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When a person walks, the only force acting on him is the force of friction between him and the ground (neglecting air resistance and all). The magnitude of acceleration due to this force is independent of the mass of the object (longer legs have more mass). Hence the person should move with with a velocity independent of the length of his legs.

But I have heard (also observed) that people with longer legs walk faster than ones with shorter legs. If that is true, then why?

One can argue that the torque about the pivot due to friction is more in case of longer legs, But then the torque due to gravity (when one raises his leg to move), which opposes the frictional torque, is also more for longer legs. And why would these torques make a difference anyway, as they have no effect on the acceleration of the center of mass?

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  • $\begingroup$ It is a relevant question, as students often don't discriminate between maximum static friction and actual static friction. Unless it is duplicate, leave it as it is. $\endgroup$
    – Pygmalion
    Commented Apr 17, 2012 at 17:03

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I think the simplest model that may be useful here is to treat the legs as simple pendula. In "steady state" comfortable walking, it is reasonable to assume that the legs oscillate close to their natural frequency. That is, the forward contacting leg lifts allowing the rear to swing forward freely over the stride. For a (simple) pendulum with:

$$\omega = \sqrt{\frac{g}{l}}$$ the velocity along the ground will be: $$v \propto l\omega = \sqrt{lg}$$

Note that this result is independent of the mass of the walker and the ground contact forces.

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    $\begingroup$ I like this answer - it gets to the point in the simplest way. The effort of moving the legs forward is minimized if we let them swing freely, at which point they move with a speed proportional to $\sqrt{l}$. Not only does this show that longer legs work better - it even gives a "natural walking speed". Note your expression is missing a factor $\frac32$ (see hyperphysics.phy-astr.gsu.edu/hbase/penrod.html) since a leg is a rod, not a point mass on a weightless string; also, it is a compound pendulum which changes the math further (but not the principle). $\endgroup$
    – Floris
    Commented Aug 15, 2014 at 12:59
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Think about the limiting cases. An ant-sized marching band would take a long time to march the length of a football field. The reason they take so long has nothing to with friction - it's just that their legs are smaller and so each stride moves them a shorter distance.

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  • $\begingroup$ Yes, but their legs weight less and the torque required to moev them is much less, so why can't they simply move their legs faster? $\endgroup$
    – John Rennie
    Commented Apr 18, 2012 at 8:02
  • $\begingroup$ From their perspective their bodies are proportioned exactly as ours, and so the timing of their strides are the same as ours. Naturally this is an extreme example - it is just meant to indicate that someone with shorter legs (say, a child) will walk slower compared to someone with longer legs. They could, of course, hasten the timing of their strides, but that would require a noticeable extra input of effort $\endgroup$
    – kleingordon
    Commented Apr 18, 2012 at 18:38
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    $\begingroup$ @JohnRennie, they do. But it's not enough to overcome the scaling. kleingordon, the timing of miniature people's footstrides would not be proportional. It is nonlinear. See tmac's answer. $\endgroup$
    – geometrian
    Commented Oct 30, 2014 at 17:43
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Interesting question. I had a Google around and came across http://silver.neep.wisc.edu/~lakes/BME315ScalingWalk.html, which seems a reasonable discussion of the mechanics (very simplified). The conclusion is that the walking speed is proportional to the square root of leg length, so taller people do walk faster but the square root dependance means it's not not much faster.

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    $\begingroup$ From 12 to 14 I was very athletic, and excelled particularly in the long jump, my style chosen as an example for other students! Then the growth spurt came and I realized I could never compete since other girls grew much longer legs and could start jumping up to 5 meters, whereas I, with my short legs, good style or not, could only reach 4m or so :( . I then concentrated in mathematics and physics :). $\endgroup$
    – anna v
    Commented Apr 17, 2012 at 17:18
  • $\begingroup$ But you could outperform them by jumping over pole, leaving your center of pass under pole ;) $\endgroup$
    – Pygmalion
    Commented Apr 17, 2012 at 17:23
  • $\begingroup$ The simple pendulum mechanism works well if the person is walking casually but if we consider an athlete running a race, factors like muscle power have to be taken into account. $\endgroup$
    – Megha
    Commented Apr 18, 2012 at 14:52
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Note, in your problem you have static friction, and for static friction

$$F_\text{sf} \le F_\text{sf,max} = \mu_\text{sf} N = \mu_\text{sf} m g.$$

Note "less or equal" in the equation above, which means that actual static friction is less or equal than maximum static friction. So when you are walking, most of the time "less" sign applies, which means that mass of the body isn't relevant for the friction!

(Example: if you are standing still, $F_\text{sf} = 0$, despite the fact that $\mu_\text{sf} m g > 0$.)

People with longer legs walk faster simply because they are making bigger steps. I guess period between two steps for casual walking is some kind of a constant for all people... maybe has something to do with our heart beats.

Dynamic friction is different, there is always "equal" sign:

$$F_\text{df} = F_\text{df,max} = \mu_\text{df} N = \mu_\text{df} m g.$$

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  • $\begingroup$ Oh right! I don't know why I neglected that(maximum and actual static friction acting). $\endgroup$
    – Megha
    Commented Apr 17, 2012 at 17:08
  • $\begingroup$ No problem. Students of traffic sciences have problem with this difference, and it is a very very important issue for them. $\endgroup$
    – Pygmalion
    Commented Apr 17, 2012 at 17:15
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Sorry to contribute almost 2 years after question was asked, but I can't help it the Google brings me to this discussion, right now ! ... ;-)

The question is strange or weird, in the way that asker start presuming something about "The magnitude of acceleration", as if with the moving legs, one must make abstraction of the whole body ?

Accelleration & speed ? - While walking, as we look ONLY at horizontal movement.. (see end of contribution), there is only an accelleration at start, then at constant walking speed, horizontal accelleration of the total mass = zero. We all know that, so, why starting with the confusion ?
- Yes, legs "(speudo) swing" involve constant accellerations (+X to zero in absolute coordinates, from +X/2 to -X/2 relative to the total body mass gravity point). It is clear that if these legs had a futile mass of their own (like with birds legs, or insects..) the lost momentum capacity to get the energy delivered by the muscle to move the total body forward (as a non efficient fraction of total enery used), would be much lower. However, there we've got nothing more, nothing less then a personal muscular strenght level, interferring with the motion. If longer + heavier legs provide more power at the same time, the loss to total speed must be compensated... equally, OR partially, OR in excess ... we can't say much on the individual muscularity of a person !!

Skeleton ! You cannot change that. At the other hand, it's completely logic that walking speed is determined "most" by the lenght of the legs, and the proportion between upper & lower legs. - Total leg lenght is offering the classic lever based speed increase at a constant power supply. (given : air resistance is futile at low speeds) (That's why we also drive a bike, to go much faster with the same leg power available) - If a classic range of proportions between lower and upper leg is about 5 (lower: floor to knee) to 3.4 à 4,0 ... (upper: knee to hip joint), then it is clear that not just the total leg lenght plays it's role, but also this often disregarded proportion, allowing for better or worse mechanics to get the momentum in a "double coordinated rolling motion" be transformed in linear motion. In other words : some people are better build then others, purely mechanically, for walking or running, and nothing you can change about that.) - Now, even more spectacular can be the actual joint sizes and angles, specifically at the hip. A 0.5 inch, 1 inch, 1.5 inch... more sidewards placement of leg bone axis, related to the joint movement center, can make a fenomenal difference in the way the muscular contractions are transformed in momentum. You could fail to use optimum power, practically, just because the momentum hits the (strain & pain) limit. Again, lever forces, but here in a rather negative way, to many of us. Reason why actually MOST people are slow walkers and runners, EVEN when having the same lenght of legs or same amount of muscle. - Same spot : the actual attachment/movement angle at the joint, differs a lot by individual, and contributes to the former effect (better or worse transformation in momentum) - The global flexibility at this hip joint, also determines the pace length we COULD do, even making abstraction from the leg length. It's clear, that a person which skeleton offers a larger "comfortable" moving angle, will be favoured.

Pendulum ?? ... complex muscular machine. - The link given above ( http://silver.neep.wisc.edu/~lakes/BME315ScalingWalk.html ) with the conclusion "the walking speed is proportional to the square root of leg length" is very interesting indeed, from (however) the mechanics within a "dead machine" model. Legs only profit little from the pendulum mechnics, and are driven with the power from within, not from outside the so called pendulum system. - We all know what happens with a constant swinging pendulum : stop the energy supply, and it's swing speed will "die out" only very, very slowly, on the little friction there is. The comparison with legs misses the point there completely, because : in walking, apart from moving forward at a mostly constant speed + a friction, we are CONSTANTLY lifting ourselves up, at every step ! Lifting, falling, lifting, falling. Both the whole body, and the legs individually. We forget about it because it's visually not a great deal (unless you start filming it !!) Most of our energy is lost in "climbing" a never ending sinus. 4 legged animals make less vertical sinus. 6 legged again less ... A bike is the ideal ! ... - So where the pendulum mechanics model got lost completely, thus, is that legs do NOT swing free, they are lifted in double semi circular motion, placed on the ground, then LIFTING the whole body weight on one leg while letting it continue "falling forwards", etc etc I think that mr. Rod Lakes , from the linked article/research , should review his complete model... het just started with a false comparison.

I think, it must be clear that walking speed has so MANY parameters, and that the 4-fold dimentional approach of the skeleton mechanics is determining the baseline !! Speed then, could be augmented in training (muscle and metabolics) but will never be able to drive away much from what the mechnical provision basically offered. A shortlegged person, with bad upper/lower proportion, and negative influencing hip joint axis extention + angle, will never walk "fast" even if he is bulking with muscle ...

Cheers

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  • $\begingroup$ Hi Giwreh, welcome to Physics.SE. It's fine to answer old questions. Your non-standard punctuation and spacing makes your answer a bit hard to read though. $\endgroup$
    – Brandon Enright
    Commented Feb 26, 2014 at 21:38
  • $\begingroup$ Thank you, Brandon. I've really no idea what you mean with 'non-standard punctuation and spacing'. If there exists such a standard, please offer me a link. English is not only NOT my native language (only 3rd language), I'm also dyslectic. $\endgroup$
    – Giwreh
    Commented Feb 26, 2014 at 23:22

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