[Physics] Why, when calculating work done when a person climbs stairs, the distance is the height of the stairs but not the distance the person travels

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I was thinking, since

Work Done = Force * Distance moved in the direction of the force ,

the distance moved in the direction of the force would be the distance of the slanted stairs instead of the vertical distance of the stairs, as shown in my school textbook. I had a little idea about this, that since the 'Force' is the weight of the person, it wouldn't make sense if the distance will be the slanted stairs' distance. The height is probably only used in that case if another force (e.g. machine pushing person up stairs) is used. However, I still can't quite get my mind around why height of the stairs is used.

Best Answer

Yes you are right, $W=f.d$, but this is a scalar product and $f$ and $d$ are vectors. Which means that the norm is $W=|f||d|\cos(\alpha)$ where $\alpha$ is the angle of the stairs. One can notice that $h=|d|\cos(\alpha)$ is the height of the stairs.

In other words, if you project all the vectors on the vertical axis, it becomes $W=|f||d|\cos(\alpha)=|f|h$

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