How Do You Find the Centroid and Hydrostatic Force in Given Scenarios?

  • MHB
  • Thread starter ineedhelpnow
  • Start date
  • Tags
    Applications
In summary: A}\int_{a}^{b} \ x[f(x)-g(x)],dx$In summary, to find the centroid of the region bounded by the given curves, the student uses the formulas \overline{x}=\frac{\displaystyle \int_{a}^{b} xf(x) \,dx}{\displaystyle \int_{a}^{b} f(x)\,dx} and \overline{y}=\frac{\displaystyle \frac{1}{2}\int_{a}^{b} \left[f(x)\right]^2\,dx}{\displaystyle \int_{
  • #1
ineedhelpnow
651
0
4. find the centroid of the region bounded by the given curves. $y=x^{3}$, $x+y=2$, $x=0$

5. a gate in an irrigation canal is constructed in the form of a trapezoid 3 ft wide at the bottom, 5 ft wide at the top, and 2 ft high. it is placed vertically in the canal, with the water extending to its top. find the hydrostatic force on one side of the gate.
 
Physics news on Phys.org
  • #2
5.) First, let's determine the width of the plate as a function of the depth $x$. We know the width decreases linearly, and we know two points on the line:

\(\displaystyle w(0)=5\)

\(\displaystyle w(2)=3\)

So, knowing two points on the line, can you determine $w(x)$?
 
  • #3
no my instructor hasnt explained any of this yet but we still have to do it and I am totally lost. i don't get what the book is saying. :/
 
  • #4
ineedhelpnow said:
no my instructor hasnt explained any of this yet but we still have to do it and I am totally lost. i don't get what the book is saying. :/

What I suggest doing is first finding the slope $m$ of the line:

\(\displaystyle m=\frac{\Delta w}{\Delta x}=\frac{3-5}{2-0}=-1\)

Now, use the point-slope formula you learned in algebra or pre-calculus to determine the equation of the line representing the width of the plate as a function of the depth $x$. What do you get?
 
  • #5
w(x)=5-x
 
  • #6
ineedhelpnow said:
w(x)=5-x

Good! Now, we may compute the force along one horizontal elemental rectangle having area $A(x)$ as:

\(\displaystyle dF=\rho xA(x)\)

The area of this elemental rectangle is its width at $x$ times its height $dx$, hence:

\(\displaystyle dF=\rho xw(x)\,dx\)

Note: $\rho$ is the mass density of water, usually given as \(\displaystyle 62.5\frac{\text{lb}}{\text{ft}^3}\)

So, substitute for $w(x)$, and then integrate from the surface $x=0$, to the bottom of the plate $x=2$. Can you set up the resulting integral?
 
  • #7
ill try but i don't think ill do it right
 
  • #8
ineedhelpnow said:
ill try but i don't think ill do it right

I bet you can...you have a constant $\rho$ times $x$ times the width function $5-x$ times the depth differential $dx$...:D
 
  • #9
$\int_{0}^{2} \ 62.5x(5-x),dx$ ?
 
  • #10
ineedhelpnow said:
$\int_{0}^{2} \ 62.5x(5-x),dx$ ?

Excellent! Now, pull that constant out front, distribute $x$ to the width function, and then use the power rule and the FTOC (fundamental theorem of calculus) to compute the definite integral. :D
 
  • #11
thanks for your help!
 
  • #12
ineedhelpnow said:
4. find the centroid of the region bounded by the given curves. $y=x^{3}$, $x+y=2$, $x=0$

First, let's get a diagram of the region:

View attachment 2716

Now, let's dig out the formulas we need:

\(\displaystyle \overline{x}=\frac{\displaystyle \int_{a}^{b} xf(x) \,dx}{\displaystyle \int_{a}^{b} f(x)\,dx}\)

\(\displaystyle \overline{y}=\frac{\displaystyle \frac{1}{2}\int_{a}^{b} \left[f(x)\right]^2\,dx}{\displaystyle \int_{a}^{b} f(x)\,dx}\)

Can you identify $a$, $b$ and $f(x)$?
 

Attachments

  • centroid.jpg
    centroid.jpg
    7.5 KB · Views: 71
  • #13
would it be from 0 to 1? I am having trouble identifying the f(x) because my book does it kinda different and it's confusing me.
 
  • #14
ineedhelpnow said:
would it be from 0 to 1? I am having trouble identifying the f(x) because my book does it kinda different and it's confusing me.

Yes, we are given the left limit, the line $x=0$, and to find the right limit, we may equate the two functions and solve for $x$:

\(\displaystyle 2-x=x^3\)

\(\displaystyle x^3+x-2=0\)

\(\displaystyle x^3-x^2+x^2-x+2x-2=0\)

\(\displaystyle x^2(x-1)+x(x-1)+2(x-1)=0\)

\(\displaystyle (x-1)\left(x^2+x+2\right)=0\)

We see that the quadratic factor has a negative discriminant, and so the only solution is:

\(\displaystyle x=1\)

Thus, we have $a=0,\,b=1$.

Now, $f(x)$ is the upper function minus the lower function...what would that be?
 
  • #15
$2-x-x^3$
 
  • #16
ineedhelpnow said:
...im having trouble identifying the f(x) because my book does it kinda different and it's confusing me.

Sorry for the delay...a brief summer storm knocked out my connection (satellite internet).

How does your book compute the centroid? I ask because I see in my old book that I may have given you a formula that does not apply in this case. :D
 
  • #17
$x=\frac{1}{A}\int_{a}^{b} \ x[f(x)-g(x)],dx$

$y=\frac{1}{A}\int_{a}^{b} \ \frac{1}{2}([f(x)]^2-[g(x)]^2),dx$
 
  • #18
ineedhelpnow said:
$x=\frac{1}{A}\int_{a}^{b} \ x[f(x)-g(x)],dx$

$y=\frac{1}{A}\int_{a}^{b} \ \frac{1}{2}([f(x)]^2-[g(x)]^2),dx$

Yes, that's what you want to use...I apologize for my mistake of posting a formula that applies only when $g(x)=0$. (Tmi)
 
  • #19
oh ok i think i get where to go from here. thanks :)
 

FAQ: How Do You Find the Centroid and Hydrostatic Force in Given Scenarios?

What are some common real-life applications of integrals?

Integrals have a wide range of applications in fields such as physics, engineering, economics, and statistics. Some common real-life applications include calculating areas and volumes, computing the distance traveled by an object, finding the center of mass of an object, and determining the amount of work done by a force.

How are integrals used in physics?

In physics, integrals are used to calculate quantities such as position, velocity, acceleration, and force. They are also used to find the work done by a force, the energy stored in a system, and the center of mass of an object. Integrals are essential for analyzing and solving problems in mechanics, electromagnetism, and thermodynamics.

What are some common techniques for solving integrals?

Some common techniques for solving integrals include substitution, integration by parts, trigonometric substitution, and partial fractions. These techniques are used to simplify and evaluate integrals in order to find their exact values. Other methods, such as numerical integration, can also be used to approximate the value of an integral.

How are integrals used in economics?

In economics, integrals are used to analyze and model various economic phenomena, such as supply and demand, production and cost functions, and consumer behavior. Integrals are also used to calculate measures of economic welfare, such as consumer and producer surplus, and to make predictions about future economic outcomes.

What role do integrals play in statistics?

In statistics, integrals are used to calculate probabilities, expected values, and moments of random variables. They are also used to find the area under a probability density function, which is essential for calculating probabilities and making statistical inferences. Integrals are also used in regression analysis and other statistical models.

Similar threads

Back
Top