- #1
amcavoy
- 665
- 0
I am having a bit of trouble deriving the moment of inertia for a disk with uniform density:
[tex]I=\int r^{2}\,dm=\int \rho r^{2}\,dV[/tex]
For a disk, I just used dA instead of dV. Now, to calculate the density:
[tex]\rho = \frac{\text{mass}}{\text{volume}}=\frac{m}{\pi r^{2}}[/tex]
So now we have:
[tex]I=\int \rho r^{2}\,dA=\frac{m}{\pi}\int \,dA=\boxed{mr^{2}}[/tex]
However, I know that the moment of inertia for a disk is [itex]\frac{1}{2}mr^{2}[/itex]. Where did I go wrong?
Thank you.
[tex]I=\int r^{2}\,dm=\int \rho r^{2}\,dV[/tex]
For a disk, I just used dA instead of dV. Now, to calculate the density:
[tex]\rho = \frac{\text{mass}}{\text{volume}}=\frac{m}{\pi r^{2}}[/tex]
So now we have:
[tex]I=\int \rho r^{2}\,dA=\frac{m}{\pi}\int \,dA=\boxed{mr^{2}}[/tex]
However, I know that the moment of inertia for a disk is [itex]\frac{1}{2}mr^{2}[/itex]. Where did I go wrong?
Thank you.