# A cylinder has inner and outer radii of 8 cm and 16 cm, respectively, and a mass of 6 kg. If the cylinder's frequency of counterclockwise rotation about its center changes from 2 Hz to 7 Hz, by how much does its angular momentum change?

Dec 18, 2016

The nasty part of this problem is in coming up with the moment of inertia for a thick-walled cylinder rotating along an axis through its centre. This is

$I = \frac{m}{2} \left({r}_{1}^{2} + {r}_{2}^{2}\right)$

where ${r}_{1}$ and ${r}_{2}$ are the inner and outer radii, respectively.

In this case, $I = \frac{6}{2} \left({.08}^{2} + {.16}^{2}\right)$ = $0.096 k g {m}^{2}$

Now, the angular momentum is defined as the product of its moment of inertia multiplied by the angular momentum (in radians/s don't forget!).

$L = I$$\omega$

Since 1 Hz is equivalent to 2$\pi$ radians, the angular velocity changes from 4$\pi$ to 14$\pi$, and so, the change in angular momentum is found as follows:

L_i = 0.096 xx 4"pi = $0.384$pi

L_f = 0.096 xx 14"pi = $1.344$pi

$\Delta L = 0.960 \pi$ $k g {m}^{2} / s$

(Hope you don't mind that I held back on the units until the last line. I thought they would make things a bit messy!)