Confusion with Calculating Centroid and Center of Mass for a Cone

[WiFO215]

NOTE: THIS IS NOT A HW PROBLEM
While calculating the moment of inertia of a cone, I made a mistake and got confused. Although I juggled the numbers and got the right answer, I have a doubt.

Say I have to calculate the centroid of a cone z² = r².

0 < z < h

It is quite obvious that the x and y co-ordinates of the centroid are zero. As for the z co-ordinate it can be calculated as follows:

\int \int \int_{V} r² dV =

\int \int \int_{V} r² r dr d\Theta dz =

\int \int \int_{V} r³ dr d\Theta dz = \Pih⁵/10

But instead, in step 2 if I substituted r = z (from the equation of the cone), I get a completely different answer.

\int \int \int_{V} z² z dr d\Theta dz =

\int \int \int_{V} z³ dr d\Theta dz = 2\Pih⁵/5

I got a similar doubt while calculating the center of mass, wherein I plugged in z = r in the equation and got a different (and wrong) answer.

Shouldn't I get the same answer either way? Why don't I get it?

Anirudh

 

[tiny-tim]

Shouldn't I get the same answer either way? Why don't I get it?

Hi Anirudh!

(have a theta: θ and a pi: π )

You've probably got the limits wrong …

what limits did you use?

 

[WiFO215]

have a theta: θ and a pi: π

For the first one I used:

\int^{2\Pi}_{\Theta = 0}\int^{h}_{z = 0}\int^{z}_{r = 0} r³ dr d\Theta dz

For the second one I used:\int^{2\Pi}_{\Theta = 0}\int^{h}_{z = 0}\int^{z}_{z = 0} z³ dr d\Theta dz

I used the same limits because r = z. Is that wrong? (Obviously, it MUST be as the answer isn't right)

 

[tiny-tim]

hmm … looking again, these are different integrals …

\int \int \int_{V} r² r dr d\Theta dz =
…
\int \int \int_{V} z² z dr d\Theta dz = …

∫∫∫ r² is not the same as ∫∫∫ z² …

z² = r² only on the boundary, not everywhere in the middle …

(and you can't transform away r dr anyway)

 

[WiFO215]

(and you can't transform away r dr anyway)

Okay I understood that r = z only on the boundary. What does the above statement mean though?