91Ó°ÊÓ

1. Mass of ring 1 is$0.120\text{kg}$
2. Mass of ring 2 is$0.24\text{kg}$
3. Inner radius of ring 1 is$0.016\text{m}$
4. Outer radius of ring 1 is$0.0450\text{m}$
5. Inner radius of ring 2 is$0.09\text{m}$
6. Outer radius of ring 2 is$0.14\text{m}$
7. Magnitude of tangential force is$12.0\text{N}$

Use the formula in terms of force and radius to find the torque. Using another formula for torque in terms of rotational inertia and angular acceleration, find the angular acceleration.

Formula:

$\mathrm{τ}=\text{r}×\text{F}$

$\mathrm{Ï„}=\text{I}\mathrm{Î\pm }$

For ring,the inertiais l,

${\text{I}}_1=\frac{1}{2}{\text{M}}_1\left({\text{R}}_{1i}^2+{\text{R}}_{1o}^2\right)$

is the mass of the ring, and ${\text{R}}_{1i}$and ${\text{R}}_{1o}$is the innerand outer radius of the smaller ring respectively.

$$\begin{gathered} {\text{I}}_1=\frac{1}{2}×0.120×\left({0.0160}^2+{0.045}^2\right) \\ {\text{I}}_1=1.3686×{10}^{-4}\text{kg}-{\text{m}}^2 \end{gathered}$$

For ring 2,

$$\begin{gathered} {\text{I}}_2=\frac{1}{2}{\text{M}}_2\left({\text{R}}_{2i}^2+{\text{R}}_{2o}^2\right) \\ {\text{I}}_2=\frac{1}{2}×0.24×\left({0.090}^2+{0.14}^2\right) \\ {\text{I}}_2=3.324×{10}^{-3}\text{kg}-{\text{m}}^2 \end{gathered}$$

$M_2$is mass of the ring and ${\text{R}}_{2i}$and ${\text{R}}_{2o}$is the inner and outer radius of the larger ring respectively.

Total inertia is as follows:

$$\begin{gathered} \text{I}={\text{I}}_1+{\text{I}}_2 \\ \text{I}=1.3686×{10}^{-4}+3.324×{10}^{-3} \\ \text{I}=3.460×{10}^{-3}\text{kg}-{\text{m}}^2 \end{gathered}$$

Now, torque as follows:

$$\begin{gathered} \mathrm{τ}={\text{R}}_{2o}×\text{F} \\ \mathrm{τ}=0.14×12 \\ \mathrm{τ}=1.68\text{N}.\text{m} \end{gathered}$$

We know

$$\begin{gathered} \mathrm{Ï„}=\text{I}\mathrm{Î\pm } \\ 1.68=3.46×{10}^{-3}×\mathrm{Î\pm } \end{gathered}$$

So

$\mathrm{Î\pm }=485\text{rad}/\text{sec}$

Now, angular speed is as follows:

$$\begin{gathered} \mathrm{Ó\neg }=\mathrm{Î\pm }×\text{t} \\ \mathrm{Ó\neg }=485×0.3 \\ \mathrm{Ó\neg }=145.5\text{rad}/\text{sec} \\ \mathrm{Ó\neg }=146\text{rad}/\text{sec} \end{gathered}$$