249

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

Towels on hooks

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.

Andrew T.
  • 123
Caleb Stanford
  • 1,979

8 Answers8

201

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.

no penis jokes

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):

yes that's a *Hitchiker's* reference

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: gamma = 0 The above graph shows what $\mu_s$ would have to be with a $\gamma = 0$ (no end cap, just a 1D stick). eta = 0 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. alpha = pi/4 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.

cms
  • 3,998
  • 3
    This is amazing, thanks! Exactly the kind of extended analysis I was hoping for. I didn't read the calculations in detail, but I like the plots and I think I understand the meaning of them. Can you explain the "where your towel is at" diagram? What are $L_1$ and $L_2$? – Caleb Stanford Jan 24 '18 at 04:51
  • 4
    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. – Caleb Stanford Jan 24 '18 at 04:55
  • 60
    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
  • What software did you use to make the plots? – Ruslan Jan 24 '18 at 14:24
  • Looks to me like you've made a couple of small errors there. Seriously, you gotta love SE. – Strawberry Jan 24 '18 at 14:58
  • 5
    @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
  • 1
    @Ruslan Gnuplot 4.4 and some perl. – cms Jan 24 '18 at 15:47
  • @Strawberry I wouldn't doubt there are errors. Would you mind telling me which you saw? – cms Jan 24 '18 at 15:49
  • 3
    @cms I was just teasing. I wouldn't know an error if it was wearing a neon sign. – Strawberry Jan 24 '18 at 15:51
  • Gnuplot (checked on 4.6, but I suppose 4.4 too) can show Greek letters. Just be sure to set termoption enhanced, and then change e.g. gamma to {/Symbol g}. – Ruslan Jan 24 '18 at 15:58
  • 19
    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
  • 1
    Upvote because gnuplot+perl instead of some fancy-schmancy newfangled python+matplotlib. – pipe Jan 24 '18 at 18:57
  • 1
    One simplifying assumption made here is that the density of the towel is uniform. When the towel has recently been used to dry something or someone, the wet spots on the towel may not be uniformly distributed making the density of the towel much less uniform. This would explain why a towel could hang in a stable manner in the morning right after a shower but dry out and fall off during the day, a rare but occasionally frustrating occurrence. – Todd Wilcox Jan 24 '18 at 20:02
  • @ToddWilcox I've assumed that a damp towel has more friction with the hook. You see, this is what we need answered now: how do the folding properties of material X change as it becomes saturated with water and what effect does that have on its ability to hang on a hook? – Carl Jan 24 '18 at 22:15
  • 1
    @Mick finally someone got my Hitchhiker's reference. – cms Jan 25 '18 at 03:26
  • @cms Douglas Adams would never had used "where you towel is AT" - that's an Americanism! –  Jan 25 '18 at 09:15
  • 8
    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
  • 4
    Example: In the limit $R=\gamma=0$, $\alpha=0$, You have $dT = f = \mu N = \mu T d\theta$, so $dT/T = \mu d\theta$ and $\Delta\log T=\mu\Delta\theta = \mu\pi$. We have $T_1 = L\rho g (1-\eta), T_2 = L\rho g\eta$, so $T_1/T_2 = \eta^{-1}-1$, and therefore $\mu = \log(\eta^{-1}-1)/\pi$, rather than infinite as your model predicts. – Meni Rosenfeld Jan 25 '18 at 12:21
  • 4
    For $\mu=0.5$ you have $\eta=0.17$, which more closely matches experience. With a good $\mu=1$ you only need $\eta=0.04$. The calculation for other slopes is similar but more cumbersome. But for a downward sloping hook you will, of course, need huge $\mu$ and $\theta$, since there is no tension on the corner. – Meni Rosenfeld Jan 25 '18 at 12:31
  • @MeniRosenfeld yes I was aware of the error in my calculations rather early on (via the “capstan problem”) but I haven’t gotten around to revising the post. Will do soon. – cms Jan 26 '18 at 14:57
  • 1
    Without offense, this answer would seem to be "completely wrong" :O It's conceptually totally wrong. (There's just an overall mass vector of the imbalance, and the system has a macroscopic static friction coefficient. Could not be simpler.) {When you figure the friction of (say) a specific tire on a specific road, you don't sort of figure the friction of, like, the knobs of gravel, tread sides, etc.} Secondly as Meni points out it's all wrong specifically. :O – Fattie Jan 29 '18 at 16:50
  • I guess thirdly, as Ilmari explains clearly, this "micro" analysis has no connection at all to the overall central point of the (specific) bunchy-towel arrangement photographed. (Which in itself anyway - means nothing. There are innumerable different rail, hook, etc arrangements. Notice the photo one in my answer which is ENORMOUSLY unbalanced, and indeed that type of hook happens to have absolutely no "back ruffle" concept.) Again, no offense meant here. – Fattie Jan 29 '18 at 16:56
  • @Fattie: I disagree. The system is way more nuanced than just an "overall mass vector of the imbalance" and a "macroscopic static friction coefficient". True physics is about making predictions, and the conceptual model you are offering will be quite useless in predicting whether a specified arrangement will remain static or not. Will your model successfully predict things such as the exponential relationship between the coefficient of friction and the % of the towel that is allowed to overhang? See also - http://lesswrong.com/lw/iq/guessing_the_teachers_password/. – Meni Rosenfeld Jan 29 '18 at 23:14
  • hi @MeniRosenfeld, (by "mass" i just meant "bulk". I can assure you there's an overall vector of the imbalance. (It's probably "forward, and a bit to the right.") Very straightforward.) I don't know what you mean by "more nuanced". If I told you a car has X torque, and it needs a certain amount to move an object up a hill - that's a "fact". It either does, or doesn't, have X torque. (By all means, there are innumerable fascinating engineering aspects to building cars and engines.) If you said "oh, your view is not nuanced", you would be incorrect. – Fattie Jan 30 '18 at 01:18
  • All the points have been made perfectly clearly, no need to copy and paste. Cheers! – Fattie Jan 30 '18 at 01:18
  • Can you explain how you calculated the gravitational force $F_{g,x}$? I dont quite understand all components. An additional sketch would be great! – Merlin1896 Jan 30 '18 at 20:41
  • Nice pictures but the theory disagrees with everyone's practical experience of hanging a towel on a hook. If the towel falls off, you would bunch it up more to get it to hang. – jwg Feb 01 '18 at 10:13
126

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.

Steve
  • 2,689
  • 7
    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. – Caleb Stanford Jan 24 '18 at 04:58
  • @6005, indeed. I've noted another thing about towels is that they usually have a reinforced rim which is less flexible to stretch than the main fabric - the smaller the tip of the hook, and the more inclined it is, the more it tends to "pierce" the main fabric out of shape which causes the fabric to cup over the tip of the hook under its own weight, and the reinforced edge prevents this cup from merely sliding off the edge of the fabric. (1/3) – Steve Jan 24 '18 at 08:50
  • ...With a horizontal, "throat-less" hook, as well as preventing the weight of the wings from adding a counterbalance to the fabric in front of the hook tip, the reinforced fabric edge comes into contact with the arm of the hook (bearing an increasing proportion of the overall weight as the main fabric slides over the edge), and this causes the cup to dissipate as the fabric-edge approaches the hook-edge in a way that it doesn't with throated hooks. (2/3) – Steve Jan 24 '18 at 08:50
  • 9
    ...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 08:50
  • 14
    +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.. GitHub STOP HELPING ICE Jan 24 '18 at 22:39
  • 6
    @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 00:22
  • 1
    I can hang a towl on recessed hook... – Yakk Jan 25 '18 at 16:02
  • 2
    @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 is on Codidact now Jan 25 '18 at 20:12
  • @Yakk, in the case of a recessed hook, the towel fabric still needs to be appropriately balanced on either side of the hook. – Steve Jan 25 '18 at 20:52
  • The final paragraph simply asserts the reality: the towel fundamentally stays there because of friction. In the exact photo shown, if both surfaces were totally frictionless: *it would slip off*. Utterly straightforward and unavoidable fact. (Of course - obviously - "so what?" if it was "perfectly" balanced it could sit there, theoretically, in delicate equilibrium, until an atom touched it. This means nothing.) – Fattie Jan 29 '18 at 01:18
  • 2
    @Fattie, but the point is that the friction only has to oppose the imbalance - it does not have to bear the entire weight of the towel. Also, the balance contributes to friction, by pulling the towel taut over the hook-tip (in situation predisposed to slide-off, friction reduces as the slide-off occurs and imbalance grows, causing the slide-off process to run away with itself). An analysis that treats only friction (or treats it as predominant) would require absurd amounts of friction for the average towel, approaching treating hook and towel like velcro surfaces which they clearly aren't. – Steve Jan 29 '18 at 01:54
  • @Fattie, I just thought I'd add another point also - an imbalance in favour of remaining on the hook obviously isn't delicately balanced, so it is possible to over-balance the towel substantially against its front-fall by placing extra material in the throat. It is true that this still leaves the side-to-side balance in question, but it is easier to gauge the lateral balance of the towel by eye, and the creasing and deformation of the towel into the hook-throat (which is what creates its drooping wings) adds a margin of resistance to further sideways movement once the crease is imposed. – Steve Jan 29 '18 at 10:04
  • hey @Steve .. " but the point is that the friction only has to oppose the imbalance" - yes, of course. Obviously. You'd have to help me understand what you mean by that. Say we asked "How to measure the power of a car engine!" and you said "It has to do with the power coming from the engine." I'm just not sure what you mean Steve: what other possible forces can you be thinking of? If you have a wooden block on a slope, the force is .............. the imbalance. – Fattie Jan 29 '18 at 19:45
  • "friction reduces as the slide-off occurs and imbalance grows" ????? surely you are famililar w/ the difference between static and dynamic friction?? – Fattie Jan 29 '18 at 19:45
  • Again, "It is true that this still leaves the side-to-side balance in question, but it is easier to gauge the lateral balance of the towel by eye" - absolutely no offense, but (as I already explained) this is tremendously misguided. There's just one imbalance - it's just a force pointing in a certain direction. (Notice the image I posted in my answer: if North is towards the wall, in fact (I checked) the sliding-off force ("the imbalance") was SSE. I will try to make a post showing how you determine this, which is easy. – Fattie Jan 29 '18 at 19:47
  • @Fattie, you need only ask. When I say the friction only has to oppose the imbalance (not the entire weight of the towel), it ought to be obvious that the relative balance of the towel affects the amount of sliding force that any friction has to resist - you acknowledged this yourself, when you accepted that a perfectly balanced towel could be held entirely without friction ("until an atom touched it" in an unbalancing manner). So it's a bit late to be claiming you don't understand the role played by balance. (1/3) – Steve Jan 30 '18 at 02:45
  • 1
    Also I wasn't referring to static vs dynamic friction - I'm referring to the fact that an equally balanced towel uses its own weight to maximum effect to pull itself taut over the hook (adding to both static and dynamic friction), whereas an imbalanced towel does not. As the imbalance grows when the towel is in the process of sliding off, the friction against the hook is variable with the balance of the towel - and this variability is not covered by the concepts of "static" or "dynamic" friction. (2/3) – Steve Jan 30 '18 at 02:46
  • Finally, your point about there being "just one imbalance", that is true if we describe the balance in all directions in aggregate, but the nature of this situation makes it convenient to talk about the balance on two separate axes - not dissimilar to how when driving a car, we communicate about "acceleration", "braking" and "steering" differently, and we don't talk about "accelerating left" (in a steering scenario) or "accelerating backwards" (in a braking scenario), even though all these operations can be conceived thus . (3/3) – Steve Jan 30 '18 at 02:48
  • hi @Steve thanks for taking the time and we better let that be the "last word", cheers. It's a shame that SO is really not set up to handle the (few) "debated" QA like this. – Fattie Jan 30 '18 at 13:36
37

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.

1D towel

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.

2D sheet model

real towels

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.

James
  • 1,702
12

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:

enter image description here

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.

Floris
  • 118,905
  • 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. – Caleb Stanford Jan 26 '18 at 20:34
6

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:

enter image description here

  1. The hook has to be rough if straight. The rough surface provides friction against the slipping off of the towel.
Wrichik Basu
  • 2,870
  • 4
    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. – Caleb Stanford Jan 24 '18 at 04:39
  • @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 05:01
  • "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
5

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!

kamran
  • 1,659
3

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.

Chinese bronze garment hook (Daigou) Eastern Zhou Dynasty

Rob
  • 2,273
  • 1
  • 11
  • 31
-1

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:

enter image description here

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).

There are some really incredible howlers on this page, in the answers and comments:

"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!

Community
  • 1
Fattie
  • 1,048
  • 4
    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). – Caleb Stanford 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. – Caleb Stanford 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
  • 1
    Right you are :) But it illustrates one of the flaws in your arguments. – Caleb Stanford Jan 28 '18 at 23:25
  • hi @6005 - sorry I didn't realize you were the OP. This is now one of the most notorious set of incredibly incorrect answers on the site. I have tried again to explain it properly, in this answer. There are some astonishing "howlers" on this page! – Fattie Jan 29 '18 at 12:49
  • Take a deep breath and consider that you may be interpreting the question differently than everyone else. I will say that as the one who asked the question, I did not mean "what is the force that ultimately holds it up", but rather, "how do towels stay on hooks". I was looking for a detailed analysis of as many factors as possible. Just saying "friction" doesn't answer my question or help anyone else. – Caleb Stanford Jan 29 '18 at 19:14
  • Hi @6005. There's no need to "take a breath" :) "how do towels stay on hooks" Because of friction. If you are looking for an "inter-system" analysis of frictional forces (and that would only be on one particular hook shape and setup), the question is highly misguided. You would instead have to ask, on this particular system (with diagrams, etc), what are some of the frictional forces contributing to the overall system static friction coefficient. – Fattie Jan 29 '18 at 19:51
  • 1
    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
  • You're definitely right about that other answer (cms). It's got a ton of problems. Yeah, I am looking for a more detailed analysis of the frictional forces (how much friction is required). Just for a "typical" hook is fine. – Caleb Stanford Jan 29 '18 at 21:42
  • I see, perhaps a more specific pic. of the holder, or something like that. Purely as a curiosity, FYI in multiphysics systems (so, bullet physics) and even just in PhysX now (so, just run Unity and use their "cloth" simulation) the (essentially, finite element analysis lite!) simulations of such things are astoundingly advanced, and can run on your game PC's gpus. – Fattie Jan 30 '18 at 01:42