Finding Horizontal force on a gate!!! please help!!?

Question

I'm learning Calculus and i ran into a problem i don't get. okay here it goes. A control gate in the form of a parabolic segment of base 12 and height 4 is submerged in water so that its base is 2 units below the surface of the water. Find the horizontal force on the gate if the density of the water is w. in the picture the equation of the parabolic gate is 9y=x^2 I don't know how to find dA, and 9y=x^2 am i supposed to find the sqroot of x or divide by nine? please just help me set up the problem. Thanks

Answer 1

the integral is going to be lower limit= 0, and upper limit=12 because the parabola is extending and has base 12, and the upper restrain on the parabolic graph is y=4.Your parabola equation will look like y=x^2/9 with bound y=4.

Using fundemental theorem of calculus this is how integral should look like.

S[4 - (x^2/9)] upper limit:12 and lower limit:0

= S(0,12)4x - S(0,12) 18x/81

=[4(12)-4(0)]-
[18(12)/81-18(0)/81]

=48-(8/3)=144/3-8/3
=136/3=45.333333....

if Area is meant to be translated to mean work,

then 136/3 units of work was commited on the gate
horizontally, though work is measured in Joules you have to be careful because we can't attain from the question if it is kilojoules or joules or 10E-10 joules.

Answer 2

I can understand that there is a net vertical force which is capable of being calculated from what you have provided. To do so I have to make some assumptions about what you are seeing.

I assume that there is a surface of revolution formed by spinning a parabola about the line perpendicular to the lowest point in the parabola. Dipping such an item into the water would displace an amount of liquid equivalent to the volume above the surface and below the level of the liquid. The force would be equal to the volume times the density of the liquid times g(the acceleration due to gravity) g would have different values depending on the units of measure being used. With feet then g is the weight of a cubic foot of water. The answer will then be in pounds. With centimeters then g is about 9.8 meters/second/second. the answer will be in newtons.

You used the term horizontal. If that is what is really being asked then my assumptions can not all be true, or the answer is zero.

Answer 3

You need to know little bit physics to understand the problem.

First, you should know that water pressure = wgh, where w is the density of the water, g is the acceleration due to gravity, and h is the depth below the top surface of the water.

Second, you should know force = pressure x area.

Since the pressure is a function of the depth, you have to include force as a function of y. If you set the vertex of y = x^2/9 at the bottom of the gate, then h = 2-y.

Now, you can write an integral to do the problem,

F = ∫PdA
= ∫wg(2-y) 2xdy [y: 0...2], (dA = 2xdy)
= ∫wg(2-y) 6√y dy [y: 0...2], (x = 3√y)
= 9.0501 wg

Hope it helps you.