Physical Model of Exercise

Question

Trying to model caloric effects on the body when exercising. I think I have something for lifting weights, but at a loss for running or bicycling.

1. Lifting Weights. I think this is pretty easy. Raising a $x$ lb bar $y$ ft takes $xy$ ft-lbs of energy. Lowering it slowly also takes energy, the previous quantity, minus kinetic energy at end of drop, just before impact at bottom. So best case scenario for a $50$lb weight lifted abd lowered $2$ ft, that's 0.064 Calories or kilocalories per lift cycle.

2. Suppose you are walking at constant speed $v_0$ for time $\tau$. What's the energy output from your body? I'm thinking $E=v_0\tau F_{avg}$ where $F_{avg}$ is the average of the net forces working against the motion. Are there other forces acting besides wind resistance and friction? I'm thinking wind resistance is $F_w=-bv_0$ for appropriate $b$. Friction is $F_f=w\mu$ where $w$ is weight and $\mu$ is co-efficient of static force.

Typical values are $v_0=4$ mph, $\tau=$ 3600 s, $E=350$kcal$=1.08e^6 $ftlb $\implies F_{avg}=\frac{E}{v_0\tau}=51.13$lbs.

For vertical forces in the running/walking scenario, just remaining erect is not letting gravity drop you to the floor. So you input as much as gravity takes per second. $\frac{wh}{2}\sqrt{\frac{g}{h}}=w\sqrt{\frac{gh}{4}}$ is calories burned just from standing up. That's weight falling from center of mass to the ground divided by the time of free fall for an object at that height, half of $h$, the height of the person in question.

If you could maintain a lift every two seconds for an hour, the calorie burn would be over 100 times that due to running, right? Two seconds an hour is 1800 lifts burning 0.064 kcal each.

Are my forces at all accurate for the walking scenario? Does weight of the person only factor in for friction and standing?

Answer 1

There are a lot of questions here. In general you should try to just ask one at a time.

The short answer to all of your questions is that this is not a problem that is amenable to a first-principles solution. You will need empirical data from which you can extrapolate approximate functions for your variables (or approximate functions that someone else has extrapolated from data already).

Much of the energy expenditure of exercise is chemical inefficiency: that is, for every watt produced by combining oxygen with glucose, much less than a watt of muscle power is produced.

Some of what remains is biological inefficiency. Keeping a muscle tensed, especially under load, does no mechanical work, but requires continuous chemical power. Holding up a weight, for instance, does no mechanical work, but is a taxing exercise.

Some of what remains is mechanical inefficiency. For example, when walking, one must lift one leg (doing work against gravity), move it forward faster than the body, stop it (most of the stored kinetic energy is wasted as heat), put it down (most of the stored gravitational potential is wasted as heat), and then repeat the process with the opposite leg. These can in principle be represented as a variety of forces acting opposite the directions (plural) of motion, but in practice a first-principles approach to a machine as multijointed, stretchy, springy, and squishy as a human being is exceedingly difficult, even for a relatively simple motion like walking.

Remaining standing takes very little power, as most of the load is taken by the skeletal system, with only a little bit of muscle power required to keep the skeleton properly aligned. What you've estimated is more in line with the power required to do nonstop box jumps, ignoring inefficiency.