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I am trying to help my daughter do her physics homework. Her assignment includes calculating Calories worked off while exercising.

The formula she is using is Work (Joules) = Force (Newtons) * Distance (Meters).

I calculated that my 65 kg daughter 4.448 is 618 Newtons. And her walking up 3 floors of stairs was 9.144 meters. Multiply those and I get 5,653 joules. Divide that by 4.18 and I show that 1,352 calories were burned per trip. Equating to 1.3 Calories (Kcal) per trip.

She did 15 trips in 30 min, I get about 20 Calories burned for the whole activity. (Note, this is just "up the stairs" time. Down the stairs time is not represented in these numbers.)

Here is the part I am not understanding. If I go to any diet and exercise site I get told of a Metabolic Equivalent number. (It is 7.5 for climbing stairs.) It says to mulitply this by her weight to get Calories burned per hour. So, 7.5 * 65 = 487.5. Divide by 2 because it was only a half hour = 243.75 Calories burned.

So 20 Calories by the physics way, and 243.75 Calories via the "exercise calculation" way.

Why does the Newtonian Physics result differ so greatly from the "exercise calculation"?

I assume that the "exercise calculation" is likely including the inefficiencies of the human body in it, but it still seems like a significant difference.

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A person is not as simple as a spring or rolling ball. We’re complex bio-chemical systems that have to consume metabolic energy to do things:

  • you use calories holding up a weight at arms length. A table does that without any energy use at all
  • you use calories going down stairs; not as much as going up, but some. A mechanical system needs energy to go up, but can get it back coming down.

Your muscles consume chemical energy to move; that’s the “metabolic” part of the calculation. From a physics point of view, that can be positive or negative mechanical work (I.e. energy transferred), but some also goes into heat (temperature rise of muscle plus latent heat to perspiration and respiration), and some is just consuming chemical energy to create chemical by-products like lactic acid that don’t produce further useful work.

N.b. Although it seems inefficient, your muscles are in many ways remarkably efficient. If you compare them to a (perfect I.e. Carnot) heat engine, you’ll discover that muscle chemistry does a better job of creating useful work than a combustion-based heat engine ever could. For example, the Carnot efficiency for an engine working between 40C muscles and 20C outside temp is about 20/300 or 1/15th. So even a perfect Carnot engine (an ideal combustion engine, that ideal being unreachable) should have taken 300 Calories to do the step-climbing work. That’s more than the metabolic rate.

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  • $\begingroup$ @sammygerbil yes, they are. The question was why it was different than the straight mechanical number, and explaining those inefficiencies was the answer to that. Or did I misunderstand the question? $\endgroup$ – Bob Jacobsen Mar 26 '18 at 7:10
  • $\begingroup$ @sammygerbil the Carnot efficiency for an engine working between 40C muscles and 20C outside temp is about 20/300 or 1/15th. So even a perfect Carnot engine (an ideal combustion engine, that ideal being unreachable) should have taken 300 Calories. Note that’s more than the metabolic rate. $\endgroup$ – Bob Jacobsen Mar 26 '18 at 9:07
  • $\begingroup$ @sammygerbil Thanks for clarifying, and sorry for my initial confusion. $\endgroup$ – Bob Jacobsen Mar 26 '18 at 9:34
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The cumulative effect of a number of small factors might account for a significant part of the discrepancy.

  1. According to p 559 of the 1990 paper by Jette, Sidney & Blumchent the ME number for climbing stairs varies between 4 and 8 depending on the intensity of the activity. You have taken a figure close to the upper extreme, which may have been equivalent to performing the same activity in (say) half the time which your daughter took. When I performed this activity it took me 30s per round trip for 2 floors (45s for 3 floors), compared with your daughter's round trip time of 120s.

  2. The figure given assumes constant upward climbing for the entire 30 minutes. Your daughter walked downstairs for half the distance, during which her ME number would be lower, nearer 2-3 for light walking. When I have performed this activity I ascended more quickly than I descended. In fact I took two steps at a time on the way up (probably close to the extreme ME number of 8) and came down one step at a time, so I spent about twice as long at the lower metabolic rate. There were horizontal sections in the stairs in which I was not climbing, and I spent a couple of seconds of 'turnaround time' regaining my breath at the bottom of the stairs.

  3. The ME number gives the overall power consumption of the activity as a multiple of base (resting) metabolic rate, so 1 should be subtracted to get the extra power consumed for the activity itself.

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  • $\begingroup$ Thank you for the response. One correction, the 30 min is all "up the stairs" time. The "down the stairs" time was calculated separately. $\endgroup$ – Vaccano Mar 26 '18 at 13:48
  • $\begingroup$ That makes the intensity very low indeed (point #1 in my answer). Climbing 2 floors two steps at a time took me less than 15s (high intensity). Your daughter climbed 2 floors in 80s - which is 5 times slower (low intensity). $\endgroup$ – sammy gerbil Mar 26 '18 at 21:14

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