Wednesday, April 2, 2014

Making a (very small) Virtue of Necessity

As I climbed to the third-floor classroom yesterday, I got to wondering how much energy I was burning.  I was predisposed to think of stair-climbing in term of calories (rather than, say, heart health) because I have put on a few pounds lately.  (Okay--more than a few.) 

This could be quite complicated as a biology question, but in terms of physics, it can be done--very roughly but adequately--in five minutes on the back of an envelope.  In fact, with some simplification it can be done in your head.

The first thing to realize is that height is a kind of energy (potential energy).  Consider a car about to roll down a hill in neutral: when that car steam-rolls you at the bottom, you will have experienced that potential energy converted into the form of motion (kinetic) energy.  By the same token, energy is "stored" in an object as it is raised to a height.  That means the energy I burn in walking up the stairs is roughly equal to the amount I gain in potential energy in going up two floors. 

A moment's thought will convince you that potential energy depends on both the mass of the object, and how high it is.  (An anvil falling on your head will affect you differently than a marble from the same height.)  It also depends on the "acceleration of gravity" (symbolized "g"), which describes how quickly a falling object on earth accelerates, and is equal to 9.8m/s2.  (This translates: a falling object increases its speed by 9.8 meters per second for each second that it falls.)  The actual equation for potential energy (Ep) is:


--where "m" is mass in kilograms, and "h" is height in meters.  (Ep is in Joules; a Joule is the amount of energy needed to accelerate a 1kilogram object by 1 meter per second every second.)

I'm guessing the floors of the school to be 4 meters apart, so 8m total height.  Taking g to be nearly 10, and my mass to be almost exactly 100kg, we have:


Sounds like I'm burning a lot of energy on my climb!  Now to turn that into a more familiar unit: a calorie is four-point-something Joules--call it four even for simplicity.  Now I have burned about 200 calories on my climb.  But wait! in one of the stupider coincidences in science, there are two kinds of calories**: the regular sort used by physics, and the Calorie (big "C") used in considering food.  A Calorie is equal to 1000 calories.  So my climb actually only burned about 2 Calories. 


According to the sugar bag in my pantry, a teaspoon of sugar--not much more than I put in my coffee--has 15 Calories.  So every time I walk up to the third floor, I burn through only a few sips of my morning brew.**

Deep funk.

I suppose the take-home lesson is one I already knew: you can't exercise yourself into weight loss (unless maybe you're the athletic type, in which case you probably don't have to); you have to control your eating.  (Yes, there's more to it, but it's still unavoidable.)  Probably I should stop making hermit cookies, full of deadly brown sugar, molasses and butter.  From the scientific point of view, it's pretty cool (and scary) how much energy food contains. 

*Another definition: a calorie (small c) is the amount of energy needed to warm one gram of water by one degree Celsius.  Therefore a Calorie can warm a whole kilogram by the same one degree.

**Okay, so it isn't really that simple: since I am assuming our bodies are 100% efficient at converting chemical (food) energy--via muscles, joints, etc.--into potential energy.  What if our bodies were only 25% efficient? or 15%?  I still don't burn that whole cup of coffee!

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