# Determining coefficients of friction of paper

#### Omega234

1. Homework Statement
My friend and I are attempting to figure out what the coefficient of friction is of a stack of paper (say, 100 sheets). Through our research, we've found that the coefficient of friction ($$\mu$$) is the maximum possible static friction force (F) divided by the normal force (Fn). Normal force is the opposite of the mass (ma) of the object times that object's gravitational acceleration (g).

Our problem is that we're not sure how to find the "maximum possible static friction force" (or what, exactly, that is), and how to account for the multiple pieces of paper. The paper on the bottom of the stack will be weighed upon by all the paper above it, which would affect how frictional the papers are. How exactly would we go about finding this out?

2. Homework Equations
$$\mu$$ = F / Fn
Fn = - ma * g
(that's not an "m" because that would be "meters")

3. The Attempt at a Solution
We're not really sure how to begin - that's the problem.

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#### Stovebolt

A better definition of a Normal force is one that acts perpendicularly to a surface. On a flat surface, an object that is acted upon only by gravity will have a normal force = (mass * gravitational acceleration), as you noted. For an inclined surface, you will need to take only the component of the gravitational force that acts perpendicularly to the surface.

Direction is not necessarily relevant for an object at rest when looking at friction, so you do not need to use the negative sign in "- ma". The force will have an equal and opposite reaction force if the object is at rest, so magnitude is important, not direction.

The "maximum possible static friction force" is the most force that can be applied on an object in a direction parallel to the working surface that will not cause acceleration. In other words, how hard can you push the object before it will move.

Are you looking to design an experiment? There are several ways I can think of off the top of my head to do this.

If any of the above is not clear, please let me know.

#### Omega234

Okay, I understand what you mean about Normal Force - that helps a bit. And thanks for that about the maximum friction thingy (too long to type again...).

Um... we're kind of in need of help designing an experiment... but not really. Like... we have what we want to test, but we're just not exactly certain how to go about doing it. Let me explain what we've thought of so far:

We want to test how the amount of interleaved paper affects the coefficient of friction, so if we take 100 sheets of paper and interleave them, what is the coefficient of friction of the paper (that would probably have to be an average, because the top would likely have a lesser coefficient than the bottom, due to having less weight pressing down). If this were to be graphed, the x-axis would be the amount of paper (100 sheets, 250 sheets, et cetera), and the y-axis would be the coefficient of friction.

We're just not exactly certain how to go about testing this, so help with that would definitely be... well... helpful. Thanks for anything!

#### Stovebolt

In your example, the coefficient of friction will not be changing. The coefficient of friction between two surfaces is a constant (provided you are not deforming the surfaces or changing the material properties), but you are correct that there would be less weight (Normal force) and therefore, less frictional force.

To calculate the coefficient of friction in your case, you would need to calculate the Normal force acting between each interfaced surface. One thing that might provide difficulty here is the amount of each sheet that overlaps the sheet below it. If you have a large overlap, it may be reasonable to consider the entire weight of each sheet, but the less overlap there is between each leaf the less accurate this will be.

You will also need to exert a force to pull the papers apart. Have you thought about how you want to approach this?

What ideas do you have for setting this up?

#### Omega234

Ah, I see. That makes sense. I think we meant to determine the amount of friction being created, or something to that effect...

So you're saying that if we have a large overlap, we ought to account for the weight, but if it's a little overlap we don't. However if there's small overlap the experiment won't fully test what we want, right? I think I know what you mean.

We were considering creating brackets of some sort to hold the papers from either end, probably with one bracket mounted on something so it doesn't move. Then we'd have the other bracket have a cord or rope of some sort with a meter that measures Newtons and pull it (or find something to pull it). We've found a... uh... "Newtometer" (that's my new word for the day, I suppose) that marks a separate dial for the highest number it recorded (so no need to watch it closely while performing the experiment). It will either test how much force pulled the papers apart, or how much force the papers withstood (depending on whether or not the papers fell apart).

Would this be a good way to approach the experiment? We're trying not to add in too many extra variables (I say "too many" because there are always variables like temperature, pressure, and those that we can't afford to control), and we don't have a whole lot of money to devote to this (although we're willing to spend some to do it, we'd rather not have to sell everything we own to fund the whole thing). Thanks for your help!

#### Mentallic

Homework Helper
This is awfully similar to the experiment that mythbusters conducted. For the leaves of 2 phones books interweaved in each other, it took approx 8000 pounds of force to tear them apart. (10 people doing a tug of war against a wall with the phone books in between produced 4000 pounds of force). Mind you, don't plan on doing it with phone books, else you would need 2 tanks like they had to use :surprised

#### Stovebolt

A
So you're saying that if we have a large overlap, we ought to account for the weight, but if it's a little overlap we don't. However if there's small overlap the experiment won't fully test what we want, right? I think I know what you mean.
Not exactly. The point I'm making is that the amount of weight that contributes to the normal force will vary depending on how much overlap you have. If you had only two sheets, you would need nearly full overlap to have the full weight of the top sheet applied to the second sheet. The less overlap you have, the lower the amount of normal force between the surfaces.

We were considering creating brackets of some sort to hold the papers from either end, probably with one bracket mounted on something so it doesn't move. Then we'd have the other bracket have a cord or rope of some sort with a meter that measures Newtons and pull it (or find something to pull it). We've found a... uh... "Newtometer" (that's my new word for the day, I suppose) that marks a separate dial for the highest number it recorded (so no need to watch it closely while performing the experiment). It will either test how much force pulled the papers apart, or how much force the papers withstood (depending on whether or not the papers fell apart).

Would this be a good way to approach the experiment? We're trying not to add in too many extra variables (I say "too many" because there are always variables like temperature, pressure, and those that we can't afford to control), and we don't have a whole lot of money to devote to this (although we're willing to spend some to do it, we'd rather not have to sell everything we own to fund the whole thing). Thanks for your help!
That sounds like a good setup.

One question - is your goal to find the force necessary to pull X number of interleaved sheets apart, or is it to experimentally determine the coefficient of friction? Or is it both?

If you are trying to find the coefficient of friction only, I would suggest reducing the number of sheets to two. I would fix one sheet of paper to a hard, smooth surface, then put the other sheet on top, then apply a weight on top of that, and pull on the upper sheet slowly (without pulling up on the sheet) using the "Newtometer" to find the peak pulling force.

If you are trying to find the actual force to pull a certain number of sheets apart, I would still suggest making an attempt at determining the friction coefficient as above for a comparison to your results.

One other suggestion that might be worth considering would be to orient the sheets vertically so gravity will not contribute to friction. Instead of using gravity to provide the normal force for friction, perhaps you might try small spring clamps or even paper clips (the heavy duty sort that act like spring clamps). This should make calculations easier as the force applied would be more consistent - but if you do this, be careful not to deform the sheets or cause too much pressure in a single point (which would also deform the sheets). I'd probably attempt to clamp the sheets between two flat surfaces (rulers, perhaps) to better distribute the force.

Regardless of how you choose to arrange the experiment, two bits of advice - 1. Try to determine your answer in several different ways, and try each way several times. If the results are inconsistent, you likely have a problem with one or more of your experimental setups. 2. When you pull using the "Newtometer", be sure to pull slowly, smoothly, and in a direction parallel to the surfaces of the sheets. If you jerk the meter, it will read much higher than the actual force needed to overcome friction.

Good luck!

#### minger

If at some point you plan on testing rigid bodies, there is an extremely easy test that one can do to test for the coefficient of friction between two surfaces.

If you draw a free-body diagram of a body on a surface at an angle, you'll see that friction is of course preventing the object from slipping. At some angle $$\theta$$ however, there will exist a large enough component of the gravitational force to overcome the frictional force.

Through some clever algebra, it can be shown that this angle is independent of the mass of the object. It can then be plugged back into the equations to solve for the frictional coefficient. This angle is related to the coefficient of friction as
$$\mu = \tan\theta$$
Where $$\theta$$ is the angle at which an object 'just' begins to slip.

Note that this only really works for dry Coulumb friction; however a protractor, a large flat surface and few test runs can get you pretty close.

p.s. Teflon has an approximate coefficient of friction of 0.04, which means that a Teflon object can start to slip on a surface with as little as a 2.3° incline!

#### Omega234

I see what you mean now. I like the idea of doing it vertically - that would reduce the amount of calculations we'd have to do (very good for us - not exactly physics majors). To determine the coefficient of friction, we could use minger's idea, couldn't we? The only problem I see with that is that the type of surface would change $$\mu$$, so what kind of surface would we use? Or would we secure a piece of paper to the surface (one that doesn't move) and have another paper on top, because $$\mu$$ for us would be between two pieces of paper, right? So we just have to find $$\theta$$ at which the top paper begins to slip, and then, find $$\mu$$ as tan$$^{-1}$$$$\theta$$, right?

Oh, and to answer your other question: our overall goal is to determine the amount of force to pull apart X sheets. But to get there we have to find our $$\mu$$, so I guess we need to find that first and then worry about the rest of the experiment.

By the way, I need any sort of references you guys use (preferably online). No offense, but it's not very easy to cite that "I asked a guy on the Internet, and this is what he told me, so it must be true" and not get questioned a bit. Thanks!

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#### Stovebolt

I see what you mean now. I like the idea of doing it vertically - that would reduce the amount of calculations we'd have to do (very good for us - not exactly physics majors). To determine the coefficient of friction, we could use minger's idea, couldn't we? The only problem I see with that is that the type of surface would change $$\mu$$, so what kind of surface would we use? Or would we secure a piece of paper to the surface (one that doesn't move) and have another paper on top, because $$\mu$$ for us would be between two pieces of paper, right? So we just have to find $$\theta$$ at which the top paper begins to slip, and then, find $$\mu$$ as tan$$^{-1}$$$$\theta$$, right?
Good ideas, you should be able to get good results with that. You are right, if you are looking for $$\mu$$ between two surfaces of paper, you need to have motion between only paper surfaces, so you will need to secure the paper to hard, smooth surfaces. One thing to be careful of - you will likely get better results if the top sheet is secured to a heavier object, such as a book. A single sheet of paper might be subject to other forces, such as static electricity or an air current in a drafty room, which might increase or decrease the force it takes to initiate motion. By using a heavier object, you will make those forces negligible.

Oh, and to answer your other question: our overall goal is to determine the amount of force to pull apart X sheets. But to get there we have to find our $$\mu$$, so I guess we need to find that first and then worry about the rest of the experiment.
Good plan. Once you find $$\mu$$, try to predict how much friction will exist in your experiment before actually doing the experiment. Remember, in your calculations you will need to look at the number of surfaces in contact, not the number of pages.

By the way, I need any sort of references you guys use (preferably online). No offense, but it's not very easy to cite that "I asked a guy on the Internet, and this is what he told me, so it must be true" and not get questioned a bit. Thanks!
That is a good move. "Guy on the internet" is a step below "Wikipedia" as a reference.

There isn't too much to reference in this case, as most of what you will be doing will be focused on experimental results. (In effect, this means you will be referencing yourself). But you should be able to find references for the definitions of Static Friction and Normal Force in pretty much any physics text book. And while I don't recommend using Wikipedia itself as a reference, you can often find good resources in the "References" section on cited articles - just make sure you actually use any references you cite.

If there are any other concepts you feel you would like resources for, let me know and I will see if I can find anything good online for you to review.

#### Omega234

Stovebolt said:
Good ideas, you should be able to get good results with that. You are right, if you are looking for LaTeX Code: \\mu between two surfaces of paper, you need to have motion between only paper surfaces, so you will need to secure the paper to hard, smooth surfaces. One thing to be careful of - you will likely get better results if the top sheet is secured to a heavier object, such as a book. A single sheet of paper might be subject to other forces, such as static electricity or an air current in a drafty room, which might increase or decrease the force it takes to initiate motion. By using a heavier object, you will make those forces negligible.
I see... so the mass of the object on top of the slanted surface doesn't affect the angle at which it starts to give? But a heavier mass would make it slide faster once it did start moving, wouldn't it? I suppose that makes sense.

Stovebolt said:
That is a good move. "Guy on the internet" is a step below "Wikipedia" as a reference.
Sad, but true. I usually use Wikipedia whenever I want to learn about something not school-related, but if I need school research... teachers don't like the free online encyclopedia.

[Y]ou should be able to find references for the definitions of Static Friction and Normal Force in pretty much any physics text book.
Oh, thanks for reminding me - my friend has a huge physics textbook his dad let him borrow. He just ingeniously left it at school over our "winter recess"...

#### Stovebolt

I see... so the mass of the object on top of the slanted surface doesn't affect the angle at which it starts to give? But a heavier mass would make it slide faster once it did start moving, wouldn't it? I suppose that makes sense.
On the first part, correct. Think of it this way - the greater the force pushing down (perpendicular to the object and the surface), the greater the friction - but this force will be proportional to the force trying to cause motion parallel to the surface. If you increase one, the other increases equally.

On the second part, that is not correct (with one potential exception that I can think of). The acceleration on the object, regardless of mass, will be due to gravity. Heavy, light, makes no difference, it experiences the same acceleration.

The idea behind using a heavier mass than a single sheet of paper is to make the frictional force very big relative to other potential forces - like static electricity. A static electricity charge is not going to change as you increase mass, but as a percentage of the net forces acting on the object it will have a much lesser effect.

Sad, but true. I usually use Wikipedia whenever I want to learn about something not school-related, but if I need school research... teachers don't like the free online encyclopedia.
A lot of that has to do with the nature of Wikipedia. There is a lot of opinion-based information, and a lot that is inaccurate or uncertain. Some articles are very good, some very bad, and many in between. I like it, but it is not good for any serious research except as a starting point to find other resources.

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