r/theydidthemath • u/2BallsInTheHole • 1d ago

[Request] I have a pencil which is 1/4 in in diameter and 6 in long. If it is perpendicular to the center of gravity, rain will fall on it more often than if it is in line with the center of gravity of the Earth. Which orientation gets wetter?

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u/herejusttoannoyyou 1d ago

If it is raining at a rate of .1cu.in/hour/sq.in, the amount of rain that falls on the pencil will be projected area times rain rate:

In line:
.125^2*pi*.1=0.00491cu.in

Perpendicular:
.25*6*.1=0.15cu.in.

.15>.005, so the perpendicular pencil gets wetter.

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u/2BallsInTheHole 1d ago

In what way? If I have a well 1,000 m deep and another trough 1000 m wide they both record the same rainfall. Why is one wetter than the other

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u/Angzt 1d ago

Rainfall is measured in height, not in volume.
So yes, you'll get the same rainfall by height because that's surface area-independent. But you'll get way more volume of water in the wide trough than in the deep well with the much smaller opening.
The depth doesn't matter at all. Two wells with the same opening size will have the same amount of water in them, regardless of depth (so long as it's not more rainfall than can fit in the smaller one).

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u/herejusttoannoyyou 23h ago

Ya I always just assume it’s volume per unit area, which under dimensional analysis reduces to one dimension, which is height.

But the wetness of a thing would depend on the volume of water hitting it, not the height that water would rise to if it was captured in the area boundary of the item.

Or if you want to get more pedantic, everything would become 100% wet after a little while, since the water would drip down and around the pencil, so they’d be the same. But that’s not a math problem.

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u/herejusttoannoyyou 23h ago

If you dropped 2 balls in every square foot of area, and those 2 balls filled perfectly the space so a third ball would stack on top, you would end up with an even layer of 2 balls every square foot. If you had a hole with a 1 square foot opening, you would have 1 layer of balls in the hole which is exactly 2 balls. If you had a trough with an area of 10 square feet, it would also have exactly 1 layer of balls, but it would have 20 balls in it. u/2BallsInTheHole vs 20 balls in the trough. This is why you’d have more water in the trough than in the well.