I am not able to wrap my head around this. Why does taking average pressure and multiplying it with the area given total hydrostatic force? In many books, the reason was because pressure is linearly varying with height but I don't see the connection. I can find this by integrating but what is the proof to this?I can prove it for a vertical rectangular plate but how can this be generalized for a surface of any shape like told in the cengage textbook.
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
If you plot pressure vs depth, you get a straight line of positive slope. The average value is the area under that line divided by the x-length of the line. The area under any sloped line is the area of a trapezoid:
$$\rm Area = \frac 1 2 (base~1 + base~2)(height)$$
And the x-length of the line is the "height" of the trapezoid, so:
$$\rm Avg Value = \frac 1 2 (base~1 + base~2) \frac{(height)}{(height)}$$
$$\rm Avg Value = \frac 1 2 (base~1 + base~2)$$
Thus it is an algebraic fact that the average value of a linear function $f(x)$ over the interval $a$ to $b$ is simply the arithmetic mean of the endpoints $f(a)$ and $f(b)$.
Yet another way to see this is:
$$\frac {f(a)+f(b)}{2} = \frac {(ca+d)+(cb+d)}{2} $$
$$= \frac {c(a+b)+2d}{2} $$
$$= c\left(\frac{a+b}{2}\right)+d $$
$$=f\left(\frac{a+b}{2}\right)$$
