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H2Bro

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**Thought experiment:When is it "cost-effective" to add a staircase Vs more floorspace**

This is a thought experiment I, erm, thought of.

Imagine a cylindrical (upright) office building with a central staircase at the focus. Workers exiting the staircase walk to their desks, and eventually walk back to the staircase. At what point does the energy required to walk from the staircase to the outermost desk, i.e. building circumference, exceed the energy required to walk up an additional flight of stairs?

I posed this while thinking about whether there are "optimal" proportions to human habitats. I lack any schooling in physics, so after buggering about in wikipedia I came up with the following calculations. Do please look it over/offer revisions because the value I found surprised me.

Assumptions

- office worker is 70kg

- 70kg person walking @ 4km/hour burns 216 cal/hour, or 54 cal/km (taken from http://www.brianmac.co.uk/energyexp.htm)

- This translates into walking efficiency of .226 Watts per meter (

- horizontal length of the staircase, i.e. length of stairs added together, is 10m

- floor-ceiling height is 2m with no discernible floor thickness

- People travel 1/3rd as fast when going up stairs (as viewed from above)

So! Let it begin.

__Energy to go up the stairs__

F(stairs)= energy to ascend 2m + energy to traverse 10m

Lifting a 700N person up 2M requires 1400J of energy.

A person approaching the staircase at 4km/h would drop down to 4/3km/h, meaning it takes 0,0075hours or 27 seconds to travel the 10m (seems about right).

1400J over 27 seconds = 51 Watts

Plus the energy required to traverse 10m, as defined above = .226 Watts X 10m = 2.26 Watts

Energy to go up staircase = 51 W + 2.26 W = 53.26 W

__Energy to travel to desk__

Now, it would be most efficient if a person only walked up the flight of stairs if it took more energy to walk to their desk and back (a distance of 2r), in other words when:

F(desk) - F(stairs) [itex]\geq[/itex] 0

Since we know the value of F(stairs) = 53.26W, and that it takes .226W to walk 1m at 4km/h, finding how far one could walk for 53.26W of energy is:

53.26W / .226W = 235 meters

Because this distance is a round trip, our ideal worker would not want to have a desk further than 117 meters from the stairs.

I did a similar calculation for a person walking at 7km/h, which burns 411 calories/hour, or 58cal/hour = 245 W/km = .245W/m. Using this value I find a maximum distance of 397/2 or 198m (!).

Now, this value seemed quite high (and an immensely large office building, at that), and I have a suspicion I didn't correctly translate/convert the energy units all the way through. Also, perhaps you can see places to refine the calculation, i.e. energy cost of standing up/sitting down, stopping/starting, or what-have-you.