What is Centroid: Definition and 178 Discussions

In mathematics and physics, the centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure. Informally, it is the point at which a cutout of the shape could be perfectly balanced on the tip of a pin.The definition extends to any object in n-dimensional space: its centroid is the mean position of all the points in all of the coordinate directions.While in geometry the word barycenter is a synonym for centroid, in astrophysics and astronomy, the barycenter is the center of mass of two or more bodies that orbit each other. In physics, the center of mass is the arithmetic mean of all points weighted by the local density or specific weight. If a physical object has uniform density, its center of mass is the same as the centroid of its shape.
In geography, the centroid of a radial projection of a region of the Earth's surface to sea level is the region's geographical center.

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  1. CivilSigma

    Calculating Centroids of a Triangle: Why Do Formulas Switch?

    Homework Statement Hello. In class we learned how to calculate centroids of a triangle. For the y-coordinate we use the formula height/3 and for the x we use 2/3 * base Homework Equations In the diagram, If I want to calculate the y coordinate , I do 72 (which is the proposed height)/ 3 ; only...
  2. J

    Find Centroid of Bounded Region: Accurate to 0.001

    Find the centroid of the region bounded by y = 1.5x2 − 14x + 23.5, and x − y = 8. You should enter the coordinates of your answer either as decimals or fractions. Your answer must be accurate to within 0.001. I got a wrong answer of (7/3, 59/52), I'm having problems plugging in the numbers. Any...
  3. M

    MHB Vector algebra- centroid of tetrahedron

    How to find out the position vector of the centroid of tetrahedron , the position vectors of whose vertices are a,b,c,d respectively. I am familiar with the result, namely a+b+c+d/4 but want to know how to derive it without using the 3:1 ratio property. Any help would be appreciated. Thank you.
  4. O

    Find centroid of catenary (y=cosh(x))

    Homework Statement A uniform chain hangs in the shape of the catenary y=cosh(x) between x=−1 and x=1 Find \bar{y} Homework Equations ∫\bar{y}\rhods=∫y\rhodsThe Attempt at a Solution I can define ds (some small segment of the arc) by \sqrt{1+sinh(x)^2}dx. Also, y is given as y=cosh(x) If I...
  5. A

    Calculating the Centroid of a Triangle: Tips and Tricks

    Hi friends, I have been trying to calculate the centroid of triangle (show in attachment) . I have got centroidal Y= h/3 Not able to get centroidal X = (1/3)*(a+b) Could anyone help on this stuff? Thanks..
  6. I

    MHB What's the Correct Way to Calculate the Centroid of the Given Region?

    $y=x^3, x+y=2, x=0$ $A=\int_{0}^{1} \ (2-x-x^3) dx = [2x-\frac{x^2}{2} - \frac{x^4}{4}]_{0}^{1} = 5/4$ I found x and it was correct but there's something wrong with my y. Here's my work. $\frac{4}{5} \int_{0}^{1} \ \frac{1}{2}[(2-x)^2-(x^3)^2]dx = \frac{4}{10}\int_{0}^{1} \...
  7. B

    Centroid of composite area (one question)

    I was doing this example from my textbook and looked at the solution guide. I'm confused as to where the textbook got a certain number. In the numerator of their work, they got (1)(6)(2), which I'm assuming is the width of the skinny rectangle, the total length of the bottom section...
  8. J

    How Does Centroid Affect Hydrostatic Pressure Calculation?

    Homework Statement Can you please help me I think I have confused something.We did this in class. A triangular plate with base = 10 m and height 12 m is placed under water with the base at the top at a depth of 8 meters under water and the apex 20 meters under water. Find the hydrostatic...
  9. C

    Finding the Centroid of a Coke Bottle

    I'm trying to find the centroid of a coke bottle, which is basically like a cylinder with varying radius. I have a 1x500 vector of radius points and I have the volume of the bottle, any thoughts on how to calculate the centroid numerically?
  10. J

    Find Centroid of Parabola Bounded by x Axis

    Homework Statement y = 16 -x^2 find centroid bounded by x axis Homework Equations x = (1/A) ∫ x(f(x)) dx and (1/A) (1/2)(f(x))^2 dx = y The Attempt at a Solution I just applied it. It is a weird because I would of thought that x would of been at 0. But I didn't get that I x =...
  11. J

    Find Centroid of Region: No Integration Needed

    Homework Statement Wondering if I did this right. Find the centroid of the region shown , not by integration, but by locating the centroids of the rectangles and triangles and using additivity of moments. Homework Equations I will give you the coordinates since I can't draw it. For...
  12. J

    Find Centroid of Bounded Curves

    Homework Statement Find the centroid bounded by the given curves. y = x^3 , x+y = 2, y = 0 So if you graph this you will see that if you did it with respect to x you would have the break the graph at the point (1,1). Or at least I think you would have to. So in my book they give the...
  13. A

    A Question related to triangle and centroid .

    Homework Statement The following is a geometry question I can't seem to get. "Consider an acute angle △ABC. Points D, E, F are mid points of sides BC, CA and AB respectively. G is the centroid of △ABC. Area of △AFG = 14, EC = 15/2. Perpendicular distance of F from BC = 6. Find BC2−AB2 "...
  14. Y

    What is the Centroid of a Wire and How to Calculate It Using Geometry?

    Homework Statement Homework Equations ƩxiLi=0 The Attempt at a Solution Using geometry to find the lengths and rotating so AB is horizontal, I have: ƩxiLi=0: [((64/80)*L)/2]*L + (-64/2)*80 + (-14/2)*100 = 0 L = 90.28mm
  15. X

    Centroid of a bounded region: Help

    Homework Statement Find the centroid of the region bounded by the graphs of y = sqrt (x) and y = (1/2) * x Homework Equations A = [f(x)-g(x)]dx from point a -> b The Attempt at a Solution x = [0,4] ; p(0,0) and p(4,2) I am just checking on if I did the integral correctly. A...
  16. J

    Properties of the centroid of an area

    Hello, I am calculating the centroid of an area using this equation for composite bodies: sum(indivudual area times individual centroid)/sum(individual area) = distance from axis to centroid this is just y bar for a cross section that is symmetrical about the y axis I was under the...
  17. S

    Centroid of a circular segment

    Homework Statement I need to find the centroid of a circular segment. I know nothing of calculus, and this is part of an analysis for statics that goes beyond the material covered in class. Homework Equations I've seen this equation for calculating the centroid...
  18. G

    Centroid of non-right triangle

    Homework Statement See attachment Homework Equations The Attempt at a Solution If the y-coordinate of the center of mass is given by (1/A)*∫y dA, how come the solution uses bh/2 as the area? This triangle isn't right angled so that area formula should not hold. What am I missing?
  19. T

    What is the Limit of the Centroid of an Area Bounded by the x-Axis and 1-x^n?

    Homework Statement Okay, so the idea here is to take the centroid bounded by the x-axis and 1-x^n. N should be an even and positive integer. We should take the limit as n approaches infinity of both the x- and y-coords of the centroid, hopefully ending up with (0, 1/2). Homework Equations...
  20. B

    MHB Centroid problem with y vector

    [SOLVED] Centroid problem with y vector So, I've been having a problem with this part here, the equations are y=4x^2 and y=2x3. So, the intersections are (0,0) and (2, 16) The area between the two is 8/3. I know I'm doing something correct because I get the x vector correct with ease, but...
  21. J

    Calculate Centroid of Hemispherical Dome w/ Top Removed

    7) Pile foundations are the part of a structure used to carry and transfer the load of the structure to the bearing ground located at some depth below ground surface. A concrete pile for a domestic dwelling has a radius of 1000 mm at its base, and 2000 mm at the top. It is 2000 mm deep and...
  22. P

    Find Centroid: x(bar), y(bar) = (?, ?)

    1. Find the centroid of the region determined by the graphs of the inequalities. y ≤ (1/3)x^(2) ; (x-3)^2 + y^(2) ≤ 9 ; y ≥ 0 x(bar) , y(bar) = (?, ?) I did the problem, but i do not know the answer because the problem is an even number and the book only have answers for the odd...
  23. H

    Finding coordinates of a centroid

    Homework Statement Find the exact coordinates of the centroid for the region bounded by the curves x=5-y^ 2 and x=0 I am not sure about this one because it uses dy instead of dx i think. I tried to set it up like this x= [0.5∫(5-y^2)^2 dy from -5 to 5]/[ ∫(5-y^2)dy from y=-5 to 5]...
  24. T

    Finding the combined centroid of two regions given the centroids of both regions

    Homework Statement The Attempt at a Solution I tried to do this problem by taking the average of the y-bar centroid values but that gave me the wrong answer. I am only interesting in knowing why this method is incorrect. Thanks!
  25. thekey

    Finding the centroid of the region

    Hi VVVVVV Find the centroid of the region bounded by the curves y = 2x - 4 , y = 2 Sqr x, and x = 1. Make a sketch of the region.
  26. A

    Proving the Centroid of ABC Triangle

    Homework Statement let ABC be a triangle where I divides angle BAC(angle A) => BAI=IAC Prove that I is the centroid of (B,AC)and (C,AB) Homework Equations i think phitagors wil come in handy but don't know how to use it The Attempt at a Solution let ac = a and AB = b aIB+bIC=0 (vectors)...
  27. L

    Find the centroid of the solid in part (a).

    Homework Statement Find the volume of the solid that lies above the cone ϕ=pi/3 and below the sphere ρ=4cosϕ . Homework Equations Find the centroid of the solid in part (a).. The Attempt at a Solution For the volume I got 10pi which I am fairly sure is correct. I attempted trying to...
  28. M

    Find the centroid of the solid

    Homework Statement The solid bounded by the surface z=y2 and the planes x=0,x=1,z=1 I have a question regarding the limits of integration, would it be incorrect, if when I graphed z=y2 I changed it to a familiar xy graph instead I just graphed it as if z was y and x was y. Pretty...
  29. C

    What Is the Centroid of the Region Bounded by y = (4x-x^2) and y = x?

    Homework Statement Find the centroid of the region bounded by the curves y = (4x-x^2) and y = x. Homework Equations Centroid = (x-bar, y-bar) x-bar = My/M y-bar = Mx/M M = ∫ρ[f(x) - g(x)]dx My = ∫ρ(x)[f(x) - g(x)]dx Mx = ∫(1/2)ρ[{f(x)}^2 - {g(x)}^2]dx The Attempt at a...
  30. 3

    Centroid Problem: Find xbar, ybar in Quadrant 1 of xy-plane

    Homework Statement Find the centroid of the region in the first quadrant of the xy-plane bounded by the graphs of the equations: x=0, x=1, y=x-x^2, and y^2=2x Homework Equations xbar: integral of x dA/ integral of dA ybar: integral y dA/ integral of dA The Attempt at a Solution...
  31. C

    Composite Bodies: Finding the Centroid

    Homework Statement A single-scoop ice cream cone is a composite body made from a single scoop of ice cream placed into a cone. Assume that the scoop of ice cream is a sphere with radius r = 3.65 cm that is placed into a 9.90 cm tall cone. The interior height of the cone is 9.00 cm. The cone...
  32. J

    How Do I Find the Centroid of a Cone Using Integration?

    Homework Statement Determine the centroid of the conical volume using integration. Height h and radius r. No numbers given.Homework Equations V = ∫dV = (from 0 to h)∫∏r2dzThe Attempt at a Solution I know from looking around in my book where zbar is (xbar at zero and ybar at zero) but need to...
  33. P

    How Is the Centroid of a Triangle Determined Mathematically?

    We got given the formula for finding the centroid of a triangle which was to (\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}) but we the teacher said he didn't have time to show the proof, however I'm still curious. Could anyone possible hint on how I could go about to prove this? My...
  34. mnb96

    Question on center of mass (centroid)

    Hello, I was thinking of the following question: suppose we have a body in two-dimensional space and its density is described by the function f(x,y) (Note! in Cartesian coordinates!). The x-coordinate of the center of mass is given by C_x= \frac{\int\int xf(x,y)dxdy}{\int\int f(x,y)dxdy}...
  35. N

    What is the centroid of a C-shape?

    Homework Statement http://prntscr.com/fcbm8 Find the centroid - All dimensions are in mm Homework Equations xbar = (A1X1+A2X2) / (A1+A2) Similarly for Ybar I assume The Attempt at a Solution I got the y co-ordinate to be 20.428mm, and would assume that the x coordinate...
  36. G

    What does the Centroid of area under Rate vs Time plot represent?

    What does the "Centroid" of area under Rate vs Time plot represent? Does the 'centroid' of the area under a rate vs time plot represent anything? I have a bunch of rate vs time plots and was trying to think of a way to compare them, other than just cumulative volume which is area under the...
  37. S

    How Do You Calculate the Centroid of a Sideways Parabola?

    Homework Statement Find the centroid of the region bounded by the curve x=5-y2 and x=0. Answer is (2,0). Homework Equations A = ∫[f(x)-g(x)]dx a->b xbar = 1/A ∫ x[f(x)-g(x)]dx from a->b ybar = 1/A ∫ 1/2[f(x)2-g(x)2]dx The Attempt at a Solution I know that the graph is a...
  38. B

    How to Prove the Weighted Centroid Formula for Parallel Forces?

    Given the formula for the weighted centroid, C of a system of parallel forces Fi acting at points Pi, show that two weights W and 3W, acting respectively at points P1, P2, are together equivalent to a single weight 4W acting at P, where P lies on the line segment joining P1 to P2 and P1 P : P P2...
  39. D

    Centroid of a TILTED Semi circle

    Centroid of a "TILTED" Semi circle! Guys, could anyone help me out how to solve the centroid of these "tilted" semicircle. I really don't have any idea how to get its centroid since it's tilted..Please I really want to understand so kindly do it step by step, Thanks.. Just Find the x-bar and...
  40. J

    Finding Centroid Coordinates Under Degree 3 Bezier Curve

    Does anyone have an easy formula for finding the centroid coordinates under a degree 3 bezier curve? The whole parametric two variable thing isn't working with the formulas I know of. Thanks
  41. D

    Centroid of a right angle triangle

    Hi i have a right angle tringle, base=1200m and h=400. I have trouble understanding on how to determine the centroid i know one side will be 2/3*x and another side would be 1/3*y. But i don't understand how to write it out. A friend said its 1/3*1200*400*2/3*1200, but i have no idea and...
  42. O

    Find centroid two-dimensional region

    Homework Statement Find the centroid of the region bounded by the given curves. y = x^3x + y = 10y = 0 Homework Equations \bar x = \frac{1}{A}\int^b_a xf(x) dx\bar y = \frac{1}{A}\int^b_a \frac{1}{2}(f(x))^2 dx Where A is the area of the region containing the centroid. The...
  43. M

    Finding the y Component of the Centroid: Is My Reasoning Valid?

    The test question was... We are given a region bounded by the functions √x and x. We were also given the density and mass of the region and asked to only the component y of the centroid. I did not know how to implement these given variables, density and mass into my equation for y bar so I...
  44. K

    Bending Moment & Neutral Axis: Why Not Centroid?

    Hey guys, as the title says I can understand why the neutral axis does not pass through the centroid after reaching a point where the bending moment is max before the beam collapses. Say you have a straight structure supported by two double I-Beams and a weight in the middle. Why does the...
  45. M

    What is the centroid of the given root locus?

    i just want to make sure I'm doing this correctly as there is some discrepancy in my textbook. i know the centroid is equal to (Ʃpoles - Ʃzeros) / (# of poles - # of zeros) the equation i have is s2 + 2s + 8 / s(s2 + 2s + 10) so the zeros are -1 + i√7 and -1 - i√7 and the poles are 0, -1...
  46. C

    Moments,Center of Mass, & Centroid

    Homework Statement Find Mx,My, & (x bar, y bar) for the laminas of uniform density ρ bounded by the graphs of the equations. (Use rho for ρ as necessary.) x=-y x=5y-y2 Homework Equations m= ∫f(x)-g(x) dx my= ∫x(f(x)-g(x)) dx =>x bar my/m mx= 1/2 ∫ (f(x))2-g(x))2dx => y bar=mx/m...
  47. H

    Centroid of an irregular shape

    Homework Statement As part of a test I was given an irregular shape to find its centroid. It was a rectangle with a right angled triangle on its right side (I don't have a picture to upload for ye unfortunately). Homework Equations dA=y.dx X(bar)=intrecal x.y.dx Y(bar)=intrecal y^2/2.dy The...
  48. M

    What is the centroid of a region D lying above a sphere and below a paraboloid?

    Homework Statement Find the centroid of the region D, lying above the sphere x2+y2+z2 and below the paraboloid z=4-x2-y2. Homework Equations The Attempt at a Solution I decided it might be easier to change it to polar coordinates soo I did M=∫∫∫ dzrdrdθ where z goes from...
  49. D

    Finding the Centroid of the surface of a hemisphere

    Hi gang, I was given this question, but I can't quite seem to get it all the way... 1.Determine the coordinates of the centroid of the surface of a hemisphere, the radius of which is r, with respect to its base. Homework Equations A=2\pi r^{2} A: Surface Area of the Hemisphere...
  50. F

    Question in centroid for given figure

    hi all .. Hello .. I am a new member .. I have a test next week .. I have some difficulty in such matters .. http://store2.up-00.com/Nov11/Eac12548.jpg [PLAIN][PLAIN]http://store2.up-00.com/Nov11/UKV12548.jpg thanks
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