There are a couple of things at play here.
One, and probably the most relevant, is that you’re actually decreasing the angle you have to climb at, by extending the total distance you’re climbing. This reduces the amount of work you need to do at any given moment, but doesn’t reduce the total amount of work you do by the top of the stairs. In fact, it actually increases the amount of work you do by the top, but it’s broken up over many smaller increments.
Second, it allows you to place more of your foot on each stair, which assists you in transferring that work energy from your body to the stair, so it’s a bit of a mechanical assist over only using the ball/toes of your foot.
And finally, it slows you down, so that helps in reducing the amount of work you’re doing at any given moment. Again, we’re not reducing the total amount of work, but we’re spreading it out over smaller increments.
It’s a lot like trying to pick up a box full of stuff. You might not be strong enough to lift it when it’s full, but you are strong enough to pick up each individual item by itself.
Aside from all the physics that others have addressed, there is also the matter of which muscles you are loading as you climb. Depending on the dimensions of the stairs and an individual’s natural gate, the hamstrings, quads, and butt may be activated more or less as one climbs which can affect fatigue.
It has to do with energy vs. power. It takes the same amount of energy either way, since you are lifting yourself up the same distance. The slower you go, or the longer the path you take, you are spreading that energy out over a longer period, which makes it feel easier.
Humans can expend a lot of energy, but the rate (power) at which they do it is very limited. For instance, you could lift a 50 lb weight 50 times at the gym. But you couldn’t lift a 2500 lb weight once, even though it is the same total amount of weight being lifted.
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