# Why is it easier to walk diagonally upstairs

Try it yourself on a set of wide steps! Work is given by

$$\int_C \mathbf{F} \cdot d\mathbf{x}$$ where $C$ is a path integral. In this case I think $\mathbf{F}$ is a rotational vector field because the stairs are essentially a set of discontinuities. This would mean that the integral is path dependent.

Here is why this seems like a paradox: if you where to climb the same height over a smooth hill, then the integral is path independent since $\mathbf{F}$ is "smooth" and irrotational. This means that the amount of work done never changes given the path you take, hence if you take a longer (diagonal) path you must exert less force over that distance than if you where to take a straight path. In the case of walking up stairs, you must always cross the same number of stairs no matter the path you take. All other distance you make is along the horizontal parts of the tops of stairs and hence requires much less force. This would seem to suggest to me walking up stairs always requires the same force to scale the vertical parts of each stair.

Am I hallucinating that walking diagonally up stairs is easier?

• I don't know whether walking diagonally makes it easier, but if it does, it may be because one's step length is closer to a standard walking pace length, so it feels more natural. Feb 23 '14 at 8:28

Ascending a hill diagonally means that while the work is the same, the power expenditure is lower (but it takes longer), which is why it is easier.

• But you cross the same number of stairs and each stair is about the same height so wouldn't you need to exert the same amount of force walking up each stair no matter how you walk?!
– ldog
Nov 29 '13 at 19:20
• It is work per unit time ( power) that tires the muscles, they have a limit on their response. It takes more steps but it is easier on the legs. Nov 29 '13 at 19:41
• I'm agreed with lionelbrits and anna on this. The biomechanics of the actual work done by a human body when moving around is a bit more subtle than a simple path integral definition, and the reason it seems easier is that the climb is spread out over a longer distance, and thus a longer time, which makes it feel like you're exerting less effort (because you are exerting less effort per time). Nov 29 '13 at 20:28
• Whether power is lower depends on speed. It needs to be explained why the speed will not increase on the shallower slope to make up for the reduced force. Aug 30 '14 at 22:23

This is an interesting observation and I reckon it just shows that there's a difference between physical work and what we subjectively consider as work.

Of course, you assume that the force field is conservative, neglecting any friction, but even when we take friction into account, it does not solve the mystery.

I find the example of the bicycle driver who's using the whole road and goes diagonally up a steep hill even more appealing, but maybe that's just me.

Consider two bicycle drivers going up and down, both put the same altitude difference behind them. Every experienced cyclist will tell you that it is more exhausting to go steep up rather than have mild but long ascents. But physically, this is not true! While downhill you don't need to pedal anything, it is only uphill that counts. But the longer you go uphill, the more you suffer from friction. So physically it would be best to do the ascents as short as possible and therefore as steep as possible.

• It is a matter of the response of the muscles to the power expenditure ( work per unit time). Their output is limited and fatigue sets in. Nov 29 '13 at 19:44

I would say this is physiological and psychological rather than dynamics phenomenon.

The total work done is slightly higher when you use longer path. The work needed to increase the potential energy is the same and a tiny bit extra is taken for overcoming the (comparably very small) drag.

So it feels easier rather than really being so. I can think of two reasons:

• Muscles can provide higher immediate power output than what the breathing cycle can supply. But when they do, they quickly fatigue and hurt due to accumulation of the intermediate product of the breathing cycle, lactic acid. The peak power when climbing stairs is almost always higher than the long term limit. By going diagonally, you reduce the frequency of the up-steps giving the muscles more time to take up oxygen and complete the breathing cycle, thus reducing fatigue.

Or in other words, it is the average power that matters and it does not matter as much whether the power output is smooth. When walking it never is and the muscles are used to that.

• Humans are not equipped with gauges for monitoring the exact power they exert. It is much easier to estimate speed, so that tends to be used as a substitute. The effect is that you probably tend to put more effort (power) to climb in steeper path (no matter whether smooth or stairs) it attempt to maintain forward speed.

This can be often observed when trekking. Inexperienced trekkers will usually rush up hills and be very tired at the top and need long rest while experienced ones will slow down to maintain power and eventually faster because they won't need rest.

Basically, the answer is that you have to do more work when you go up a set of stairs vs going up a wheel chair ramp or like you mentioned, a smooth hill. I would think this is caused by the up and over pattern of stairs where you would have to go straight up against gravity and then over instead of moving your feet like one would on a ramp. The total distance for the stairs, over all and including the horizontal and vertical axis together, would be greater for the stairs than for the smooth hill where your feet only go up, and the dist. horizontal is barely noticeable. Perhaps I'm confused on your question though.

• No. You have to do less work the shorter the path is. The work that goes into climbing is the same in both cases, because only the vertical component contributes to that and longer horizontal path means more work needs to go into overcoming drag (which is small comparable to the work needed for climbing). Apr 1 '14 at 20:42

It's because of how the body is oriented when going up diagonally which allows for the lateral thigh muscles to contribute to the effort, versus only the frontal muscles when going straight up. Since the effort is spread over more muscle areas, it feels less strenuous.

• Is this because your foot makes contact differently with the step? (if you can place your foot fully on the step vs. just the toes / front of the foot making contact)
– Joe
Apr 2 '14 at 0:57