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## Main Question or Discussion Point

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 z

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:

[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r

[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r

[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r

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

[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] z

[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] z

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

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 z

^{2}= r^{2}.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:

[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r

^{2}dV =[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r

^{2}r dr d[tex]\Theta[/tex] dz =[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r

^{3}dr d[tex]\Theta[/tex] dz = [tex]\Pi[/tex]h^{5}/10But instead, in step 2 if I substituted r = z (from the equation of the cone), I get a completely different answer.

[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] z

^{2}z dr d[tex]\Theta[/tex] dz =[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] z

^{3}dr d[tex]\Theta[/tex] dz = 2[tex]\Pi[/tex]h^{5}/5I 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