Centre of mass of a solid cone

[shreyaacharya]

im actually bugged of finding a solution for d topic mentioned can any 1 pleasezzz help me

 

[shreyaacharya]

pleasezz help me solve dis problem

 

[arildno]

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: $\vec{r}_{v}=(x_{v},y_{v},z_{v})$.

Thus, any point within the cone will lie on some line segment from $\vec{r}_{v}$ to a point (X,Y) in I, so we can therefore represent all points in the cone with the following function:
$$\vec{r}(X,Y,u)=(\vec{r}_{v}-(X,Y,0))u+(X,Y,0), 0\leq{u}\leq{1},(X,Y)\in{I}$$
$$\vec{r}(X,Y,u)\equiv(x(X,Y,u),y(X,Y,u),z(X,Y,u))$$
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]<br /> and similar expressions for the y-and z-coordinates to anyone point in the cone.[/tex]

 

[Curious3141]

Search under my name and using the keyword "cone". I've gone through the entire derivation in an earlier discussion in the HW forum.

 

[shreyaacharya]

this waz a very difficult derivation ...can u be kind enough o explain wid a diagram...thank u

 

[arildno]

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 $\hat{x}$ is given by:
$$\hat{x}=\frac{\int_{O}xdV}{\int_{O}dV}=\frac{\int_{O}xdV}{V}, dV=dxdydz$$
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]

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]<br /> 2. We also have the coordinate representations:<br /> $$x(X,Y,u)=(x_{v}-X)u+X,y(X,Y,u)=(y_{v}-Y)u+Y, z(X,Y,u)=z_{v}u$$<br /> <br /> Use these relations to derive the coordinates for the centroid![/itex]