How to Determine the Centroid of a Solid Using Cylindrical Coordinates?

In summary, the conversation discussed using cylindrical coordinates to find the centroid of a solid bounded by a sphere and a paraboloid. The summary provided clarification on solving for the limits of integration for r and the correct solution for r in terms of the given equations. The conversation also addressed the process of finding the centroid in the z direction and any potential difficulties with the integrals.
  • #1
Phil Frehz
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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 limits of integration for r.

I have been able to get the sphere in terms of r as z = (2-r2)1/2 and the paraboloid as z = r2

I understand that in cylindrical coordinates the region is occupied from 0<theta<2pi and r originates from origin n to the uppermost part of the sphere.

I also found that when you equate both surface z equations. you get r2 = (2-r2)1/2 . Solving for r you get radical 2 and 1. r being the furthest distance would then have the upper limit of 1. I'm not sure if this is the correct method of solving.[/B]
 

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  • #2
First off, ## \sqrt 2>1## and ##\sqrt 2## doesn't seem to solve your equation.

Go with z=1 for the intersection point, so z is defined by the sphere above 1<z <2 and as the paraboloid for 0<z <1.
 
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  • #3
See the attached image, this is the shape you are dealing with (revolved around the z axis to form a volume). Clearly, your centroid will be at x=0, y=0. How will you find the centroid in the z direction?
PF_1.jpg
 
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  • #4
RUber said:
See the attached image, this is the shape you are dealing with (revolved around the z axis to form a volume). Clearly, your centroid will be at x=0, y=0. How will you find the centroid in the z direction?
View attachment 86271

So what I did was found out the value of z at the intersection point by equating both surfaces in terms of r, which led z to be either -2 or 1. When inputting z =1 i get r to also be 1

The centroid in the z direction will thus be 02pi 01 r2√(2-r2) z ⋅ r dz dr dθ all divided by the volume of the region
 
Last edited:
  • #5
Phil Frehz said:
So what I did was found out the value of z at the intersection point by equating both surfaces in terms of r, which led z to be either -2 or 1. When inputting z =1 i get r to also be 1

Do you mean you solved ##r^2 = (2-r^2)^{1/2}## as a quadratic to find ## r^2 = 1 \text{ or } -2 ## and then discarded the imaginary solution?
The centroid in the z direction will thus be 02pi 01 r2√(2-r2) z ⋅ r dz dr dθ all divided by the volume of the region

Any troubles with these integrals?
 
  • #6
RUber said:
Do you mean you solved ##r^2 = (2-r^2)^{1/2}## as a quadratic to find ## r^2 = 1 \text{ or } -2 ## and then discarded the imaginary solution?Any troubles with these integrals?

Yes I discarded the imaginary solution, I have no problem solving the integrals. I only had trouble identifying the upper limit of r. Would you say that the way I solved for r was a correct solution?
 
  • #7
That is the correct way to do it.
What you did wrong on your original work posted in the image was to go from
## r^4 + r^2 =2 ## to r =1 or r = sqrt(2); you should have gotten r =1 or r = sqrt(-2).
 
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  • #8
RUber said:
That is the correct way to do it.
What you did wrong on your original work posted in the image was to go from
## r^4 + r^2 =2 ## to r =1 or r = sqrt(2); you should have gotten r =1 or r = sqrt(-2).

Ah I overlooked the proper solution for r. Thanks for taking the time to help me out!
 

Related to How to Determine the Centroid of a Solid Using Cylindrical Coordinates?

1. What is the definition of a centroid?

A centroid is the geometric center of an object, where all the mass or weight of the object is evenly distributed.

2. How do you find the centroid of a 2D shape?

To find the centroid of a 2D shape, you can use the formula: x̄ = (1/n) * Σ(xi) and ȳ = (1/n) * Σ(yi), where n is the total number of points and xi and yi are the coordinates of each point. This will give you the x and y coordinates of the centroid.

3. Is the centroid of a 3D object the same as its center of mass?

No, the centroid and the center of mass are not the same. The center of mass takes into account the distribution of mass within an object, while the centroid only considers the geometric center.

4. How do you find the centroid of a solid object?

To find the centroid of a solid object, you can use the formula: x̄ = (1/V) * ∫∫∫xρ(x,y,z) dV and ȳ = (1/V) * ∫∫∫yρ(x,y,z) dV, where V is the volume of the object and ρ(x,y,z) is the density at each point (x,y,z). This involves integrating over the entire volume of the object to find the average x and y coordinates of the object's mass distribution.

5. What is the application of finding the centroid of a solid?

Finding the centroid of a solid object is useful in many engineering and physics applications. It can help determine the stability and balance of an object, as well as its reaction to external forces. It is also important in the design of structures such as buildings and bridges, to ensure that the weight is evenly distributed and the structure is stable.

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