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. D

    Finding the Centroid of a Solid

    Homework Statement Find the volume and the centroid of the solid E that lies above the cone z=√x^2+y^2 and below the sphere x^2+y^2+z^2=49. Homework Equations The Attempt at a Solution My bounds were: \theta=0 to 2\pi \varphi=0 to \pi/4 \rho=0 to 7cos(\varphi) So my integral...
  2. K

    Finding the Centroid of a Parabola: Calculating Coordinates for y = 10m

    The equation y = x2 describes a parabola. Find the x and y coordinates of the centroid of the area bounded by this curve, and the line y = 10m
  3. P

    Finding the centroid of a cardiod curve.

    Homework Statement A cardioid or heart shaped curve is given in plane polar coordinates \rho,\phi by the parametrisation \rho=1-cos\phi or equivilantly \rho=2sin^{2} (\frac{ϕ}{2}) for 0\leq\phi\leq2\pi A shape resembling an apple is generated by revolving the upper (+y) half of the cardioid...
  4. M

    Coordinates of the centroid of the lamina - problem with finding limits

    Homework Statement The question asks me to find the coordinates of the centroid of a uniform lamina enclosed by curve y=1-x^2. X axis and Y axis are in the 1st quadrant. Can you please tell me how to work out the limit for this equation?
  5. I

    Centroid of an object in a fluid

    Hi friends just a quick question, does the centroid of an object change if it is submerged in a fluid and is on an incline? Or indeed does the centroid of an object ever change when in a fluid or inclined? Thanks! Claudius
  6. D

    Find the centroid of the region

    Homework Statement Find the centroid of the region bounded by ... y=x^3, x+y=2, y=0Homework Equations The Attempt at a Solution I think you find the area of the region which is 5/4
  7. J

    Centroid of a 3D Region using Triple Integral

    Homework Statement Compute the centroid of the region defined by x^{2} + y^{2} + z^{2} \leq k^{2} and x \geq 0 with \delta(x,y,z) = 1. Homework Equations \overline{x}=\frac{1}{m}\int\int\int x \delta(x,y,z) dV \overline{y}=\frac{1}{m}\int\int\int y \delta(x,y,z) dV...
  8. P

    Find Centroid of Shape: X & Y Dimensions from Point O

    Homework Statement CALCULATE POSITIONS OF THE CENTROID OF THE SHAPE SHOWN BELOW GIVE DIMENSIONS FROM POINT O ALONG THE HORIZONTAL X-DIRECTION, AND THE VERTICAL Y-DIRECTION. [PLAIN]http://img402.imageshack.us/img402/738/unledgj.jpg Homework Equations The Attempt at a Solution I...
  9. A

    Why do we use the distance to the centroid when finding the resultant force?

    When find the resultant force on a plane surface under uniform pressure why do we use ybar, the distance to the centroid when the resultant force acts through the centre of pressure? Shouldn't we use the distance to the centre of pressure?
  10. M

    Calculating the Centroid Position of a Symmetrical T-Section

    Basically for the last three hours I have been struggling with this question. "Calculate the position of the centroid of a symmeterical t-section having a verticle stem 40mm wide by 120mm high, and a top section which is 80mm wide and 40mm high." The answer is meant to be 92mm but I have...
  11. P

    Using centroid of object to find moment of inertia

    Hey, I was doing some problems involving finding the moment of inertia of a spun spandrel, and I came across the idea of using the centroid to find the moment. For example, if you have a parabola, find the centroid. If you're rotating around the x-axis (y=x^2), then find y_bar and multiply...
  12. T

    Find centroid of region - triple integrals, please

    Homework Statement Find the centroid x,y,z of the region R cut out of the region 0<=z<=5sqrt(x2+y2) by the cylinder x2+y2=2x. Homework Equations x2+y2 = r2 x= rcosθ y= rsinθ The Attempt at a Solution Centroid x being Mx/m I'm guessing I've been working on this problem...
  13. I

    Statics - Centroid of a 2-D figure

    Homework Statement Problem Homework EquationsArea of a semi-circle, area of a semi-sector of a circle, centroid of a semi-circle, centroid of a sector, general equation for centroid of a plate: (X-Bar) * \SigmaArea = \Sigma(x-bar * Area) The Attempt at a SolutionAttempt. I tried my...
  14. I

    Centre of Pressure and Centroid Fluid Mechanics

    1. Homework Statement a 2m by 4m window is to be installed under water at a depth of 50 m with seawater on one side and atmospheric pressure on the other. Calculate and plot the distance between the centroid and the centre of pressure starting from when the window is installed at surface...
  15. I

    Centre of Pressure and Centroid Fluid Mechanics

    Homework Statement a 2m by 4m window is to be installed under water at a depth of 50 m with seawater on one side and atmospheric pressure on the other. Calculate and plot the distance between the centroid and the centre of pressure starting from when the window is installed at surface such...
  16. F

    Can You Solve for Theta in Terms of a Known Expression?

    This is part of a bigger problem, I need to find an expression in terms of θ for the expression below. 1. x=(2r sin⁡θ)/3θ where x=r/2 2. r/2=(2r sin⁡θ)/3θ 3. r=(4r sin⁡θ)/3θ 4. 1=(4 sin⁡θ)/3θ 5. 1/4= sin⁡θ/3θ 6. 3θ/4= sin^-1(θ) I don't know where to go from there. All...
  17. S

    Is the centroid always the centre of gravity for a triangle of uniform density?

    Centroid...centre of gravity! How to prove that centroid is the centre of gravity of a triangle...
  18. M

    Finding Centroid of Triangle in First Quadrant

    Homework Statement Find the centroid of the triangular region in the first quadrant bounded by the line 4x+y=4, the x-axis, and the y-axis. Homework Equations \int\int\deltadA xbar = My/M ybar = Mx/M The Attempt at a Solution Is the boundary for the dy from 0 to 4, and dx from 0 to...
  19. W

    Statistical Error of Centroid of Gaussian Distribution

    Statistical "Error" of Centroid of Gaussian Distribution If I have L data samples, distributed randomly (3D real Gaussian distibution, unity variance) about the origin in 3D real space, how can I derive an expression for the "origin estimation error" (i.e. the distance between the true origin...
  20. T

    Distance from centroid to surface of polyhedron as a function of frequency

    Hello members, I am a bit stumped on a mathematical problem. If this is in the wrong section I apologize. Recently, I've become interested in polyhedrons. However, I am having a bit of trouble with their analysis. Consider a given dodecahedron. If we consider the solid to be of uniform...
  21. A

    Calculating the Centroid of x=2-(y^2) and the Y Axis

    Homework Statement Easy question: Find the centroid of x=2-(y^2) and the y axis Homework Equations The Attempt at a Solution Already calculated, I got an area of (4/3)(√2) Cx = 4/5 Cy = 0 When calculating Cx, in the book they have: Integral (x)(√(2-x)) And go...
  22. W

    Finding Centroid Location using Symbolic Math in Matlab

    I'm at a complete loss on this one, so far. This problem needs to be solved symbolically with Matalb (which won't be a problem once I know what to do), but I'm not even sure where to begin. Any help would be greatly appreciated [PLAIN]http://img227.imageshack.us/img227/9344/prob18.jpg
  23. S

    Difference between centroid and break away/in point

    Hi, I am studying control engineering and we are covering root locus concept right now. My knowledge of this concept is varied..I get some things and others I don't. One thing I don't understand is the difference between centroid and breakaway/in point. So for example for this function...
  24. T

    Deriving centroid of quarter circle.

    Homework Statement Find the centroid of the region cut from the first quadrant by the circle x^2+y^2=a^2Homework Equations I know that y' = (I y dA)/(I dA)The Attempt at a Solution Taking a strip dy with length x, I obtain dA = dy.x dy(a^2-y^2)^(1/2) So I y DA = I y*(a^2-y^2)^(1/2) dy from...
  25. D

    Calculate 90 Degree Pipe Elbow Centroid with Excel or MathCAD

    Looking for 90 degree pipe elbow centroid Excel or MathCAD calculation.
  26. R

    Finding Centroid and Second Moment of Area for Semi-Circular Section

    Homework Statement I need to get the centroid of this shape http://img713.imageshack.us/img713/5607/section.jpg it's a semi-circular section basically. On the left most side, the distance from the origin to to where it cuts the y-axis is 'R' and the distance from the x-axis to the end of the...
  27. B

    Strange thing i noticed about centroid of a cone vs triangle

    To calculate the centroid of a cone, it seems that you have to use calculus. It comes out to be h/4, where h is the height from the base of the cone. But intuitively I thought that the centroid would have been h/3 because that's a triangle's centroid, and the cone can be obtained by rotating a...
  28. P

    How do I locate the centroid and the center of gravity of a square pyramid?

    Height=104 Base=220 x 220 Density=2500
  29. C

    Centroid Problem Homework Solution

    Homework Statement A blade on an industrial fan has the configuration of a semicircle attached to a trapezoid (see figure). Find the centroid of the blade. Homework Equations The Attempt at a Solution My plan was to solve for the centroid of the trapezoid, then of the semi circle, then...
  30. D

    Finding the Centroid of a Spherical Crescent

    Hello, What I really want to do is deal with spherical "crescents" and incomplete annuli and see how well they are approximated by spherical caps, but here is my question: How would you go about finding the centroid of a spherical crescent (one spherical cap minus the other) in the case when...
  31. L

    What is the correct centroid for a semicircle?

    The centroid of a semicircle is noted at being (4r)/(3 pi) - http://en.wikipedia.org/wiki/List_of_centroids. However, when I did the work myself using the integral of y da over the area, I came up with (2 r)/(pi). I figured I was doing something wrong so sought out someone else's work and found...
  32. Z

    What is the Centroid of a Cylindrical Cone?

    Homework Statement Determine the centroid of volume for a right circular cone with base diameter of 100mm and an altitude of 200mm. Homework Equations I know that if the my xy-plane is parallel to the base of the cylindrical cone then the x and y coordinates of the centroid must be...
  33. J

    Lower Level integration problem (Find the centroid)

    EDIT - Solved. Thank you. Homework Statement Find the Centroid of the region bounded by the x-axis and \sqrt{9-x^2}Homework Equations So far, I found my Mx which ended up being 27p after applying this forumula: p\int(\frac{\sqrt{9-x^2}+0}{2}) (\sqrt{9-x^2}-0)dx *My interval was [-3,3] The...
  34. P

    Centroid and centre of pressure

    Homework Statement Explain why the distance between the centroid and the centre of pressure for a plane submerged fluid decreases as the depth of fluid increases. I know that the centre of pressure for a plane submerged fluid is located below the centroid because pressure increases with...
  35. W

    Find the centroid of the solid bounded below by the cone

    Homework Statement Find the centroid of the solid bounded below by the cone z = \sqrt{3(x^2+y^2)} and bounded above the sphere x^2+y^2+z^2=36. Homework Equations Let G be the given solid and denote its volume by V_{G}=\int \int \int_{G} 1 dV. \frac{\bar{x}= \int \int \int_{G} x...
  36. C

    Centroid of a triangle using Green's theorem

    Homework Statement Given a curve C that starts from the origin, goes to (1,0) then goes to (0,1), then back to the origin, find the centroid of the enclosed area D. Homework Equations \bar{x} = {1/(2A)}*\int_C {x^2 dy}\bar{y} = -{1/(2A)}*\int_C {y^2 dx}The Attempt at a Solution Well, obviously...
  37. B

    Finding Centroid of 2D Shapes & Parabolas

    I think I'm right when I say that the centroid of a 2D shape is found by the intersection of the lines that separate that shape into two shapes of equal areas. Is that correct? I don't want to (for now) think about it in terms of moments and integrals, because frankly, it's a little confusing...
  38. J

    Need someone to check my work: Finding the exact coordinates of the centroid

    1. Homework Statement [/b] Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid: Homework Equations I don't have the answer for this problem to check my work. The Attempt at a Solution see...
  39. J

    Centroid of the region bounded by the curve

    centroid of the region bounded by the curve...need help! Find the centroid of the region bounded by the curve x=2-y^2 and the y-axis: my work shown: therefore if A= 2 times the integral of sqrt(2-x) dx is the M_x equal to the integral of (2-x) dx from 0 to 2? and the M_y equal to...
  40. N

    Can the Centroid Symmetry Argument Help with Graph Coordinates?

    I am confused with the coordinates when using the symmetry argument for this graph. Is Mx=0 using the symmetry argument ?
  41. U

    Fluid Mechanics - Distance to Centroid

    Homework Statement http://i284.photobucket.com/albums/ll12/uradnky/fluid.jpg I'm asked to solve for the force on the projected plane surface and show where it is acting. This problem seems pretty easy, but with the dimensions given I can't find Lc, so I can't find dc to begin it. Is...
  42. B

    Centroid of a semicircular arc

    Homework Statement Here, I have two ways of finding the y-coordinate centroid of a semicircular arc using polar coordinates. First one is considering a circle of radius a, centred at the origin. What I have done is \int ds = \int_0^{\pi} a d\theta = \pi a and then \int y ds = \int r^2 sin...
  43. M

    Determining the Centroid of an Area: Statics Homework Problem

    Homework Statement Determine The Centroid of The Area Homework Equations x_centroid = [(Area of each piece)(Centroid of each piece)]/(Area of each piece) The Attempt at a Solution This seems like a really basic problem, I am just reviewing for my statics final on saturday. The only problem...
  44. R

    Vectors - Prove the following relation about the centroid

    Homework Statement http://img147.imageshack.us/img147/733/vectorsba4.png Given is the triangle OAB and a variable point P. G is the centroid. Prove that: PA2+PB2+PO2=GA2+GB2+GO2+3 (PG2) The Attempt at a Solution I treat O as the origin. Vectors are denoted in bold. Position vectors...
  45. R

    Centroid of a solid of revolution

    Homework Statement The curve 3x2+2y2-12y=32 is rotated about the x-axis and forms a solid hemisphere. Verify that the weight is 8cm from the bottom of the hemisphere. Homework Equations The Attempt at a Solution Now, I can only do a little bit in centroids but that is for...
  46. R

    I with this finding a centroid of a triangle

    Homework Statement Well I just need to understand how to find the centroid of a triangle, I know it's 2/3 from the vertex, but I need to know how finding it is done. Homework Equations \overline{x}= \frac{\int x dA}{\int dA} \overline{y}=\frac{\int y dA}{\int dA} The Attempt...
  47. J

    How to Calculate the Centroid of a Half Circle?

    Homework Statement Calculate the location of the centroid (See PDF) ---------------------------------------- I am stuck on how to find the centroid of a half circle, item #5. I don't how to find the distance of that centroid on the x-axis. Any help is appreciated. Thanks!
  48. CalleighMay

    Find the centroid of the solid region ?

    Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help! I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is...
  49. F

    Find the volume and centroid of the solid

    Homework Statement Find the volume and centroid of the solid E that lies above the cone z= sqrt (x^2 + y^2) and below the sphere x^2 + y^2 + z^2 = 1. The Attempt at a Solution I found the correct volume=(pi/3)(2-sqrt2) How do I find the centers of mass if I don't know the mass and...
  50. K

    Centroid of a Solid (triple integral)

    Homework Statement Find the centroid of the solid: the tetrahedron in the first octant enclosed by the coordinate planes and the plane x+y+z=1. Homework Equations xcenter = \frac{\int\int\int_G x dV}{V} ycenter = \frac{\int\int\int_G y dV}{V} zcenter = \frac{\int\int\int_G z dV}{V} The...
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