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.
The figure and formulas is shown above. My strategy of cutting the areas/shapes is shown below:
Area 1 = Area of Triangle
Area 2 = Area of the square - Area of the quarter circle
Area 3 = Area of the larger quarter circle - Area of the smaller quarter circle
Computing for the areas, I got...
Hello!
Im given this function ## f:[-\pi/2,1] -> [0,1]## with f(x) = 1-x for x (0,1] and f(x) = cos(x) for x ##[-\pi/2,0] ##
And im susposed to find the centroid of this function so xs and ys.
For that I am given these 2 equations ( I found them in the notes)
## x_s =\frac{1}{A}...
To find the y value of the centroid of a right triangle we do
$$\frac{\int_{0}^{h} ydA}{\int dA} = \frac{\int_{0}^{h} yxdy}{\int dA}$$
What is wrong with using
$$\int_{0}^{h} ydA = \int_{0}^{b} y*ydx$$ as the numerator value instead especially since ydx and xdy are equal and where h is height of...
I recently learned how to calculate the centroid of a semi-circular ring of radius ##r## using Pappus's centroid theorem as
##\begin{align}
&4 \pi r^2=(2 \pi d)(\pi r)\nonumber\\
&d=\frac {2r}{\pi}\nonumber
\end{align}##
Where ##d## is the distance of center of mass of the ring from its base...
I found the centroid of four sections, 2 semicircles, 1 rectangle (xz plane) and the remaining right triangle.
I tabulated the individual centroids (x,y,z) of the 4 sections, multiplied each one with the Area of respective section. In the end I calculated the required (x,y,z) by dividing (sum of...
I realize that this is to be solved by breaking up the object into simple objects and using their known center of mass to find the center of mass of the entire object.
1. In the solution the circular gap is also considered in the calculations with a negative center of mass, why is this done?
2...
If I use cartesian co-ordinates, I get:
##\bar{x}=\frac{1}{A}\iint x\, dA=\frac{1}{A} \iint r^2\cos\theta\, dr\, d\theta= \frac{2a\sin\theta}{3\theta}##
##\bar{y}=\frac{1}{A}\iint y\, dA=\frac{1}{A}\iint r^2\sin\theta\, dr\, d\theta= \frac{2a(1-\cos\theta)}{3\theta}##
But if I use polar...
The first part is not a problem, I let one radius lie along the ##x## axis and then we can write down
##S_x = \frac{M}{2}f(\frac{\alpha}{2})\cos{\frac{3\alpha}{2}} + \frac{M}{2}f(\frac{\alpha}{2})\cos{\frac{\alpha}{2}} = Mf(\alpha)\cos{\alpha}##
from which we can then get the following after...
Summary:: I'm solving an exercise.
I have the following center of gravity problem:
Having the function Y(x)=96,4*x(100-x) cm, where X is the horizontal axis and Y is the vertical axis, ranged between the interval (0, 93,7) cm. Determine:
a) Area bounded by this function, axis X and the line...
Homework Statement: Defining Centroid, Centre of Mass, Centre of Gravity for 2D/3Dshapes
Homework Equations: Defining Centroid, Centre of Mass, Centre of Gravity for 2D/3Dshapes
Hello all;
I am trying to understand the terms:-
- Centroid for a 2D shape and 3D shape
- Centre of Mass for a 2D...
Hi,
In one of the standard calculus textbooks, source #1, the formula for y-coordinate of center of gravity for a homogeneous lamina is given as follows.
In another book of formulas, source #2, the formula is given without the factor "1/2" as is shown below. Personally, I believe that source...
Homework Statement
Where is the center of mass of an isoceles triangle?
Homework Equations
xcm=∫xdV/V (where V is the volume of the triangle)
The Attempt at a Solution
The representation of the sides is what I'm confused with. Flipping the triangle to it's side is what's recommended to be...
How do i suppose to determine the uncertainty for the slope of my Static friction against normal reaction graph?
My data for static friction and normal force has the uncertainty of +/- 0.0001
The uncertainty is too small for me to draw airbox/bar in the graph to draw the max and min slope...
Homework Statement
determine the center of mass of a thin plate of density 12 and whose shape is the triangle of vertices (1,0), (0,0), (1,1). Then, using the appropriate pappus theorem, calculate the volume of the solid obtained by rotating this region around the line x = -2.
Homework...
Homework Statement
[/B]
I am having a problem understanding a calculation performed as part of a bigger solution in the design of Slabs.
That is, how to determine the centroid of the critical shear section, which consist of 3 planes intersecting to form a 3D model (please see attached...
Find the mass and centroid of the following thin plate assuming constant density
Sketch the region corresponding
to the plate and indicate the location
of the center is the mass
The region bounded by
$$y=ln x$$
$$x-axis$$
$$x=e$$
\begin{align}\displaystyle...
The Problem is #16 in the attached picture. Essentially, I need to find the length of BC using information about congruency and the location of the centroid. I've been able to show a whole bunch of things, but nothing that gets me close to actually finding out the missing side length.
I began...
Hey! :o
We have a triangle $ABC$ with $A(a_1, a_2)$, $B(b_1, b_2)$ and $C(c_1,c_2)$. I want to show that the coordinates of the centroid S is $\left (\frac{1}{3}(a_1+b_1+c_1),\frac{1}{3}(a_2+b_2+c_2) \right )$.
$S$ is the intersection point of the midpoints of AB, BC and CA.
We have that...
Homework Statement
For this shape , it's clear that the centroid and the horizontal line of equal axis lies on the same horizontal line , am i right ?
Homework EquationsThe Attempt at a Solution
I'm not sure . correct me if i am wrong . [/B]
Homework Statement
In this image :
http://ezto.mheducation.com/13252703414204806874.tp4?REQUEST=SHOWmedia&media=2.83qs.jpg
Why does the weight of the gate have a centroid at 2R/pi away from the force F?
Homework Equations
The centroid for a half circle in both x and y directions is =...
g(x)= √(19x) = upper curve
f(x)= 0.2x^2 = lower curve
Firstly, I found the point of intersection, which would later give the upper values for x and y.
x=7.802
y=12.174
Then I found the area under g(x) and took away the area under f(x) to get the area between the curves.
31.67 units^2
This is...
Homework Statement
If G be the centroid of ΔABC and O be any other point, prove that ,
## 3(GA^2 + GB^2 + GC^2)=BC^2+CA^2+AB^2##
##and,##
##OA^2 + OB^2 + OC^2 = GA^2.GB^2+GC^2+3GO^2##
Homework Equations
i m practising from S L LONEY coordinate geometry first chapter ... only the equation...
In the notes , the author stated that when the force is applied through the centorid of cross section , the channel will bend and twist.
but , on the second page , the author stated that the shear center lies on an axis of symmetry of member's cross sectional area...
So, i am confused whether...
Homework Statement
In the notes , the author stated that when the force is applied through the centorid of cross section , the channel will bend and twist.
but , on the second page , the author stated that the shear center lies on an axis of symmetry of member's cross sectional area...
So, i am...
Homework Statement
No actual work, could just use some assistance in understanding formulas involving the centroid of an object, specifically with integrals. For example, how would you understand the following formula(s) (as seen in part 2)? I understand that the centroid is the sum of all the...
[ Mod Note: moving this to Physics H/W ]
Homework Statement
for this question , I'm having problem with the shear stress at point E and shear stress at centorid.
normally , the shear stress at the centoid will be maximum .
But , in my working , I found that the shear stress at the centroid is...
Homework Statement
Find the centroid of solid enclosed by surface z= y^2 , plane x=0 , x = 1 and z =1 . The density is 1
Homework EquationsThe Attempt at a Solution
Here's my working .
Centoird = mass of inertia / mass
So , i find the mass first .
It's clear that the circle is on zx...
Pappus's centroid theorems were discovered 17 centuries ago, when calculus wasn't invented yet. How are these theorems proved without using calculus?
"The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C...
Homework Statement
Find the centroid of the shape formed by the equation y2=x3-x4, the x-axis, and the y-axis.
Homework Equations
A=∫f(x)dx
Mx=∫(1/2)[f(x)]2dx
My=∫x[f(x)]dx
The Attempt at a Solution
I'm stuck on the integral.
I attempted u-substitution and got du=(1/2)(x3-x4)-1/2dx; "parts,"...
Hello,
I am currently studying how to find the centroid of shapes. And I understand that to find the location of the centroid, we must analyze the distribution of the mass over the x and y-axis (i.e calculating Qx and Qy).
However what baffles me is that, given an L shaped beam, the centroid...
I'm having trouble understanding what this question is actually asking for. Is it assuming the centroid to be the origin and asking how far the bottom of the shape extends downwards for the origin to be the centroid?
Homework Statement
determine the second moment of inertia about the horizontal axis and vertical axis for the shaded area with respect to x and y axes through the centroid of the area .
Homework EquationsThe Attempt at a Solution
Since the x and y axes is drawn thru centroid , why not the y...
Homework Statement
As in the photos
Homework EquationsThe Attempt at a Solution
my working is
(6.43x10^-3)(310+10) + (11x10^-3)(155) / (6.43x10^-3 +11x10^-3 ) = 215.8 , but the ans is 112 , what's wrong with my working ?
Homework Statement
i have a few question here .
1. the y bar for I should be 16.98, am i right ?
2. The y bar for II should be 50+12+24=86 , am i right ?
3. the x bar for V should be 50 , am i right ?
Homework EquationsThe Attempt at a Solution
Homework Statement
What is the difference between the Centre of Pressure and Centroid
Homework EquationsThe Attempt at a Solution
My understanding is that the centre of pressure acts on a centroid. So, how come they can be on different positions for a submerged surface?
1. Homework Statement
The tension in the cable is 800LBs. Find the depth of water that produces this tension. The gate is hinged at B; the cable is attached at A. (Figure 1)Homework EquationsThe Attempt at a Solution
OK, so this is just 2 opposing lever arms: the tension in the cable is...
Homework Statement
Find the coordinates of the centroid of the uniform area.
Homework Equations
equations for centroid coordinates at the top of my paper.
The Attempt at a Solution
Homework Statement
Use cylindrical coordinates to find the centroid of the solid.
The solid that is bounded above by the sphere x2 + y2 + z2 = 2
and below by z = x2 + y2
Homework Equations
x = rcos(theta)
y= rsin(theta)[/B]The Attempt at a Solution
I am having trouble trying to find the...
1. Find the position of the centroid of the shaded area
http://imgur.com/ieiSPPY
2.
The black triangle with the square cut out is where the centroid must be located.I know that the object is symmetrical and the triangle can be divided into parts with two smaller right angled triangles
I have...
Homework Statement
A lamina is bounded by the x-axis, the y-axis, and the curve ##y = 4 -x^2.## Determine the centroid position ##(\bar{x},\bar{y})## of the lamina.
Homework Equations
## A = \int_a^b (f(x) - g(x)) dx ## (Area)
##\bar{x} = \frac{1}{A}\int_a^b x(f(x) - g(x)) dx ##
##\bar{y}...
Hi all,
I'm preparing for a deferred exam this semester after falling ill last year. Just looking over my course notes and have a question. I understand how this works in the big picture scheme. What I don't understand however is how my instructor simplified the original equation.
1. Homework...
In the context of centroids and moments, what do ##\bar{x}_{\textrm{el}}## and ##\bar{y}_{\textrm{el}}## represent?
For example:
$$\bar{x}L = \int \bar{x}_{\textrm{el}}dL$$
If I have an integrated area such as the blue area in the link below, what function can be written to find the location on the x-axis where half of the area is one side and half is on the other or more specifically a function that determines the x-axis location of the centroid...
Homework Statement
Find the x-coordinate of the centroid of the region bounded by the graphs of
Homework Equations
y= 5/(√(25-x2))
The Attempt at a Solution
I stuck at finding Mx
You are given an arbitrary triangle ABC. Inside ABC there is a point M such that Area(ABM) = Area(BCM) = Area(ACM) . Prove that M is the centroid of triangle ABC.
I have had very little progress with this question. I've tried connecting a line from M which bisects BC, but I cannot prove that...