“Imagine the cow is a sphere…” is the punchline for many versions of a popular allometric physics joke. **Allometry**, by the way, is the study of how organisms scale. The canonical example is the 1950s horror film *Them!*, where giant mutant ants threaten a New Mexico town. The *real *horror is that the film writers didn’t understand that ants could not possibly grow to the size Shaquille O’Neal in the off-season.

Why? Consider the heuristic ant above, which we imagine is composed of three roughtly spherical sections with rod-like connections. Now what happens when we double the scale, of the ant? The mass of the ant goes as its volume, i.e. . Most of this mass is concentrated in the head, thorax, and gaster (the three round sections) which are held together with rod-like connections (neck and petiole). The shear strength of these rod-like bits to hold up the massive parts go as their cross sectional area, . Or, said in another way, the ant’s exoskeleton scales roughly as the area.

Thus at some scale the ant becomes too massive for its support structure to keep it together. Lawrence Krauss opens his book *Fear of Physics* with this parable, explaining that (1) one cannot expect to grow arbitrarily large cows for uber-milk efficiency and that (2) this is why brontosauruses had such small heads relative to their bodies.

This kind of analysis known to physicists as dimensional analysis. While one might think that dimensional analysis is only useful for making back-of-the-envelope estimates, we will see in a subsequent post how it can be used rigorously to understand the renormalization group. Undergrads will already be familiar with `rigorous’ dimensional analysis, however, in the context of mechanics, where the use of similarity transformations. As a quick reminder, we can take the force law:

,

and note that if we scale and then the above equation is still true. Thus we can conclude that the velocity of a particle’s motion in a central force field is halved when its mass is quadrupled. (Update, 22 Dec: this, of course, only holds when is independent of !)

The above example comes from chapter 2.11 of *Mathematical Methods in Classical Mechanics* by Arnold, which contains three of the neatest problems to be found in any physics textbook, which I reproduce here. (It’s worth noting that Arnold cites Smith’s *Mathematical Ideas in Biology* for these problems.)

1. A desert animal has to cover great distances between sources of water. How does the maximal time the animal can run depend on the size of the animal?

To be explicit, one can imagine the animal is a sphere (though the scaling holds even if it weren’t a sphere). The animal fills up on water at one lake and must run to the next lake, which is *just *at the limiting distance that it can run before dehydrating iteslf. The amount of water the animal can store goes as its volume, while the rate at which water is perspired away is proportional ot the animal’s surfac (e area, . We assume that the rate of perspiration is constant and hence over a time the animal perspires a volume . Setting this equal to the volume of water stored , we see that the maximum time goes as its length . To be exact, , but we’re not interested in overall constants.

2. How does the running velocity of an animal on level ground and uphill depend on the size of the animal?

The trick here is to think about the *power* (energy per time) used. On level ground, the main resistance to motion is air resistance, which goes as the cross sectional area times the squared velocity . The power is obtained by multiplying by velocity, so that we have . What is the power output of the animal? This goes as the heat output, which is proportional to the animal’s surface area, so that . Setting these two equal we see that . The running velocity of an animal on level ground does not depend on its size! (Well, it can be proportional to the logarithm of its size… as anyone who has done “Naive Dimensional Analysis” on divergent integrals would remind you.)

In the uphill case, however, the main resistance comes from gravity, with . Taking the slope of the hill to be constant, the power is then . Setting this equal to from the surface area argument above, we get . Arnold notes that a dog will easily run up a hill while a horse will slow its pace.

3. How does the height of an animal’s jump depend on its size?

The energy required to jump to a height is , thus has the proportionality . Muscles can produce a force proportional to (e.g. the strength of bones is proportional to their cross section), while the work accomplished by this force goes as . Thus setting the energy equal to the work, we find . Arnold notes that “a jerboa and a kangaroo can jump to apporximately the same height.” (Tall and short basketball players have roughly the same leaping ability, but being taller makes it easier to dunk.)

I hope you enjoyed that as much as I did.

December 9, 2008 at 7:08 pm

Cute problems. I especially like the bit about the horse and dog running uphill, but it brings out another point: there are lots of assumptions here, so that we can’t take the results on faith, but really have to do the experiment after all. For example, I don’t think wind resistance is the biggest contributor to power requirements when running over flat ground. Think of my grandmother easily riding her bicycle next to me, while I’m running very hard. Huge difference in power output, same wind resistance. Then when going up a steep hill, I can keep up with a well-trained mountain biker while on foot, and grandma is left far behind.

December 9, 2008 at 9:00 pm

Hi there Mei! Thanks for the comment. I agree that these results aren’t meant to be taken strictly when applied to real systems. I have to disagree with your counter-example, though, because comparing a person to a person on a bike `strongly’ breaks the assumption that we are just doing scale transformations.

Implicitly (though I guess it should have been explicit) we assumed that all animals have roughly the same running mechanism (legs that swing back and forth). Under this assumption, it’s reasonable to compare a dog and a horse. They’re both animals with four legs that run in more-or-less a similar way.

A bike has a significantly different propulsion mechanism. It might make sense to talk about a small bicycle with a small rider (child) versus a large bicycle with a large rider (with correspondingly larger tires) and discuss the scaling, but I don’t think it makes sense to discuss the scaling between different classes (bikes vs animals).

As far as the wind resistance, I also think this is a bit of a shakey assumption. But (e.g. for the case for a bike) the next effect I can think of would be friction, which would go as the area (in contact with the ground). Hence one would get the power from the frictional force going as . Comparing this to the power from the animal giving off heat () one still gets that the velocity scales as .

December 10, 2008 at 2:10 am

Thanks for the reply, Sparticles. I agree that it doesn’t make a lot of sense to discuss scaling when going from a bike to a runner so long you’re interested in a quality that’s dependent on the gate, but wind resistance should be dependent mostly on just the size of the object moving (it’s a bit more complicated, because the top part of the wheel is moving twice as fast as the rider, and the bottom part isn’t moving at all relative to ground). If wind resistance is slowing down the runner, it should slow down the similarly-sized biker about as much.

Could you elaborate on why friction scales with the contact area? Isn’t the friction force mu*N, with N the weight, and “mu” some constant that depends on the materials?

December 10, 2008 at 7:32 pm

Hi there, Mei. Let me start with the second question. The frictional force (static friction) depends on the amount of surface area available to cause friction. This is why racing bikes have such thin wheels. This is also why (partially) it becomes difficult to bike on flat wheels. Off the top of my head I’m not sure how to relate this to the statement F = mu*N, but I’ll think about it.

As for wind resistance (drag force), it can be shown from dimensional analysis that the drag force must be proportional to the square of velocity. Naively: force has dimensions ML/T^2. We want to write this as a function of fluid density p (M/L^3), area A (L^2), and velocity v (L/T). The only way to do this is to write F ~ p v^2 A.

While this does give the correct behavior, a more proper dimensional analysis would also take into account the viscosity of the fluid. One can read more about the derivation using dimensional analysis on Wikipedia. (See also their treatment of the Buckingham theorem.)

-S

December 11, 2008 at 4:13 am

Sparticles,

Thanks for the info on wind resistance.

However, I have to object to your analysis of friction.

A racing bike does not have thin tires to reduce the friction between the wheel and the ground. Because the wheel does not move relative to the ground, you cannot lose energy at that interface. The relevant friction that slows down a bicycle is, for example, friction in the bearing of the axle, which does not depend on the width of the wheels. In fact, you want the friction between the wheel and the ground to be as HIGH as possible, because that prevents slipping.

I think what’s going on here is that although friction is, as you say, proportional to surface area of contact, it is also inverse-proportional to pressure. If I have a box of books that I want to push across my apartment floor, it will be easier if I take half the books out first. That way, the pressure of any sqaure inch of the box on the floor is halved, and so the friction is also halved.

On the other hand, if I try to reduce the friction by piling the books twice as high in a box with half the contact area, this won’t make things any easier. The contact area is halved, but the pressure is doubled, so pushing the books is just as hard. This means that friction scales with the total weight, not the surface area.

As for the flat bike tire, it makes the bike more difficult to ride because the tire deforms more when it is flat. As the tire rolls, a different part of the material is on the bottom, so any given segment is deformed once per cycle. The flatter the tire, the more dramatic this deformation. The deformation of the tire is highly non-reversible, meaning it sucks up the energy used to deform the tire and turns that energy into heat, like a wad of Play-doh been thrown at wall. Some people will over-inflate their car tires to minimize this effect and get better gas milage, but it increases the chances of a blowout.

As for drag from wind, according to what I have read, it is approximately quadratic in velocity in the domain of fast movement/low viscosity, and approximately linear in the domain of slow movement/high viscosity. As you mention, including viscosity can change the dimensional analysis and allow linear drag force. My source on this is chapter 2 of the book “Classical Mechanics” by John Taylor.

But in comparing the bicyclist and the runner, the dependence of drag force on wind velocity is irrelevant, because I was using the example of bicyclist and runner moving at the same speed, but clearly expending highly different energy. They use a different gate, but the wind shouldn’t care much about what gate you’re using. The bike could potentially be more aerodynamic, but you don’t have to be “tucked in” to easily outstrip a pedestrian.

December 15, 2008 at 10:40 pm

Hi Mei — apologies for my delayed response. I think I agree with your analysis of friction.

I still do not think, however, that it is appropriate within the dimensional-analysis framework to compare people-on-bikes to people.

When does this dimensional analysis/scaling argument work? When the things that we’re comparing are reasonably similar. By “reasonably similar” I mean that the essential features are captured faithfully. The assumption of the dimensional-analysis/scaling argument is that animals running (up-hill or on flat ground) are limited by their metabolism, i.e. how much power they can generate. In the example of the bicycle going uphill, however, the limiting factor isn’t necessarily metabolism but rather mechanical (muscular) strength.

Consider, for example, how you feel when you’re jogging up a hill. It’s not as easy as jogging on a flat plane, but it’s the same kind of cardiovascular workout. One is limited by (at the risk of oversimplification) one’s breathing. Running up a steep hill, the first thing to limit your velocity is whether you get out-of-breath. With a bike, on the other hand, going up a steep hill one is more likely to feel the strain on one’s legs — i.e. one is limited by muscular strength.

Your original statement was that it is necessary to do experiments to test statements. This is of course true at a very fundamental level. What I’m trying to express is that one can also take the dimensional analysis seriously if one understands the limits with with the comparisons are valid.

December 16, 2008 at 1:04 pm

Sparticles,

I’m totally with you on the uphill thing. Some of my friends are runners and others are bikers. We have one particular long climb nearby that both bikers and runners use. We haven’t organized the race yet, but the best times recorded by bikers and runners up this course are both within one minute of 20 minutes.

Empirically, for a given distance, there’s some gradient at which runners and bikers are about even. Steeper than that, runners have the advantage. Shallower, and it goes to bikers. As you say, that relationship probably won’t come out of dimensional analysis.

All I was concerned with before was the question:

“Is air resistance the limiting factor to running speed over flat ground?”

I thought the answer was, “no”, and the bike was intended to provide an example to solidify that point.

Thanks for the week-long exchange – I learned a lot from it. It’s the first time I’ve checked out this blog, but I’ll be sticking around.

Mark

December 19, 2008 at 4:01 pm

Hi,

This is a nice post. The subject touches on several topics that interests me. Here are a couple of questions/comments.

In the case of your force law equation you write that

I don’t understand how you arrived at this conclusion. As far as I know in orbital mechanics the motion of the particle is independent of its mass m. The orbit is independent of mass m.

If you want to change the period you need to change the radius of the orbit. You can scale the orbit by raidus r and period t but not by m. So I am not sure your force scales the way you wrote it. Please correct if I’m missing something.

On the second problem, on the level ground, you cancel L but I don’t think they are the same L. One is cross section of the surface facing motion, the other total surface area. So if this animal were a cube the cross section would be the surface area of the face in the direction of motion.

So taking this into consideration and if D1 and D2 are distances per unit time for the animal 1 and animal 2 and C1 and C2 are corresponding cross sections we have

D1/D2 = C2^2/C1^2

If animals have the same aerodynamic profile C1 = C2 and D1 = D2 and both will cover the same distance in unit time.

You made power proportional to total surface area, not to the cross section, if S1 and S2 are total surface areas

D1/D2 = C2^2/C1^2 S1^2/S2^2

So the bigger animal will go faster unless he has really a bad aerodynamic profile.

Let me know any errors.

The third case with animals moving upward, we only have L (not C and S as above) so the power and bulk are surface and volume

D1/D2 = L1^2/L2^2 L2^3/L1^3

and cancelling

D1/D2 = L2/L1

The same result as yours, the bulkier animal will go slower. But this solution is simpler because I didn’t need to use several terms such gmh that you introduced but did not really enter the problem.

My interest lies in understanding where physics is in a problem like this. In this case we model an animal with geometry, physics does not enter (I think). For instance, as I am sure you know, Galileo was the first person to notice that animals do not scale linearly. And he reasoned geometrically and not by using physics.

Thanks again.

December 19, 2008 at 6:42 pm

Hi Z,

1. The motion of a particle in a gravitational field *does* depend on its mass. For a fixed gravitational potential (e.g. the Sun), there are three things you can fiddle with: the mass of the particle (M), the period (T), and the radius of the orbit (R).

If you increase the mass, then you increase the gravitational force. Thus if you want to maintain a stable orbit, the particle must have a greater tangential velocity, i.e. a shorter period T.

So let’s understand what I was trying to say. Consider the equation that I wrote above. is independent of . Solutions to this equation represent “physical” (i.e. allowed) dynamics. In particular, if we’re given a solution where with some trajectory, then we look at the equation and note that if we double then the left hand side is multiplied by a factor of 1/4. We can compensate by quadrupling . Thus the equation with doubled and quadrupled is still valid, and hence that represents permissible dynamics.

What dose it mean to “double “? This simply means things that have dimension of time are doubled. In particular, the velocity — which goes as the inverse of time — is halved.

2. One of our implicit assumptions is that the various length dimensions of an animal have fixed proportionality to the animals “characteristic length scale.” So indeed, the cross sectional area isn’t *exactly* , but rather some number for some constant . We are assuming that this number is the same for each animal we’re considering. This is very close to true when comparing two animals of the same species but different sizes, and still roughly-true for comparing animals of the same general type (e.g. four-legged mammals like dogs, cats, lions).

3. I’m a bit confused about what you’re doing. The tricky thing about these dimensional analysis problems is that one has to compare quantities in a way that can sometimes be a bit tricky. In problem #2 we invoked power because we wanted to compare velocities given a knowledge of how forces scale. In problem #3 we don’t use power because we don’t care about velocity and don’t know anything about it.

I’m not sure how much ‘physics’ there actually is here — the point is that this really *is* geometry. One can certainly apply these scaling laws to physics (e.g. the problem with the orbits), but this really just comes from an understanding of the physics rather than representing something new.

December 20, 2008 at 9:43 pm

Hi,

“1. The motion of a particle in a gravitational field *does* depend on its mass.”

It’s well-established that bodies of different masses all fall with the same motion and since orbital motion is free fall your statement amounts to denying the free fall law. Are you saying that we should go back to Aristotelian physics and believe that heavier bodies fall faster?

If you’ve discovered that free fall and orbital motion depend on the mass of the body and you can prove this I think you should publish it and get credit for it. This way your Nobel prize will be ensured π

December 20, 2008 at 10:03 pm

Unfortunately for my Nobel aspirations you are still mistaken.

What you’ll find in high school physics textbooks is that the acceleration of objects in a gravitational field is independent of their mass. This is the statement . This force law comes from the Taylor expansion of the inverse-square law with respect to the radius of the Earth, :

The order-1 term is precisely what we mean by . The higher order terms are tiny since is so much larger than any displacements one would work with on human scales.

To put it plainly, the “constant acceleration” law only holds when one is near the surface of the Earth, i.e. when one is at nearly-constant radius.

Anyway, I would suggest skimming through a physics textbook for a refresher on these topics.

December 22, 2008 at 8:28 am

Sparticles,

You wrote that,

βThe motion of a particle in a gravitational field *does* depend on its mass.β

Please take a look at the equations in your last comment. In physics we say that when a term — such as the mass of the satellite m in your equations — appears in both sides and we cancel that term we say that the equation is independent of the term that we eliminated. Accordingly, looking at your equation, a physicist would conclude that the motion of a satellite is independent of the mass m of the satellite. This is well established.

Your equations prove that the motion of a satellite is independent of its mass. Are you still saying that

βThe motion of a particle in a gravitational field *does* depend on its mass.β

You might also want to check this book.

Do you agree with this or do you still claim that the motion of a satellite is dependent on its mass? In that case you would be claiming new physics that is not accepted generally.

December 22, 2008 at 11:11 pm

Ak, of course — my apologies to Zeynel. I wasn’t thinking and ended up spouting rubbish! Of course Z is correct. The scaling analysis that I gave was based on the equation

.

The scaling analysis (changing mass and time) works, provided that is independent of mass; otherwise also scales. The gravitational force is of course dependent on mass, so the scaling analysis does not hold. (This does work for other central force fields, e.g. the electric force of a point charge.)

I’ll make the appropriate update to the post, but I’ll leave the string of comments as a permanent reminder that I should think before I post anything. π Thanks to Z to pointing this out.

January 2, 2009 at 2:43 am

[…] already said a few cursory words on dimensional analysis and renormalization. It turns out that we can use simple dimensional analysis to yield some insight […]

December 22, 2009 at 9:37 pm

Contrary to some notions, well explored articles still fetch in reviewers like me. You showed clear understanding of the topic matter and my opinions are now complete after reading your post. Please sustain up the sound work and i will subscribe to your rss feed to be informed of any emerging postings.

May 28, 2010 at 1:42 am

Wow am I really the only reply to your amazing writing.