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Centre of mass of a solid cone

im actually bugged of finding a solution for d topic mentioned can any 1 plzzzz help me
 
plzzz help me solve dis problem
 

arildno

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Do not post homework problems in the tutorials section!

Now, to help you along a bit, let I be the cone's base in the plane z=0, and let (X,Y) denote a point in I. Let the vertex have the coordinates: [itex]\vec{r}_{v}=(x_{v},y_{v},z_{v})[/itex].

Thus, any point within the cone will lie on some line segment from [itex]\vec{r}_{v}[/itex] to a point (X,Y) in I, so we can therefore represent all points in the cone with the following function:
[tex]\vec{r}(X,Y,u)=(\vec{r}_{v}-(X,Y,0))u+(X,Y,0), 0\leq{u}\leq{1},(X,Y)\in{I}[/tex]
[tex]\vec{r}(X,Y,u)\equiv(x(X,Y,u),y(X,Y,u),z(X,Y,u))[/tex]
This should be useful to you.

In particular, remember that the x-coordinate to any given point in the cone is:
[tex]x(X,Y,u)=(x_{v}-X)u+X[/itex]
and similar expressions for the y-and z-coordinates to any one point in the cone.
 
Last edited:

Curious3141

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Search under my name and using the keyword "cone". I've gone through the entire derivation in an earlier discussion in the HW forum.
 
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this waz a very difficult derivation ....can u be kind enough o explain wid a diagram....thank u
 

arildno

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Now, remember that the centroid coordinates are gained by averaging the point coordinates over the volume V of the object O.
For example, the horizontal centroid coordinate [itex]\hat{x}[/itex] is given by:
[tex]\hat{x}=\frac{\int_{O}xdV}{\int_{O}dV}=\frac{\int_{O}xdV}{V}, dV=dxdydz[/tex]
Now, just compute the Jacobian to the cone representation I've given, so that you may integrate with respect to the variables (X,Y,u) instead.
 

arildno

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Since it is Yule, I'll be a bit more charitable:
1. The Jacobian is readily computed to be [itex](1-u)^{2}z_{v}[/tex]
2. We also have the coordinate representations:
[tex]x(X,Y,u)=(x_{v}-X)u+X,y(X,Y,u)=(y_{v}-Y)u+Y, z(X,Y,u)=z_{v}u[/tex]

Use these relations to derive the coordinates for the centroid!
 

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