Question

A vertical dam has a semicircular gate as shown in the figure. The total depth d of the figure is 30 m, the height h of air above the water level is 6 m, and the width w of the gate is 8 m. Find the hydrostatic force (in N) against the gate. (Round your answer to the nearest whole number. Use 9.8 m/s2 for the acceleration due to gravity. Recall that the weight density of water is 1,000 kg/m3.)

Answer 1

Answer:

Should be = 6462092.108N

Step-by-step explanation:

So,

To solve this problem, you must first declare that your height, h, in a perspective. I chose to make it

h=Bottom of the gate to the top

That is one of the only ways to do this problem.

Now you can recognize that if you drew the semicircle on a coordinate plane than you can see that it splits the semicircle in half. This makes it where if you drew a slice a width, w, on each side.

From this you can start to determine the area of your slice that you just made.

$Area of Slice=2w *delta h$

Since we need w in terms of h, we can use the equation of a circle.

$x^2+y^2=r^2$

or in this case: $w^2+h^2=4^2$

If you solve for w you get: $w=\sqrt{16-h}$

Now the Area of the Slice becomes:$A=(2\sqrt{16-h})*deltah$

Now that we got area, we just need to determine the depth, which in this case is just the height of the top of the water to the slice, which is:

depth=$(24-h)$,

The 24 is from subtracting the height above the water and the total height of the dam.

Now we just set up the integral:

$\int\limits^4_0 {(9800)(2\sqrt{16-h} })(24-h) \, dh$

Which should be = 6462092.108N