- #1
SHawking
- 12
- 0
First, this is not a homework assignment even though it may seem like a homework type question.
I have no problem with the derivation of the moment of inertia of a cylinder but am having more trouble with a sphere. I completely understand the often referenced disk method, but, I would like to take a shell method approach. THe following work is wrong conceptually, I think mathematically it is fine, and I hope someone can point me in the right direction. I don't see what one would need to do it using discs, and am assuming that some slight variation on the following should work:
I=[tex]\int[/tex]r2dm
p=dm/dv
dvp=dm
I=[tex]\int[/tex]r2pdv
v=4/3[tex]\pi[/tex]r3
dv=4[tex]\pi[/tex]r2dr
I=[tex]\int[/tex]r2p4[tex]\pi[/tex]r^2dr
I=4[tex]\pi[/tex]p[tex]\int[/tex]r^4dr (Evluated between 0 and R)
I=4[tex]\pi[/tex]pR5/5
p=m/v=m/(4/3 [tex]\pi[/tex] r3)
Plugging that in and simplifying I get
I=3/5mr2, though I need 2/5mr2.
I think the problem lies in my set up, conceptually, and that I need to set up an equation somewhere subtracting two values (R from r?) though I am not sure.
If you could help me out I would really appreciate it.
And again, I understand and have no problem with the disk method for this, I am just trying to figure out what this method is not working.
Thanks!
I have no problem with the derivation of the moment of inertia of a cylinder but am having more trouble with a sphere. I completely understand the often referenced disk method, but, I would like to take a shell method approach. THe following work is wrong conceptually, I think mathematically it is fine, and I hope someone can point me in the right direction. I don't see what one would need to do it using discs, and am assuming that some slight variation on the following should work:
I=[tex]\int[/tex]r2dm
p=dm/dv
dvp=dm
I=[tex]\int[/tex]r2pdv
v=4/3[tex]\pi[/tex]r3
dv=4[tex]\pi[/tex]r2dr
I=[tex]\int[/tex]r2p4[tex]\pi[/tex]r^2dr
I=4[tex]\pi[/tex]p[tex]\int[/tex]r^4dr (Evluated between 0 and R)
I=4[tex]\pi[/tex]pR5/5
p=m/v=m/(4/3 [tex]\pi[/tex] r3)
Plugging that in and simplifying I get
I=3/5mr2, though I need 2/5mr2.
I think the problem lies in my set up, conceptually, and that I need to set up an equation somewhere subtracting two values (R from r?) though I am not sure.
If you could help me out I would really appreciate it.
And again, I understand and have no problem with the disk method for this, I am just trying to figure out what this method is not working.
Thanks!