# How do towels stay on hooks?

Towels (and coats) are often stored on hooks, like this:

To the untrained eye, it looks like the towel will slide off from its own weight. The hook usually angles upwards slightly, but a towel does not have any "handle" to string around and hang on to the hook -- this makes it seem like it will simply slide off.

Yet these hooks hold towels well, even heavy bath towels. Why?

I have three ideas:

1. There is sufficient friction between the towel and the hook to counteract the force of the towel pulling down.
2. The hook is angled such that the force is directed into the hook, not directed to slide the towel off of it.
3. The center of mass of the towel ends up below the hook, since the towel is hanging against the wall.

Which of these ideas are likely correct? I am also happy with an answer based purely on theoretical analysis of the forces involved.

• I suspect that the most widely discussed analysis that applies is the capstan problem, the solution to which is somewhat surprising because of it's exponential dependence on the angle of contact. – dmckee Jan 23 '18 at 21:23
• Why not a combination of all 3 explanations? They don't appear to be mutually exclusive. Or are you hoping that someone will entertain you by providing a complete theoretical analysis? Or even the results of a home experiment? There might be an Ig Nobel Prize in this. – sammy gerbil Jan 23 '18 at 21:44
• My towels keep falling off their hooks, except the towels that have hooks to hang them on. – gerrit Jan 24 '18 at 14:37
• Obviously, towels are programmed to stay on hooks. That way, you always know where your towel is in case you need to hitch a ride on a passing ship. – Ellesedil Jan 25 '18 at 4:40
• "it looks like the towel will slide off from its own weight." But this is totally, completely wrong. Going from back to front, about half of the towel is behind the top apex of the hook, and about half is in front. (Indeed, in the actual photo, I'd judge about 60% if in front and 40% is behind. You then have the question "How can it hang if 60/40?" The answer is: friction.) – Fattie Jan 29 '18 at 1:24

There is some contribution from the friction of the various surfaces, but the main factor is the balancing of weight.

It's important to note that the hook is set slightly away from the wall, which allows almost the entire weight of the towel to move alongside or behind the front of the hook tip.

The manner in which the towel is cast over the tip of the hook creates "wings" that droop down the sides and behind the tip of the hook.

Weight in the wings that is supported by fabric on either side of the hook tip, does not contribute to sliding off (provided the towel is hooked in its middle and the amount of weight on each side is balanced).

Therefore, the weight of the fabric forced into the "throat" of the hook (and the wings which hang from it), needs only offset the weight of the fabric that remains on the front side of the hook, which is only a very small amount of the overall weight of the towel (and therefore only needs a very small amount of fabric in the throat of the hook to offset it).

Incidentally, even silk fabric on a smooth hook can be hooked in this manner - the reduced friction simply requires more fabric to be accumulated in the throat, whereas rough fabrics on rough hooks can get away with relying less on balance and more on friction.

• Thanks! I'm inclined to think this is it. At least, the "wings" must be a huge part of it. The other answer does an extended analysis essentially without factoring in the wings, and gets extremely high required coefficients of friction. What you are saying explains it perfectly -- by pushing mass into the wings / throat, you offset the weight in front. – 6005 Jan 24 '18 at 4:58
• ...A surprisingly large number of fabric items and garments in the home have this combination of relatively stretchy main fabric with reinforced edges - a coat or body jumper for example, have reinforcement around the neck, which is probably another reason why most body garments are easily hookable (even when they are clearly not balaced on the hook). Who knew towels and hooks could be so intriguing! (3/3) – Steve Jan 24 '18 at 8:50
• +1. While the other answer (cms) has all the heavy math, it has little relation to reality. It's easy to check that a narrow strip of towel (matching the cross-sectional model) will slip right off a hook and that the wings/3D properties are needed to achieve holding. – R.. Jan 24 '18 at 22:39
• @R.., I think another factor with towels is that the fabric must usually be significantly wider than it is longer, and be hooked along the wide edge - this ensures that the two "wings" that arise due to the fabric in the throat of the hook, have enough weight between them to offset the fabric that falls over the front of the hook. Therefore, a towel that is twice as wide as it is long, will be rather easier to hook than a square towel, because the wider towel's wings are longer and proportionally heavier than the "front-fall" of the towel. (God help me for coining all these neologisms!) – Steve Jan 25 '18 at 0:22
• @Steve for a square towel I recommend folding it diagonally in half, although folding it orthogonally in half should also be sufficient (making it twice as wide as it is long, as you suggest). – trichoplax Jan 25 '18 at 20:12

Since this is PhysicsSE, I am happy with an answer based purely on theoretical analysis of the forces involved.

Oh boy, time to spend way too much time on a response.

Lets assume the simple model of a peg that makes an angle $$\alpha$$ with the wall and ends in a circular cap of radius $$R$$. Then a towel of total length $$L$$ and linear mass density $$\rho$$ has three parts: one part that hangs vertically, one that curves over the circular cap, and one that rests on the inclined portion like drawn. This is very simplistic, but it does encapsulate the basic physics. Also, we ignore the folds of the towel.

Let $$s$$ be the length of the towel on the inclined portion of the peg. I will choose a generalized $$x$$-axis that follows the curve of the peg. Note this model works for both the front-back direction and side-side direction of the peg. In the side-side (denoted $$z$$) $$\alpha$$ is simply zero (totally vertical):

Where $$\eta$$ is the fraction of the towel on the right side of the picture. Then the total gravitational force $$F_{g,x}$$ will be:

$$F_{g,x} = \rho g (L - R(\pi - \alpha) - s(1 + \cos(\alpha)) - \int^{\pi/2 - \alpha}_{-\pi/2} \rho g R \sin(\theta)\,\mathrm d\theta$$ $$F_{g,x} = \rho g (L + R(\sin(\alpha) - \pi + \alpha) - s(1 + \cos(\alpha))$$

The infinitesimal static frictional force will be $$\mathrm df_{s,x} = -\mu_s\,\mathrm dN$$. $$N$$ is constant on the inclined part and varies with $$\theta$$ over the circular cap as $$\mathrm dN = \rho g R \cos(\theta)\,\mathrm d\theta$$. Then: $$f_s = -\mu_s \rho g s \sin(\alpha) - \int^{\pi/2-\alpha}_{-\pi/2} \mu_s \rho g R \cos(\theta)\,\mathrm d\theta$$ $$f_s = -\mu_s \rho g ( s \sin(\alpha) + R(\cos(\alpha)+1) )$$

Now we can set the frictional force equal to the gravitational force and solve for what values of $$\mu_s$$ will satisfy static equilibrium. You get:

$$\mu_s = \frac{L + R(\sin(\alpha) +\alpha - \pi) - s(\cos(\alpha)+1)}{R(\cos(\alpha) + 1) + s\sin(\alpha)}$$ $$\mu_s = \frac{1 + \gamma(\sin(\alpha) +\alpha - \pi) - \eta(\cos(\alpha)+1)}{\gamma(\cos(\alpha) + 1) + \eta\sin(\alpha)}$$

where the second line $$\gamma = R/L$$ and $$\eta = s/L$$, the fraction of the towel on the peg's cap and incline, respectively. Thus $$\mu_s$$ depends on three factors:

1. The angle of the peg, $$\alpha$$
2. The fraction of the towel past the cap of the peg, $$\eta$$.
3. The fraction of the towel on the circular cap, $$\gamma$$.

Lets make some graphs: The above graph shows what $$\mu_s$$ would have to be with a $$\gamma = 0$$ (no end cap, just a 1D stick). The above graph shows what $$\mu_s$$ would have to be with a $$\eta = 0$$ (no stick, just a circular cap that the towel drapes over. The above graph shows what $$\mu_s$$ would have to be when the angle is fixed $$\alpha = \pi/4$$ and the length of the peg ($$\eta$$) is varied.

### summary

What all the graphs above should show you is that the coefficient of static friction has to be enormous ($$\mu_s > 50$$ -- most $$\mu_s$$ are close to 1) unless the fraction of the towel on the peg ($$\eta$$ and $$\gamma$$) is large, like over 50 % combined. The large values for $$\eta$$ can only be accomplished when you put the towel at approximately position $$\mathbf{A}$$, whereas its very difficult to hang a towel from position $$\mathbf{B}$$ because it reduces $$\eta$$ in both the $$z$$ and $$x$$-directions.

### 3) the towel has a center of mass below the peg

This isn't a sufficient condition for static equilibrium; a towel isn't a rigid object. As a counter-example, see an Atwood's machine. The block-rope system has a center of mass below the pulley, but that doesn't prevent motion of the blocks.

• Second question. What are the assumptions inherent in this? (1) I think you are assuming the towel rests straight down from the edge of the hook (is that necessarily true?) (2) I think this may assume only a single dimension, and that could explain the extremely high required coefficient of friction. After all, it is much harder to hang a towel on a cyllindrical bar (like over a rod inside a closet) than over a hook. – 6005 Jan 24 '18 at 4:55
• Will you publish a paper derived from these results and hope for the Ing Nobel prize? :P – M.Herzkamp Jan 24 '18 at 10:08
• @6005 $L_1$ and $L_2$ are the long and short dimensions of the towel. As far (1): yes, I am ignoring any folding action of the towel -- my intuition is that it doesn't drastically change the model much. (2) Most people hang a towel over a cylindrical bar (like a shower curtain rod) long-side down, which makes it difficult. I just did a very detailed experiment in my bathroom and hung a towel short side down and saw a wide range of stability. :-) – cms Jan 24 '18 at 15:46
• I'm pretty sure that the folding that you've ignored is actually the most significant contributor to the stability of a hanging towel in reality, since it causes most of the towel's bulk (and thus its center of mass) to be closer to the wall than the tip of the hook is. Thus, even a frictionless hook can easily support a flexible towel. OTOH, as your analysis shows, a non-folding towel-like object (such as a sheet of paper, or a narrow belt or rope perpendicular to the wall) cannot be practically supported on a hook unless either the hook is very long or it has a Velcro-like $\mu_s$. – Ilmari Karonen Jan 24 '18 at 16:19
• You have a huge mistake in your calculation of the normal force, and hence, the friction. The normal force is caused not only by the weight of each element, but by its tension as it curves around the peg. The extra force is $T\ d\theta$ and is independent of $R$, and hence is significant even in the limit $R=0$. This is why you got the clearly nonsensical result (e.g. in the 1st graph) that a peg going upwards is as difficult as a peg going downwards, and why you got coefficients of friction (or % of length) much higher than observed in practice. – Meni Rosenfeld Jan 25 '18 at 11:08

I like the top-rated answer for it's methodological approach and nice graphs, but I believe it fails to answer the question at it's heart because it misses a critical aspect: the towel folding.

If we imagine a 1 dimensional towel we can easily see that the fabric on the wall-side of the hook is insufficient to counteract the bulk of the material on the opposite side.

If we imagine this 1 dimensional model extruded into a mostly rigid sheet we will again see that the wall-side material, again, will be insufficient to hold the towel in place. (Rigid in the sense that folding is constrained to 1 dimension--imagine a hard plastic sheet).

Therefore, the critical aspect of this system is that the towel "folds" on all sides of the hook, producing a symmetric distribution across the y-axis.

Recall friction is a function of the normal force:

$f_{s}=\mu_{s}N$

If you look at the towel, you will see that it, for the most part, hangs nearly in a straight line very near the hook. This means the center of mass is located not far from the center-line of the hook. This also means that the force is mostly normal to the hook's tip. Very little lateral force is exerted in this system, thus the friction created by the towel's weight is sufficient to overcome the sliding friction.

Most of the mass is evenly distributed across the y center-line, which does not contribute a net sliding force. Also, all of the towel's mass contributes to a normal force to the tip, which provides the necessary static friction to overcome any imbalance caused by the mass distribution across the x-axis. Furthermore, the mass imbalance across the x-axis is not as extreme as it appears at first sight, as there is material both in front and behind the hook tip on that axis.

Conclusion

The mass distribution of the system is more balanced than it appears at first sight. All of the weight of the towel contributes to the normal force, which gives the system enough friction to counter any small imbalances across the x-axis.

I am going to go in a different direction here... and claim that the towel doesn't slip because it deformed when it was placed on the hook.

The weight of the towel pulls on the tissue in a generally downward direction; because most of the towel is on the outside, friction alone is not sufficient to prevent the towel from falling (as was nicely shown in @cms's answer. But tension in cloth is not just in one direction: it depends on the shape of the material. Take a sideways look at the towel-on-hook, and I believe this is what you see:

The distortion of the tissue at the top of the hook means that there is a significant fraction of the weight applied to the back of the hook: this is why a relatively low coefficient of friction is sufficient to hold the towel in place.

A simple thought experiment confirms this: if you take a piece of paper and just drape it over a hook, with most of it on the outside, it will slide off. But if you crumple the paper just a bit at the top, it will stay. This is because the paper / towel wants to maintain its distorted shape in the presence of the tension due to the weight - and this shape is what keeps it on the hook.

• Yeah. What's clear is that the "draping around" of the towel is extremely significant. This is another nice take on why that might be -- I hadn't thought about the increased friction and the tendency of objects to maintain their distorted shape. – 6005 Jan 26 '18 at 20:34

The towel is necessarily kept up by the upward force by the hook. The upward force balances the weight of the towel.

As you've yourself pointed out here, a too smooth and straight hook causes the towel to slip off. So, two other important factors:

1. The hook has to be bent upwards if it is smooth. The bend assures that a small part of the towel is on the inner side of the bend, which prevents the slipping off. Look at the picture:

1. The hook has to be rough if straight. The rough surface provides friction against the slipping off of the towel.
• This doesn't really explain how that part actually manages to prevent slipping though. – JMac Jan 23 '18 at 21:54
• Thanks. This seems right. So how do the forces work out? Supposing the hook goes upward, why does that make a difference? It seems like the hook could be at any angle, and the weight of the towel would still outweigh the friction. – 6005 Jan 24 '18 at 4:39
• @6005 it is actually a game of equilibrium. It's true that the towel can slip at any angle, and that's why you've to put that small part of the towel in the inner side of the bend, which will prevent the slipping off by increasing weight in another direction. If the towel is rough, unlike mine, then that part can be small, because friction plays a role here. On the other hand, a silk cloth would require a greater portion inside the bend so as to prevent the slipping. – Wrichik Basu Jan 24 '18 at 5:01
• "The upward force balances the weight of the towel." Not relevant at all to the question. A human is standing on the Earth. Sure - obviously - the force upwards of the ground stops the human falling towards the center of the Earth. But it has nothing at all to do with questions such as "why does the human not slip left-right". – Fattie Jan 29 '18 at 12:26

Those towels remain on that particular style of hook because the majority of the weight is behind and underneath the ball of the hook, if the towel was damp and only a small portion of it placed over the hook the towel would likely slide off once it had dried sufficiently.

From a point of view of functionality one ought to choose a deep throat and longer neck with a square head hook, as opposed to one of the opposite design.

During ancient China's Eastern Zhou Dynasty (770-256 BC), where it is thought that some of the first coat hooks (Daigou) were made, the hook had a narrow throat with a long neck and a square head; this was prior to 770 BC and modern Patents or engineering.

These were sometimes made from bronze or made from stone and usually had an animal head (Dragon) to provide friction. Compare the ancient design with modern technology, to me modern design seems less expensive and less effective.

The towel and most fabrics will yield and deform under a concentrated load. The threads have some play to slide laterally in both orthogonal directions and make enough slack to allow a bump in an otherwise flat surface.

Many of us old timers who used to wear pocket shirt at work and sometimes carry small erasers or what not in that pocket remember the permanent indentation they left.

How many times you had to get rid of a decent pair of jeans only because the knee part has permanently turned into an ugly half ball?

This sagging in is combined with friction and holds the towel on the hook, sometimes even when one hanging side is much longer than the other side!

In the original photo,

# the amount of towel in different directions is unbalanced.

It looks to be quite unbalanced front-rear, and a bit unbalanced left-right.

OP is asking, why then does it not slip off.

The answer is simply friction. That is all there is to it.

Note that in the exact photo shown in the original OP:

# if both surfaces were totally frictionless: it would slip off.

Utterly straightforward and unavoidable fact.

Here's yet another hook design:

It is very unbalanced front-rear, and also quite a bit to the side. Why does it not slip off? Friction. Again in this example, if both surfaces were totally frictionless: it would slip off.

The answers which detail in an engineering manner where the forces lie, for some particular hook design, are fundamentally wrong.

What stops a car sliding? The answer is: friction. If you make a detailed analysis of the surface area given different pressures, different concrete blends, etc, that would be fantastically helpful when eg. designing tires.

But it has no connection to the answer.

The "thing" that stops the car from sliding is "friction".

Consider any design whatsoever for the hook:

• hooks that are just straight unadorned sticks pointing outwards are common
• really aggressive hooks that go straight up are common
• there are "designer" hooks which are straight pencils which point down (!) somewhat, with a little nub on the end
• imagine hooks that have no wall beside,
• hooks that have strong negative or positive angled walls beside,
• hooks that come from the floor, ceiling, or anything else
• hooks on other planets ...
• hooks in elevator thought experiments
• enormous hooks, tiny hooks
• note that indeed exactly the same question can be asked about full towel rails. On a rail you don't have to hang the towel balanced - it's fine to be displaced 30-40% on a typical towel rail.

In all cases, imagine the towel being unbalanced front-to-rear, or left-to-right (or in any direction).

When unbalanced, what stops it from slipping?

It's just friction.

In all cases, very simply imagine just replacing all surfaces of the hook and of the towel, with more and more slippery surfaces. With perfectly slippery surfaces, it will slip off when unbalanced (absolutely regardless of design, center of gravity issues, etc).

"Anyway I agree it's pretty obvious that friction is necessary in one direction"

What does that even mean? We're talking about static (not dynamic) friction here. Of course, obviously, it's only "necessary" in whatever direction the imbalance is at the moment of discussion. (You could, obviously, of course, imbalance it the other way and then the friction would be "necessary" the "other way".) It's just a (no offense) very meaningless thought; it doesn't even parse in the normal way you talk about forces. one direction - what?

"The towel is necessarily kept up by the upward force by the hook. The upward force balances the weight of the towel."

Say you are discussing whether a person standing on a slope will slip sideways or not. {Which is exactly the same as the question under discussion.} Say you observed "Oh, the person will not fall to the center of the Earth - because the ground pushes upwards with equal force!" It is an incredibly misguided and confusing observation. What is under discussion is whether static friction will be overcome and the shoes will slip.

The answer which is currently ticked contains just staggeringly, amazingly, incorrect basic physics -

Therefore, the weight of the fabric forced into the "throat" of the hook (and the wings which hang from it), needs only offset the weight of the fabric that remains on the front side of the hook, which is only a very small amount of the overall weight of the towel (and therefore only needs a very small amount of fabric in the throat of the hook to offset it).

Looking from the top of the hook, you can draw (if you wish, for some reason) any normal around the 360 degrees, and note the weight imbalance on either side of that normal. But of course you wouldn't do that, you'd just have some vector, which would point in a particular direction being the current overall weight imbalance. It's (A) utterly meaningless to talk about "offsets" on some particular normal and (B) who cares? All you do is state, the imbalance is such and such, in such and such direction.

But doing any of that is remarkably unclear. Quite simply, in the OPs photo - you could move the towel around, left, right, sideways - whatever - and in many cases it would still hang there even though it is unbalanced. Why? Obviously friction.

The very question itself has a huge, huge howler (which - incredibly, given the length of the false analysis, nobody even noticed)

There is sufficient friction between the towel and the hook to counteract the force of the towel pulling down.

Heh, the force cause by the imbalance is lateral - horizontal. Nothing at all to do with "downwards". (Looking from the top - overhead - it will slide sideways (in any direction, from 0 to 360 degrees, as seen from the top) if you set it as too imbalanced in that direction.

Perhaps most amazingly -

There is some contribution from the friction of the various surfaces, but the main factor is the balancing of weight.

What does that even mean?

(i) "the main factor is the balancing of weight" well yes, this question is about the towel being unbalanced (at some angle, 0-360, looking from the top). There's no "factor" about "balancing of weight". You would simply write down, the towel is unbalanced (say, 400 grams) in some direction (say "213° East of North".)

So there is a static force supplied by X grams in direction D. OK.

(ii) "There is some contribution from the friction..." what else can "contribute" to one surface against another not slipping, other than static friction? Can anyone state anything else?

There are numerous other "howlers" on this page, and I do not have time to point them all out unfortunately!

• So your overall argument is: "if there were no friction, the towel would slide off, therefore, friction is the only thing causing it to stay on". This only proves that friction is necessarily one component -- but weight balancing strongly influences how much frictional force is needed in which direction. Anyway I agree it's pretty obvious that friction is necessary in one direction, but I think with some hook / towel designs friction is unnecessary in the other direction (it won't slide off in the front even with 0 friction). – 6005 Jan 28 '18 at 22:41
• No matter what the shape of the "hook", the amount of friction required is dependent on the balancing of weight. Since you mention "hooks on other planets", imagine a hook that is actually just a large cup, that the towel goes into. Then it stays with 0 friction no matter the direction. Anyway, you make some good points some places in there, but the overall conclusion is a big leap. – 6005 Jan 28 '18 at 22:44
• hi @6005. That wouldn't be a "hook". That would be a bucket. – Fattie Jan 28 '18 at 22:51
• Right you are :) But it illustrates one of the flaws in your arguments. – 6005 Jan 28 '18 at 23:25
• And regarding "taking a breath" :) Nobody is stressed here. But, as has been very clearly pointed out, the answer with diagrams (that has a huge number of upvotes) is just, utterly, incorrect. On all three levels: The mathematical calculation is totally in error (!), the model of the (specific) system is totally wrong (!), and conceptually it's completely wrong. Geesh! – Fattie Jan 29 '18 at 19:53

## protected by Qmechanic♦Jan 24 '18 at 6:10

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