# Measuring energy when no work is done — comparing push-ups to planks

tl;dr

Is there any meaningful (physical) way to compare the energy expended in the exercise of doing $x$ pushups in $t_1$ seconds, vs the exercise of doing the plank for $t_2$ seconds?

I'm confused about the concepts of energy, work, and power, in the case where no distance is traversed.

For instance, take a push-up. I can approximate the work I've done by using the change in potential energy $\Delta U = mgh$, ie. $$W = mg \Delta h$$ $$= m [kg] * 9.8 [m/s^2] * h[m]$$ where $m$ is some proportion of my body mass, and $h$ is a bit less that the length of my arm.

The result has the units of $Newton-meters$, or $kg m^2/s^2$, aka $Joules$. I can also measure the average power of my push-up using $\Delta W/\Delta t$, which has the units of $J / s$, or $kg m^2/s^3$, aka $Watts$.

Fine so far -- but now take a plank (holding the push-up position for some time). Obviously when you do a plank, you are expending some energy, even though you are not doing any work (since there is no change in potential energy). What are you doing then? You are counteracting the force of gravity on your body mass (that is, $9.8 [m/s^2] * mass [kg]$) for a fixed time. This would produce units of $kg m/s^3$ -- these units are not named, that I can recall.

So my question is twofold:

1. Is there an actual unit that describes the energy I've expended in doing a plank, beyond $(mass * g * time) kg m/s^3$
2. Is there any meaningful way to compare the energy I've expended by doing a plank, with the energy I would expend in doing $x$ pushups? (Beyond work or power, since those values are always zero for the plank, which is not helpful.)
• Closely related: physics.stackexchange.com/questions/1984/… and various others in the linked sidebar of that one. – dmckee --- ex-moderator kitten Feb 23 '14 at 20:22
• @dmckee ah, thanks. The top answer at the link says that muscle cells will be contracting/expanding constantly, thus doing work. The question is -- how would you measure the actual work. I wonder if there is a rule of thumb, or if it varies considerably by individual. – McGarnagle Feb 23 '14 at 20:31

1.) You lose energy by doing work. The difintion of work by itself is the product of force and dislplacement(if the force is onstant) or else $\int_{}{}F dx$ for non constant forces.In case the case of doing a plank, you do zero work though you are resisting the force of gravity. So in terms of physics, you lose 0 energy even though you get tired. The reason you are getting tired of course is completely different, it is due to the inability of the muscles to strech indefinetly, due to internal frictons and other cellular acitvities. You could think of it like this, a more well trained person of your same weight would be able to do it longer than you even though you both are exerting the same force (if ever you did any work, you both would be doing the same thing) because his muscle cells are more finely tuned to doing it.