Phy 202
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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This is a question on the test that I am taking tomorrow. I am really lost for how slope affects the height that a bottle engine will raise water. I know how to use bernoulli's equation for verticals, but I do not know how to apply slope to the equation. I was hoping you could answer this as soon as possible. If not before tomorrow I am still really interested in knowing.
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I'm sorry I wasn't able to answer this as promptly as I would have liked.
If the water level changes in a tube which makes angle theta with horizontal, with the water moving distance `ds along the tube, then the vertical change in the water level is `ds sin(theta).
There are a number of ways to see this.
In terms of vectors: The displacement vector is `ds at angle theta with horizontal; this vector has vertical and horizontal components `ds sin(theta) and `ds cos(theta), respectively.
In terms of right triangles, form a triangle with the displacement along the hypotenuse, and legs equal to the horizontal and vertical displacements. The angle theta will be opposite the vertical displacement, which is therefore `ds sin(theta).
The significance of the vertical displacement is that it gives you the change in the rho g y term of Bernoulli's equation.
Note that the slope of the tube is tan(theta), where theta is again the angle with horizontal. Thus theta is the arcTangent of the slope, and it is easy to work out, using basic trigonometry, that sin(theta) = slope / sqrt( 1 + slope^2).
These relationships are of concern mainly to General College Physics and University Physics students.