Centre of mass of a solid cone

In summary, the centroid of the cone is (X,Y,Z), where X is the x-coordinate of the vertex, Y is the y-coordinate of the vertex, and Z is the z-coordinate of the vertex.
  • #1
shreyaacharya
3
0
im actually bugged of finding a solution for d topic mentioned can any 1 pleasezzz help me
 
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  • #2
pleasezz help me solve dis problem
 
  • #3
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 anyone point in the cone.
 
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  • #4
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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  • #5
this waz a very difficult derivation ...can u be kind enough o explain wid a diagram...thank u
 
  • #6
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.
 
  • #7
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!
 

1. What is the formula for finding the center of mass of a solid cone?

The formula for finding the center of mass of a solid cone is (1/4)h, where h is the height of the cone.

2. How does the shape of a cone affect its center of mass?

The shape of a cone affects its center of mass because the center of mass is located closer to the base of the cone than the tip. This is because the base of the cone has a larger mass compared to the tip.

3. Can the center of mass of a cone be outside of the object?

Yes, the center of mass of a cone can be outside of the object. This is possible when the cone has a non-uniform density or when the base of the cone is not circular.

4. How is the center of mass of a cone related to its stability?

The center of mass of a cone is directly related to its stability. A cone with a lower center of mass will be more stable compared to one with a higher center of mass. This is because the lower center of mass creates a larger base of support, making it harder for the cone to tip over.

5. Can the center of mass of a cone change?

Yes, the center of mass of a cone can change depending on its orientation or if additional mass is added or removed from the cone. However, the height of the cone will always remain a factor in determining the center of mass.

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