trautlein
- 17
- 1
Homework Statement
A hollow cylinder has mass m, and outside radius r_{2}, and an inside radius r_{1}. Use intergration to show that the moment of inertia about its axis is give by I=\frac{1}{2}m(r^2_2+r_1^2).
Uniform composition is assumed.
Homework Equations
I=\int r^2dm
dm=\sigma dA
and
A=\pi r^2
(note, later on I constrict the lower and upper bounds of integration to account for the now entire area of the circular side, which sounds as though it is assumed here)
The Attempt at a Solution
Note: Using equations above.
I=\int r^2dm
I=\int r^2\sigma dA
dA=2\pi rdr
I=\sigma 2\pi\int r^3dr
(here I insert my upper and lower bounds)
I=\sigma 2\pi\int_{r_1}^{r_2} r^3dr
I=\sigma 2\pi \left(\frac{r^4_2}{4}-\frac{r^4_1}{4} \right )
I=\sigma \pi \left(\frac{r^4_2}{2}-\frac{r^4_1}{2} \right )
I=\frac {1}{2}\sigma \pi \left(r^4_2-r^4_1 \right )
\sigma=\frac{m}{A}
and since
A=\pi r^2_1-\pi r^2_2
then
I=\frac{m}{2(\pi r^2_1-\pi r^2_2)} \pi \left(r^4_2-r^4_1 \right )I=\frac{\left(r^4_2-r^4_1 \right )m}{2 (r^2_1-r^2_2)}This is as far as I venture into the problem before I realize that I've done something wrong if I'm supposed to get the answer they assume at the beginning of the problem. Where did I go wrong?
Thank you very much for the help, it's greatly appreciated.