 4 pages, then there are 3 contact surfaces, so that's what you should be averaging over. That "half the total weight" approximation for the normal force is pretty good when there are many pages.

 

[Atham]

I see. I do have a question about the different sizes of paper (area). If it was A5, instead of A4, how would that affect the friction force?

 

[nasu]

Hello all,

I am working on a science project based upon paper, and I have chosen the affect of friction between two interlaced books. The attachment below is something I have found on the internet, as part of my research.

I don't understand why the R_av = 0.5 * (mass₁ + mass₂) *9.81. Why is the total mass of the books halved? Could you please explain this?

Thanks in advance,

A.T.

EDIT: I am not sure if this is the correct section. If it isn't the correct section, please do let me know and I will delete this post and put it in the appropriate place.

If you try the same thing with the books in a vertical position, you will see that the effect is still "there". You can even hold one book with the spine horizontal and the other one will hang on.
But this time the weight of the paper is not normal but (almost) tangential to the sheets of paper.
So the calculation of average weights may be a little irrelevant anyway.

 

[Atham]

Because for the experiment, I will hang (vertically) the two books onto a pole and attach different masses, and trying to find the point at which the books will separate. What would be the formula for the force now?

EDIT: Or could I use pulleys to keep the books in a horizontal position like this:

===(Force Meter)===[BOOK 1][BOOK 2]====={Pulley}
dddddddddddddddddddddddddddddddddddddddddd||
dddddddddddddddddddddddddddddddddddddddddd||
dddddddddddddddddddddddddddddddddddddddddd||
dddddddddddddddddddddddddddddddddddddddddd||
[Mass]

Would that work?

Better picture here (there should be a rectangle at the bottom of the pulley string with the object providing the mass:

 

[Curve Shifter]

If the books are unsupported, and in the vertical position, it will be quite difficult to ascertain the normal forces. And indeed, the normal force would probably only be able to be determined indirectly and even then only if the coefficient of friction between the contact surfaces is accurately known.

You might want to have them horizontal, and resting on a flat surface. But what about the friction between the book resting and the surface it's resting on? Just test that independently, and then back it out of the amount of force required to separate the books. Then you have known force in the horizontal direction, and known normal force (or at least a reasonable average from the 0.5(m of B1 +B2)*g.

In ideal conditions, the surface area in contact doesn't affect the coefficient of friction. But smooth planar surfaces are not ideal, as there may some electrostatic interaction between the pages. So you may have to do some additional experimentation using books with various sized pages to determine if you need to attach some surface area multiple to determine a friction coefficient.

*There also may be some force from the binding coming into play. So possibly it might be easier to determine the friction coefficient first? And then your design would be fine for determining the average normal force acting on each page.

 

[Atham]

The assumed COF of paper is 1. I do agree that there is some force from the spine of the book; I have tried pulling normal exercise books, were the pages are stapled in place, and the binding did tear off; but only the binding of the spine.
In order to have the books in a horizontal position, would it be a good idea to use a pulley, or not?

Part of the science project is a presentation which gives an explanation. It has to be explained in simple terms, as the panel of judges is a group of non-science teachers and head masters, which is why I don't want to complicate things with putting the books in a vertical position.

 

[Curve Shifter]

The pulley is a very good idea. If the height of the pulley (and the attachment to the force meter) are such that the tensions are perfectly horizontal, then your normal forces will be more correct. I wouldn't assume a COF, or that it would not vary with surface area (see previous comment). You could eliminate the the force from the binding as a variable by using loose-bound paper in 3 ring binders or something similar where the binding isn't actively pressing the pages together.

Determining the COF experimentally (and whether it varies with surface area) should be fairly simple with the apparatus you're describing.

Some things to consider:

Unless the force being measured is enough to pull your force-meter perfectly horizontal, there will be a downward component of your tension. Is it possible to mount it in a fixed horizontal position?

What is the object of the experiment? Are you trying to mathematically predict the total Ff produced? If that's the case, (and really always) the fewer variables the better. And the more precise your givens (COF), the more accurate your overall equation will be. For instance, if your COF is off by .01, your average normal force is 10N, and you have 500 pages in each book, your estimated number will vary from experimental by several dozen Newtons.

I've judged a few math/science/physics fairs. The concept of COF shouldn't be beyond the understanding of non-technical educators. Just explain it conceptual, rather than mathematical terms. Maybe a diagram of what's happening microscopically at the contact surfaces to produce friction?

Anyway, sounds like a fun project!

 

[Atham]

I got the COF from the attachment, which it assumed to be 1.
We are being judged upon our creativity, and originality of the experiment, as well as the actual science facts of the experiment and correct methodology.

I will be using books with less pages as I want to achieve the point at which the books will actually separate, and thus proving the mathematical formula that I have found out. I am either thinking of using the pulley and add masses onto it, or have one end of the books attached to a stationary heavy object, like a tree or a post, and the other end will be pulled upon by groups of people.

My initial idea was to keep on adding masses until we reach the separating point, but know I am thinking of trying books with different amounts of pages, making the number of pages per book the independent variable, and to see how the number of available surfaces affects the force required for separation. The latter would make more sense as it would also give a graph. The experiment could then be be mathematical calculations backed up by experimental data, proving the formula.

EDIT: Curve Shifter, you mentioned working out the COF experimentally. How could I do that?

 

[Curve Shifter]

the best way to determine COF would be to use a single contact surface (paper to paper), with a known mass pushing down on them. Determine how much force it takes to cause slip, divide by the normal force (mg), and that will yield your static COF. Repeat several times.

You could then vary the area of your contact surface and repeat the experiment to see if the surface area affects the COF.

In the case of paper, the surface area will almost certainly come into play because paper is very flat, so electrostatic forces between the sheets will be a factor. Ever notice how two very flat surfaces seem to adhere to one another? Lay two CLEAN glass slides one atop the other, and take notice of how difficult it is to make them slip. In this case, nearly all the normal force causing friction is supplied by electrostatic attraction between the atoms/molecules of one glass sheet and the other rather than by gravity acting on the mass of the slides. Not as noticeable for paper, but still a factor.

There will be some friction in your pulley, so you will still need a force meter to accurately gauge the tension being applied to the books.

If you want to bounce ideas for equations around, feel free to post up. Detailed diagrams or pics of your apparatus would be good.

Good luck!

 

[Atham]

Thanks for the tips, Curve Shifter. I have to give in a plan of what equipment I need by Tuesday, and then I can do some testing, as the experiment is going to be conducted in about one month's time. Before and after the final tests I will do some equipment testing and I will post some pictures for better communication.

 

[Atham]

I have tested the books with a pulley and the rope would work with it.

For the COF of paper, let's say that I have 2 A4 paper sheets. What would be the known mass? The mass of the paper above it? How would I determine the force required to separate them? Would I pull very gently?

 

[Curve Shifter]

For determining the COF, you'll want to test a SINGLE paper-to-paper contact surface. All mass above the contact surface will contribute to the normal force. I'd use maybe a notebook to press the contact surfaces together. Or you can use a laboratory mass. You don't want to go too heavy, because then you won't really be able to determine the effect of the electrostatic force between the sheets. Don't want to go too light either, or your force meter may not be sensitive enough to yield an accurate COF.

Probably 100g is as heavy as you want to go. Much lighter if your force meter is of good quality.

 

[Atham]

I see. So it would look something like this:

--------
| Mass |
--------
======= Sheet one
--------- Sheet two

 

[Martin Yang]

had you guys tried any other parameters outside of pages overlapped?

And oh, i found out that you had better pull it with a string instead of holding on the spine with hands, or else the normal force couldn't be caculated