# Climbing Stairs and Calories Burnt

I climb stairs to work (20 floors) every day . The least amount of work my body is doing by gaining potential energy (P.E) = mgh where m is mass , g is acc. due to gravity and h is height . Assuming every floor is 1 metre high . Mass -> 70kg Work Done Comes out to 70*10*20 = 14000J . 1 calorie = 4.2 Joules . So I am burning somewhere around 3000 calories. Right? But this calculation seems way off. I do not feel this tired after climbing 20 floors. What am I missing?

• " Assuming every floor is 1 metre high", seems a tad small!
– jim
Apr 10, 2016 at 18:57
• @jim Conservative Estimates . Makes math easy :) Apr 10, 2016 at 19:01
• Work in SI units. Apr 10, 2016 at 19:42

Due to an accident of history there are two different units called the calorie and the Calorie - yes, the only difference is the capital C.

The calorie is 4.2J but the Calorie is 4.2kJ, and the calories counted in diets are actually Calories even though they are invariably written on the food packaging with a small c.

So you only used 3 Calories walking up the stairs. You'll need to do a lot more walking to lose weight :-)

• Thanks . I thought I could just do it with climbing stairs . Dang :( Apr 10, 2016 at 19:02
• @RamanSingh: humans aren't very mechanically efficient, so you used a lot more than 3 Calories. But it takes a surprising amount of exercise to burn calories. From memory running uses about 100 Calories per mile, but that's proper running not jogging. Apr 10, 2016 at 19:04
• One important caveat to this is that, to a reasonable first-order approximation and for typical exercise regimes, the amount of energy you burn is completely unrelated to the kind of naïve calculations physicists like to do: That's because, under the same conditions, your body has an efficiency of approximately zero. So the trick of exercise is not to do useful work, but to force your body to dump more energy as heat. So $mgh$ is not very useful because there's an enormous 'work dumped into heat' term which is missing.
– user107153
Apr 10, 2016 at 21:36

John Rennie has explained the problems with units here. Now, you'll burn about a factor 4 more than the work you perform, due to losses when glucose or fats are burned to allow the muscles to do the work. The Gibbs free energy change when glucose or fat reacts with oxygen and changes into water and carbon dioxide gives you the maximum amount of work that can in principle be extracted; it turns out that the human body manages to come quite close to the theoretical maximum of about a quarter of the total energy released in the reaction).

So, you'll burn approximately 12 Kcal. A rule of thumb is that going down requires about half the amount of energy, so that's about 6 Kcal. You can make a decent cardio exercise out of this by running up and down the stairs 25 times during e.g. the lunch break. The 450 Kcal you'll burn is similar to running at a decent pace for half an hour.

Some good points have been made here. What I think is missing is the step in physics where, after getting an answer, you ask yourself, "Does this make sense given my question?" The reason why 3000 calories is wrong was explained, but 3 kcal doesn't make sense either. Trying to improve upon that and taking the mechanical efficiency of the body helps, but that still doesn't make sense. 20 floors is a long way. 12 or 15 kcal is too low.

Cross-check it this way: How long does it take to climb 20 flights of stairs and what is your average heart rate? A floor typically has 13 stairs. So you climb 260 stairs. By my quick test, that task would take me about 4 minutes and my heart rate would be about 160bpm. For my age and using one of the "standard formulas" (http://www.calories-calculator.net/Calories_Burned_By_Heart_Rate.html --But note: calorie burn is a predictive measurement subject to quite a lot of error), it predicts 68 calories. This would mean if I did it for an hour straight I would burn a little over a thousand calories. That seems high, but then I don't think I could keep that pace for an hour. So we are in the right ballpark.

-->Update: a developer on our team tested this at the gym. He used a stair climber and did 20-floors at 179 watts, burning 64 calories according to the machine. (The machine didn't account for his weight, so there is a margin of error here too.)

So the physics and math question has us stumped by a factor of about 4x-5x.

For weight loss, I suggest using a heart rate monitor and tracking app like FitTrip (disclaimer, my company makes that app, but it is free to track HR and calories). This will help with the practical matters of managing calorie intake and burn. It looks to be the case that, for each pound you want to lose, you may have to climb about 1,200 flights of stairs without increasing food intake.

But, if the point is to continue the rather enjoyable theoretical discussion then we must start looking for energy expenditure not accounted for using a deeper insight.

So, what is the thing we are missing? The gap is not pure mechanical inefficiency (already accounted for) nor heat (that is what mechanical inefficiency turns into, and therefore is also accounted for). This is my hypothesis: Most mechanical efficiency tests are done on bikes, where the body is mostly at rest and the legs are isolated as the generator of work. My guess (which is reinforced by the difference in walking and biking efficiency in this article: https://www.exploratorium.edu/cycling/humanpower1.html) is that a lot of mechanical work is required to hold a body upright, balance, alternate legs and arms, and otherwise move a gelatinous mass up and across stairs in a bipedal fashion. --Very different from if food calories could power lab-created muscles to turn an elevator crank and lift a mass in a purely linear fashion.

If this is true, then the conclusion is that the human body is actually only about 4%-5% efficient at moving itself up flights of stairs.

Does anyone have another hypothesis? It would be interesting to expand our collective understanding.

mgh/Energy conversion efficiency (η) is the right formula if you think in mechanical terms. An example: 70kgx10m/s2x20m (for 20 meters height) divides to energy conversion efficiency ( about 25% is maximum for humans) = 14000J/25=560J = 133.84 cal = 0.13 kcal for only 20 meters of height. I am sure that 20 floors is more than that :). The example is correct only in physics terms because the human is a complex biological system.

• Thanks for posting your first answer. Have a look at how you canformat math here to improve the look and readability of your answers. And in generaltry to be A clear as possible two questions i have for this one is: what did you use the efficiency for here? And there is an s on your calculations that I don't see what it is, also Iam not sure what the first calculation relates to. Greetings. Aug 21, 2022 at 10:11