# A cylinder has inner and outer radii of 12 cm and 18 cm, respectively, and a mass of 6 kg. If the cylinder's frequency of rotation about its center changes from 7 Hz to 6 Hz, by how much does its angular momentum change?

Apr 17, 2016

$\Delta P = - 28 , 8 \pi$

#### Explanation:

$\text{1-calculate the changing of the angular velocity}$
$\text{2-calculate the moment of inertia for cylinder}$
$\text{3-calculate changing of the angular momentum}$

$\text{1).....................................................................}$
${f}_{i} = 7 H z \text{ initial frequency}$
${f}_{l} = 6 H z \text{ last frequency}$
$\Delta \omega = {\omega}_{l} - {\omega}_{i} \text{ changing of the angular velocity}$

${\omega}_{l} = 2 \cdot \pi \cdot {f}_{l} \text{ "omega_l=2*pi*6" "omega_l=12pi " } \frac{r a d}{s}$

${\omega}_{i} = 2 \cdot \pi \cdot {f}_{i} \text{ "omega_i=2*pi*7" "omega_i=14pi" } \frac{r a d}{s}$

$\Delta \omega = 12 \pi - 14 \pi$

$\Delta \omega = - 2 \pi \text{ } \frac{r a d}{s}$
2)........................................................................
$I = \frac{1}{2} \cdot m \left({r}_{1}^{2} + {r}_{2}^{2}\right)$
$\text{moment of inertia for cylinder which has inner and outer radius}$

m=6kg

${r}_{1} = 12 c m = 0 , 12 m$
${r}_{1}^{2} = 1 , 44$

${r}_{2} = 18 c m = 0 , 18 m$
${r}_{2}^{2} = 3 , 24$
$I = \frac{1}{2} \cdot 6 \left(1 , 44 + 3 , 24\right)$

$I = 3 \cdot 4 , 68$

I=14,04

3)............................................................................
$\Delta P = I \cdot \Delta \omega \text{ angular momentum change}$
$\Delta P = - 14 , 4 \cdot 2 \pi$

$\Delta P = - 28 , 8 \pi$