Regardless how deep the water wheel is dipped into the river?
Ok so let's say the water wheel is about 20 feet diameter, and the paddles are 2 foot long, so the paddles can go down as low as 2 feet.
Lower the wheel so the paddle bare touches the surface 1 inch. The wheel isn't going to turn much. Most of the water is just going to flow underneath, so the paddle isn't really blocking the water.
The paddles are 2 feet long, and also 2 feet wide, and you force the water to flow within a 2 foot wide channel into which you are lowering the paddles of the wheel. Totally lowering the wheel causes the paddles to completely block the channel.
Still doesn't change the fact that if the wheel is only 1 inch in the water if the water flows faster it will turn faster.
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