2016-01-20 21:59:57 8 Comments

(Inspired by Why are four legged chairs so common?)

I've been wondering for a while... Why do most wheeled office chairs have 5 wheels?

My guess would be that while stability vs. simplicity results in 4 legs, adding mobility to the equation may result in the need for 5 wheels.

Edit: This question is about a *mobile* chair

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## 4 comments

## @Rocco Jibraltar 2018-03-25 03:43:53

i'm pretty sure that if you check the OSHA (the U.S. Occupational Safety and Health Administration) regulations for chairs found at https://healthfully.com/osha-regulations-workplace-chairs-5916239.html you will find the ultimate answer to this question.

in addition to specifications pertaining to the back, seat and armrests, they also have regulations about the...

Base

Your workplace chair must have a strong, five-legged base with casters that are suitable for the flooring type of your workstation. Workplace chairs with four or fewer legs may give insufficient support. Chairs that have no casters can make it hard for you to position your chair close to your desk. This could increase bending or reaching to access your workstation components, leading to fatigue and muscle strain.

## @Mike 2018-03-25 04:13:38

+1 even though this doesn't address the physics underlying the recommendation. Some friends had a small engineering company and got an OSHA inspection where the inspector found just one problem: that a wheeled four-legged chair they (and even I) had all used numerous times and had never had a problem with needed to be replaced by a five-legged chair. As soon as the inspector left, the employees gathered around remarking how ridiculous it was. One of them said "I love this chair", sat down in it, and immediately fell over.

## @Karl 2017-09-20 22:31:27

I don't know the history, I am just speculating. The chair in the picture has features that you don't find on a typical 4 point chair: It has casters, it has a swivel seat, and it has a springy reclining back. All of these make the chair more likely to tip over without warning. Lets assume that you roll the chair forward so all the casters are rotated to the back, then lean back. At the moment that it tips onto the two back most casters, the base will swivel to the least stable orientation, and the casters will rotate to form a smaller circle. In a 4 point chair, the base can rotate from diamond to square, and the casters rotate to make the square smaller, with no warning. A rigid 4 point chair is very predictable, and yet we have all tipped them over. Add the swivel and casters and the chair becomes an unpredictable killer. To compensate, the base circle must be increased and/or the number of casters must be increased.

## @Rolazaro Azeveires 2017-09-20 23:12:20

Thank you, but note that the accepted reply already includes stability as a main point

## @Floris 2016-01-20 22:34:13

Consolidating some of the points made in the answers to the question you linked, and comments:

When constructing a chair, 4 legs is easy when you use traditional (wooden) construction - 90 degree angles, and easy to make stackable. A little bit harder than three legs because you have to make sure they are all the same length (or the chair will wobble).

Once you have a "office chair" with a hydraulic center post, the construction argument goes away. That leaves us with greater freedom to pick number of legs. The considerations are:

All engineering design is a question of tradeoffs; in this case, I think that the first point argues for fewer legs, and the second / third point for more legs. The question then becomes: what is the additional value, and the additional cost, of one more leg? Below, I calculate the cost of adding more legs for the same stability and cost - this makes some assumptions but concludes five is indeed optimal.

But there is another important factor (tip of the hat to my daughter for this concept): when the floor is uneven, a chair will be not be supported by all its legs - it will "wobble". Now if you have four legs, this wobble will happen along one of the diagonals of the square, and this line will be below (or very close to) the center of gravity. That makes the energy needed to go from one side to the other very small. When you have five legs, the center of gravity is always displaced relative to the line of support. So five legs provide greater stability on an uneven floor. As you add more legs, the "diagonal of support" gets close to the center. Even numbered regular polygons always have the potential of having the line of support going through the center, making them the worst choice (incidentally this shows that a trapezoidal arrangement of four legs is slightly better than a square... you will sometimes see that, and now you know why).

All of that makes five the optimal number of legs - good stability in all directions. Note that from a construction perspective, it only makes sense to give a chair five legs when you start with a (metal or plastic) center post - the older (square wooden legs) construction makes four a more sensible number as the other answer stated. Once you want the chair to have vertical adjustment, a single center post makes sense - and then you have the flexibility to choose the number of legs.

Finally, a reference from a large supplier of office furniture:

UPDATEI thought more about the problem of optimization, and think I can explain that five legs is best.

Assume that the chair has to support a constant weight $W$, and that we want a constant stability. Stability is determined by the shortest “tipping distance” $D$. For a radial distance $R$, a chair with $n$ legs has

$$D = R \cos\frac{\pi}{n}$$

So we can define a “stability factor” $S=\frac{1}{R\cos\frac{\pi}{n}}$

Thus, for constant $S$ we get $$R\propto \frac{1}{\cos\frac{\pi}{n}} \tag1$$

Next, we look at the stress on each leg. The stress will be greatest when the tipping torque $\Gamma$ is directly in line with just one leg. At that point,

$$\Gamma = W\cdot R$$

Now we want to calculate the shape (section) of the leg that can support this torque. The maximum stress $\sigma$ for a rectangular beam of width $w$ and height $h$ is proportional to $wh^2$, and the mass of the leg of length $R$ is $whR\rho$; if we assume a constant aspect ratio $\frac{w}{h}$, then mass is proportional to area times length:

$$m \propto h^2 R \tag2$$

where the first term is a function of the strength, and the second term a function of the stability.

Similarly, for given torque $W\cdot R$ we can write the bending stress as

$$\sigma = \frac{My}{I}$$

where $M$ is the bending stress, $y$ is the perpendicular distance to the neutral axis, and $I_x$ is the second moment of area about the neutral axis $x$. For a rectangular section, $y \propto h^4$.

For constant $\sigma$, the maximum will occur at the outer edge of the beam where $y=\frac{h}{2}$, leading to

$$h^3 \propto W\cdot R$$

For given weight $W$, it follows that

$$h\propto R^{1/3} \tag3$$

Substituting $(3)$ into $(2)$ we get

$$m \propto R^{5/3}$$

For constant breaking strength, we get the total mass of $n$ legs:

$$M = n\cdot m \propto n R^{5/3}$$

For constant stability, we use $(1)$ to obtain

$$M \propto \frac{n}{\cos^{\frac53}\frac{\pi}{n}}$$

We can evaluate this for n between 3 and 7, and obtain $M$ as a function of the number of legs:

This shows that indeed the structure with five legs needs the lowest mass to support a certain torque - if we can equate "mass" with "cost", and stability is indeed the main driver, this proves that a chair with five legs is optimal.

## @March Ho 2016-01-21 06:00:02

This answer would be improved by a citation that 5 legs is in fact the optimal level of cost vs comfort. It seems that the current answer merely assumes that it is the case. Furthermore, a naïve view of this theory would suggest that there should exist a market for very expensive chairs with >5 and cheap chairs with <5 legs.

## @Floris 2016-01-21 23:13:25

@MarchHo - thanks for your comment. I have added a calculation (no reference, my own work and potentially flawed) which shows that 5 is indeed better than 4 or 6, 7, ...

## @docscience 2016-01-25 00:41:56

Nice analysis. This is the kind of answer I was looking for when I posted this: physics.stackexchange.com/questions/146623/… . But it was closed as being off topic. I suspect for the chair problem that 4 legs can provide the same stability as 5 legs - but the legs would have to be

longer. And the added length perhaps conflicts with another constraint of the wheel base not banging into the base of the desk or other chairs. There are likely other design constraints than just stability.## @docscience 2016-01-25 00:44:38

And I have to ask - why didn't this question get closed down? What's the difference in analyzing stability of animal legs and chair legs? The biologists not likely to think of the problem the way physicists do.

## @Floris 2016-01-25 06:50:43

Four legs leads to "wobble" on any uneven surface; five legs means that any uneven surface may reduce you to support by three legs only, but those legs will be distributed to either side of the center of mass. That means the chair is still stable. As for closing or not - your earlier question was more (or "felt more") biological in nature. The system mostly works but it's not perfect.

## @Rolazaro Azeveires 2016-01-25 23:45:16

Wow. Looks great. I'll likely accept it, but please allow me some (maybe long) time to read it. Thank you!

## @Floris 2016-01-26 00:10:42

Take your time - it took me a while to construct... incidentally I just made an edit to better highlight the difference with mobile (vertically adjustable) chairs, further strengthening the difference with the other question (the one that isn't a duplicate)

## @Rolazaro Azeveires 2016-01-30 22:23:57

accepted - I can't be sure this is correct (if I was I wouldn't ask :-) and I still wish I'll reread the reply with pen-and-pencil, I have to check about "bending stress" which I am not at all familiar. As a note, I wouldn't be surprised if the wheels themselves (here assumed to work flawlessly) had something more to add. Very nice effort, thank you - now I can work in peace, instead of staring at the chairs next to me wondering :-)

## @Dean MacGregor 2016-01-21 16:46:49

The difference between an office chair's 5 wheels/supports and a regular chair's 4 legs is that the latter has all of its load going straight down. The legs only need to be strong enough not to shatter. In fact, a chair could easily get away with 3 legs but for the stability. In contrast the office chairs legs support load perpendicular to their orientation. They need to be strong enough not to snap. Therefore they need more supports to support the load from above.

Another reason to have 5 legs as opposed to 4 is that the wheels aren't in a fixed position relative to the chair. That means if there were only 4 wheels the tipping axis could be anywhere so a sitter could lean in a direction and tip without realizing it. In a 4 legged chair a sitter can lean diagonally and not worry about tipping. Since office chairs can't confer the certainty of where the tipping axis is, it makes sense to make it less able to tip.

## @Rolazaro Azeveires 2016-01-25 23:47:00

+1 - I'll likely accept Floris' answer, but I think you have a good point too. Thank you

## @Dean MacGregor 2016-01-26 14:16:37

@RolazaroAzeveires holy moly, yeah I can't compete with all those formulas