Centroid Definition and 178 Threads

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. P

    Proving the 2:1 Ratio of a Triangle's Medians at the Centroid Using Vectors

    I am trying to figure out how to prove the 2:1 ratio of a triangle's medians at the centroid using vectors. Example if I had a triangle ABC with midpoints D of BC, E of AC and F of AB. I know G is where the medians intersect. I have seen many proofs and understand the process that proves the...
  2. A

    Centroid ( center of mass ) of cardioid revisited

    Centroid ("center of mass") of cardioid... revisited This is actually a variation of the https://www.physicsforums.com/showthread.php?t=119622". As it happens, the cardioid I'm working with is identical to jbusc's: r = 1 + \cos(\theta) and \delta \equiv 1 The task is to find the...
  3. K

    Where is the Centroid of a Moebius Strip? [SOLVED]

    [SOLVED] centroid of a moebius strip Since the edge of a moebius strip is topologically equivalent to a circle, we can cut a disk form an euclidean 2D plane and sew a moebius strip on its place. Now we have a 2D space with a circular-edged moebius strip added. And now this strip is a 2D region...
  4. R

    Arc length help extended to surface area and centroid.

    Homework Statement A curce,C, has equation y=x^{\frac{1}{2}}-\frac{1}{3}x^{\frac{3}{2}}+\lambda where \lambda>0 and 0\leq x \leq 3 The length of C is denoted by s. Show that s=2\sqrt{3} The area of the surface generated when C is rotated through one revolution about the x-axis is denoted...
  5. W

    Finding the Centroid of a Half-Cone: Tips and Troubleshooting

    The question is this: find the centroid of the half-cone sqrt(x^2 + y^2) <(oet) z <(oet) 1 and x >(oet) 0 (oet being or equal to, I apologize for the lack of sophistocated symbols). I thought I was doing it correctly, but my answers do not match up with those in the book. I assumed...
  6. J

    What Is the Correct Approach to Find the Centroid of a Parabolic Area?

    [SOLVED] Centroid Parabolic area Homework Statement http://img227.imageshack.us/img227/7518/slowge9.th.jpg Find the centroid of the area.Homework Equations The Attempt at a Solution I'm not quite sure what I'm screwing up on this problem, I can do other problems like when y = x^2. I have...
  7. B

    MATLAB Matlab: Centroid of a cluster of points

    I have collected and plotted thousands of data points and would like to now find where the center of this "data cloud" lies. I was wondering if anybody had an idea as to how I could approach this in Matlab. Thanks in advance.
  8. N

    Multiple integration + Centroid Help.

    Homework Statement Find the volume bounded by sphere rho = rt. 6 and the paraboloid z = x^2 + y^2 and locate the centroid of this region The attempt at a solution http://www.mathhelpforum.com/math-help/latex2/img/4deb41286077aabd94b30802f0e6a68a-1.gif So Thats the integral that I...
  9. J

    Find Centroid of U: First Octant Ellipsoid

    Homework Statement Let U be the solid region in the first octant bounded by the ellipsoid (x^2)/4 + (y^2)/9 + (z^2)/4 = 1. Find the centroid of U.Homework Equations The Attempt at a Solution I tried to do this problem but I'm not sure if my answer is right. First, I find the mass and I got...
  10. P

    How can the centroid and triangle angles help prove equilateral triangles?

    I found this problem off of mathematics magazine and I want to give it a try solving it, but I'm lost for ideas. The problem states the following: Let G be the centroid of triangle ABC. Prove that if angle BAC = 60 degrees and angle BGC = 120 degrees then the triange is equilateral. My...
  11. B

    Finding the Force Exerted on a Log Using Virtual Work Calculations

    Virtual work (fireplace thong) Homework Statement http://s2.photobucket.com/albums/y31/bambidurojay/?action=view&current=fireplacetong.jpg I have been given a fireplace thong and need to determine the force exerted on a log. Homework Equations It is in the virtual work chapter of my...
  12. M

    How do you find the moment of inertia about a centroid of a isoceles triangle?

    Pls do respond...
  13. G

    Find Centroid of y=x^2: Homework Help

    Homework Statement Just studying statics in my first year of university and i do not understand the centroid topic at all. The question goes like this, given a equation y=x^2 , and the x-axis is 0 - 4 , with the lower portion of the graph is highlighted. It is given by diagram but sorry I...
  14. C

    Finding the centroid of the triangular region

    Homework Statement Find the coordinates of the centroid G of the triangular region with vertices (0,0),(a,0),(b,c). Homework Equations for the centroid x = (1 / area) * double integral ( x dA) y = (1 / area) * double integral ( y dA) The Attempt at a...
  15. S

    Find Centroid: Solve Homework Statement

    Homework Statement http://img170.imageshack.us/img170/214/centroidssk4.th.jpg Homework Equations \bar{Y}A_{TOT}=\bar{Y_1}A_1+\bar{Y_2}A_2+\bar{Y_3}A_3 The Attempt at a Solution by observation: \bar{Y_1}=7.5mm \bar{Y_2}=82.5mm \bar{Y_3}=215mm A_1=(15mm)(150mm)=2250mm^2...
  16. B

    Finding the Centroid of a Shaded Area with y=sqrt(x) Curve: Step-by-Step Guide"

    Homework Statement I need to locate the centroid of the shaded area, in my picture. The shaded area is under the line(in between the x-axis and the curve.) The curve is y=sqrt(x) And stretches a length from the y axis(0,0), along the x axis, a length of b. This is a new concept...
  17. B

    Finding the Centroid Using Green's Theorem

    Can someone help me with the following? I'm supposed to find the centroid of a region D using Green's Theorem. Assume that this density function is constant. ∫Pdx + ∫Qdy = ∫∫(dQ/dx)-(dP/dy) A = ∫xdy = -∫ydx = ½*∫xdy - ydx I know that the mass of a region D with constant density...
  18. S

    Finding coordinates of the centroid

    Sketch the region bounded by the curves and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. y=1/x, y=0, x=1, x=2
  19. S

    Centroid formula homework help

    Ok, I think I've figured this out but the book gives me different answer for my x value: y=4-x^2 y=0 A= \int_0^2(4-x^2)dx A=16/3 x bar= \frac{3}{16}\int_0^2 x(4-x^2)dx and this comes out to 3/4..The book says 0 for the X-value...where did I go wrong?
  20. T

    What is the Formula for Finding the Centroid of a Shape?

    Does anybody has the formula sheets for the centroid formula for rectangular,semicircular... or any website recommended. pls help thanx
  21. J

    Centroid ( center of mass ) of cardioid

    centroid ("center of mass") of cardioid I'm having trouble calculating the centroid of the cardioid (and various other polar-coordinate-defined lamina), i.e., r = 1 +\cos \theta I can see how it's symmetric over the x-axis, so y-bar is zero. So to calculate x-bar, then I do...
  22. E

    Finding bounds of a centroid problem

    I am having a problem finding the upper and lower (x,y) bounds for this problem. Find the centroid of r = 1 + cos(theta) which lies in the 1st quadrant. I come up with (2,0) and (1,0) or the axis intercept points. Is this the correct way to go about it? m=((&int;)[0]^2 ) (&int;)[0]^1
  23. B

    How do I find the centroid of a solid with given volume and no density function?

    Hi, I am having trouble with the following question. Can someone help me out? a) Find the volume of the solid that lies above the cone \phi = \frac{\pi }{3} and below the sphere \rho = 4\cos \phi . b) Find the centroid of the solid in part (a). For the volume I got 10pi which I am...
  24. S

    The centroid of a hemispherical surface

    I have to find the centroid of a hemispherical surface with radius r. I want to use Pappus's theorem.S=2*Pi*x*s The surface S, is 2*Pi*r^2 and the length of the curve s, is Pi*r so the distance from the centroid ,which lies on the y-axis by symmetry, to the x-axis is S/(2*Pi*Pi*r) = r/Pi...
  25. S

    Where is the centroid of a parabolic wire located?

    I know I'm making this more difficult than it needs to be. I need to find the centroid of a wire bent into the shape of a parabola, defined to be y=x^2 with -2<X>2 and 0<y>4. Obviously due to symetry X-bar =0... but what's y-bar?? No dimensions are given for the width of the wire, so I assume...
  26. agro

    How to Determine the Centroid of a Complex Plane Area in Calculus?

    This is a problem from a past final calculus test in my university (I'm studying for the finals which will come in around 1 week :) ): Find the centroid of a plane area in the first quadrant bounded by x^{2/3}+y^{2/3}=2^{2/3} \frac{x^2}{9}+\frac{y^2}{4}=1 and the x axis. I tried...
  27. C

    Centroid of a shape with a half circle hole

    I'm not sure if this is physics, but I thought I'd ask my question in case someone knows. In my effort to find the centroid of a cross section, I have a half-circle hole at the top of the shape. I know that the area is (r^2*pi)/2 and from the edge of cross-section to the center of the...
  28. J

    Centroid of irregular polygons

    Hi there, I have a bit of a problem for you. I have recently had to write a program to compute the centroid (centre of area) of a 2d shape. I used a many-point weighted triangle method. The shapes themselves are ROI's of anatomical features on SPECT and MRI scans. Im writing up my...
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