Built on Facts

Apropos of the calorie/Calorie discussion yesterday, here’s something interesting to think about with regard to the energy used in exercise.

The formula for gravitational potential energy is m*g*h, where m is mass, g is the acceleration due to gravity, and h is the height. Of course this is if the changes in h are small enough so that g can be taken as truly constant. So any change in height is going to require energy if you’re going up, and will release energy if you’re going down. This is why you hit the gas driving up a hill and hit the brake going down a hill. Actually since we’re talking exercise let’s change that to biking. Uphill is hard, downhill is easy.

Now if you’re going a long way uphill your height changes quite a bit, and correspondingly so does your potential energy. Should you decide to climb the stairs of the Empire State Building you will ascend some 1,050 feet. Now let’s say you weigh 160 pounds and that the gravitational acceleration is the usual 9.8 meters per second squared. We’re after mgh, so multiply them all together after of course converting everything to SI units. I get a total change in potential energy of roughly 230,000 joules. That works out to 54 food calories.

54! Surely something that difficult would burn a lot more calories, you’d think. And it does. The immense effort you expend in climbing is mostly budgeted to different bodily processes. You have to move extra air in and out of your lungs. You have to circulate blood at a much higher rate. You have to process the complicated chemistry required to keep your muscles moving. All of these things take energy, and by the time the shoe meets the stair most of the energy has already been lost, eventually ending up mostly in the form of heat. Your body can’t afford to overheat and so you begin sweating to carry the excess heat energy away. All that energy had to come from somewhere, and it came from the food you ate. By the time you’re on the observation deck looking over Manhattan you’ll have used up a lot more than 54 calories.

I like to spring this as a quiz question on my 201 students near the beginning of the semester. One of the most important things in physics is developing an intuition for roughly what the right answer should be, so you can tell if you’ve made an obvious mistake somewhere. But for this to work you have to have worked out for yourself how these physics problems look in terms of real life quantities. Sometimes it’s not what you’d expect, and this is one of those cases.

Comments

#1 Martin Watts
December 30, 2008

I had a very similar question in a chemistry test, 30 years ago now, but I can still remember how it started. “A chemist, mass 70kg, eats a spoonful of sugar, mass 5g…”
#2 Nelson Muntz
December 30, 2008

On the climb up, you’d be radiating well over 100 watts of heat.
#3 Julie Stahlhut
December 30, 2008

This explains how I lost 10 pounds on a three-week trip to Italy in 2007, despite indulging in lots of excellent food. Granted, only one of our daily meals was a large one, but we walked everywhere and just about every historic site we visited required long walks uphill.

Ever since that experience, my favorite workout has been a brisk walk on a treadmill at maximum incline. I know that the calorie estimates on exercise-machine readouts are not necessarily accurate, but it makes perfect sense that they register much higher when walking “uphill” than when jogging or running on a flat surface.
#4 Uncle Al
December 30, 2008

54/200 = 27% thermal efficiency. That is better than a gasoline IC engine (certainly in California with a 3-5 mpg cost of Offical air pollution abatement). America can be SAVED! with two small changes,

1) All motor vehicles are banned in favor of bicycles, and
2) All roads are rebuilt to be downhill in all directions, San Francisco already being half-way there.
#5 natural cynic
December 30, 2008

The energy you have in the calculation – 230 kJ is used to move your 160 lbs 1050 ft vertically. Another component that must be added is the energy that is needed to move the body horizontally, say 1.5 ft for every 1 ft vertically [steps plus landings]. This gives about 1500 ft. which uses roughly 100 kJ [this is the actual amount of energy needed to walk horizontally that distance].
#6 biziml
December 30, 2008

thenks i
#7 Peter
December 30, 2008

See my comment on the previous post.

The body is a very complex thing – evolution has also ensured that an organism that thas to repeatedly climb 1050 ft will have some means of trimming excess body weight that is not required under these circumstances (but might have been desirable while huddled around a fire trying not to freeze.)

For humans, exercise reduces fat accumulation and storage, while inactivity prompts fat storage. Most of us ingest well above what we burn, with absorption efficiency and fat storage mechanisms determining our equilibrium weight, not available calories. Exercise shifts that equilibrium to a leaner meaner machine much more effectively than starvation…
#8 grunchy
December 30, 2008

@natural cynic,

Can you explain how you estimate 100kJ for walking 1500ft horizontally? This seems to me to be high.
#9 natural cynic
December 31, 2008

A rule of thumb in estimating the number of Calories needed to walk 1 mile for a 160 lb person is 80-100, if the person has reasonable coordination and body composition and walks at a 3.5-4 miles/hour pace. 1500 feet ~ 0.3 miles * 80 cal/mile ~ 24 Cal * 4.1 kJ/kcal ~ 100.

This energy expenditure would only be true for a brisk walking pace and does not take into account any difference in mechanical efficiency between the horizontal component of walking and the horizontal component of stepping.

Incidentally, walking at a race-walking pace uses considerably more energy than walking at a normal pace or running at the same speed as a race walk. And slow jogging is slightly more energy intensive on a per mile basis.

[From a graph that was etched in my memory from Astrand & Rodahl Textbook of Work Physiology]
#10 jeffk
December 31, 2008

I always looked at it like this: if you try to use gravity to calculate how much energy something should take, you’d conclude that running on flat ground burns no calories.
#11 grunchy
December 31, 2008

Thanks for the explanation. That text looks interesting.
#12 Timothy
November 24, 2010

Actually, the formula is correct, but the units of measure is wrong. The E=mgh should be: m = mass in kg, g = gravity (9.81 m/s²), and h = meters. In this case, 160 pounds = 72.57kg and 1050 feet = 320 meters. Doing the math now… E = 72.57 x 9.81 x 320 = 227811.744 kilocalories or 227.811 calories, which is more like it.

Timothy
#13 Duncan
January 1, 2011

Timothy – surely that is 227.811 KJ not kcal or indeed calories? That works out at 54.42 (food) calories which is what the original post says?
#14 Shrek
October 26, 2011

Don’t forget weight gain due to relativistic effects.

A +320m delta h for a 100 kg man will mean a 3.5 * 10^-12 kg increase in rest mass and a tighter waistline! yikes!
#15 Shrek
October 26, 2011

@jeffk

Not exactly, otherwise perpetual motion machines would be available at the corner store…
#16 Shrek
October 26, 2011

@muntz

Assuming an exercising surface body temp of 40 C (313 K), emissivity of 1, Surface area of body approx. 2 m^2… using Stefan- Boltzmann you’d be radiating about 1-1.1 kW