- #1
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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:
[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r2 dV =
[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r2 r dr d[tex]\Theta[/tex] dz =
[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r3 dr d[tex]\Theta[/tex] dz = [tex]\Pi[/tex]h5/10
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] z2 z dr d[tex]\Theta[/tex] dz =
[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] z3 dr d[tex]\Theta[/tex] dz = 2[tex]\Pi[/tex]h5/5
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
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:
[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r2 dV =
[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r2 r dr d[tex]\Theta[/tex] dz =
[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r3 dr d[tex]\Theta[/tex] dz = [tex]\Pi[/tex]h5/10
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] z2 z dr d[tex]\Theta[/tex] dz =
[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] z3 dr d[tex]\Theta[/tex] dz = 2[tex]\Pi[/tex]h5/5
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