Why muscles get tired without doing work

A student's question: If I hold up an object without moving it, I'm not doing work (by the physics definition), but I still get tired. Why is it so?

The short answer: your muscles require biochemical energy to exert a force, even if there is no motion*. So you use biochemical energy, which is tiring, even if the energy used does zero work.

In the multimedia tutorial Energy and Power, we saw that work, as physicists define it, roughly overlaps with some applications in everyday life: lifting a heavier weight is more work; lifting it through a greater height is more work. But that misses the student's question: suppose I hold my arm out horizontally with a 2 kg mass in my hand. Fairly quickly, my arm muscles become tired. A muscle requires continual input of biochemical energy just to maintain a force, even if that force doesn't do any work. And that raises a further question:

If I put the mass on a table, the table requires no input energy to maintain the normal force that holds up the mass. The table is doing zero work, requires zero energy input and doesn't seem to get tired. Why the difference?

The difference is related to the different kind of molecular bonds involved, and how they vary with displacement. The molecules in a muscle fibre exerting or resisting a force are in an active state, and to stay in this state requires input of the biochemical ATP (adenosinetriphosphate , described often as a cell's energy 'currency'). ATP molecules must be supplied continuously to the myosin head of each muscle filament to maintain a molecular cross-bridge that maintains tension in the filament. To supply the ATP, you must 'burn**' fuel. Not only that, but you also get tired from lowering the brick, because the muscle is still under tension during (controlled) lowering.

The bonds in the wood of the table are also biochemical, but they are in a stable state and are only stretched or compressed under load, a bit like a spring. This tiny stretching or compression requires no new chemical reaction and is usually close to reversible***. Regarding stretching and compression of intermolecular bonds, I give a simple model relating Young's modulus to molecular forces here.

* It's worth noting that you get more tired from applying a force over a distance (and thus doing work) than applying the force with no displacement (and thus no work). For instance, it's fairly easy to apply a force of 200 N horizontally against a wall. However, pushing a small car at 5 m/s with a 200 N force requires 1 kW of power and this will tire you very quickly. (1 kW will propel a solar racing car at more than 20 m/s: see car physics.)

** Viewed from the level of the organism, the overall chemical reaction is indeed like burning: you take in some biological fuel such as a sugar and you combine it with inhaled oxygen. You then exhale CO2 and water. If we burned sugar in air these could be our final products, too.

*** Not quite reversible, there is some hysteresis in the stress strain relation.

The physiology and biochemistry behind muscle fibre force are well explained elsewhere (see here on Wikipedia). But beware the terminology. For physiologiests, a muscular force applied with no motion is called an isotonic contraction (whence isotonic exercise).

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