Question

Your $64\mathrm{cm}$diameter car tire is rotating at $3.5\mathrm{rev}/\mathrm{s}$ when suddenly you press down hard on the accelerator. After traveling $200\mathrm{m}$, the tire’s rotation has increased to $6.0\mathrm{rev}/\mathrm{s}$. What was the tire’s angular acceleration? Give your answer in $\mathrm{rad}/{\mathrm{s}}^2$ .

Short answer

The tire’s angular acceleration is$0.75\mathrm{rad}/{\mathrm{s}}^2$

Step-by-step solution

Given information

Diameter of car tire is $\mathrm{D}=64\mathrm{cm}$

Car tire is rotating at ${\mathrm{\omega }}_{\mathrm{i}}=\mathrm{initial}\mathrm{angular}\mathrm{velocity}=3.5\mathrm{rev}/\mathrm{s}$

After traveling $\mathrm{d}=200\mathrm{m}$, the tire’s rotation has increased to ${\mathrm{\omega }}_{\mathrm{f}}=\mathrm{final}\mathrm{angular}\mathrm{velocity}=6.0\mathrm{rev}/\mathrm{s}$.

We have to find the tire’s angular acceleration$\mathrm{\alpha }$ in $\mathrm{rad}/{\mathrm{s}}^2$.

Formula

Perimeter of circle is $2\pi r$

${\omega }_f^2={\omega }_i^2+2\alpha ∆\theta$

Explanation

By putting values in formula

$\mathrm{P}=2\mathrm{\pi r}$

For radius

$$\begin{gathered} \mathrm{r}=\frac{\mathrm{d}}{2} \\ \mathrm{r}=\frac{64}{2} \\ \mathrm{r}=32\mathrm{cm} \end{gathered}$$

Putting values:

$$\begin{gathered} \mathrm{P}=2\times 3.14\times 32\mathrm{cm} \\ =200.96\mathrm{cm} \end{gathered}$$

Converting the car tire in revolutions:

$$\begin{gathered} \mathrm{\theta }=\frac{\mathrm{d}}{\mathrm{P}} \\ \mathrm{\theta }=\frac{200\mathrm{cm}\times 100}{200.96\mathrm{cm}} \\ \mathrm{\theta }=99.52\mathrm{rev} \end{gathered}$$

For angular acceleration

$$\begin{gathered} 6^2=3.5^2+2\mathrm{\alpha }\times 99.52 \\ \mathrm{\alpha }=0.1193\mathrm{rev}/{\mathrm{s}}^2 \end{gathered}$$

Now, convert $\mathrm{rev}/{\mathrm{s}}^{\mathrm{s}}$in $\mathrm{rad}/{\mathrm{s}}^2$

localid="1648908260302" $$\begin{gathered} \mathrm{\alpha }=0.1193\mathrm{rev}/{\mathrm{s}}^{\mathrm{s}}\times 2\mathrm{\pi } \\ =0.1193\mathrm{rev}/{\mathrm{s}}^{\mathrm{s}}\times 2\times 3.14 \\ \mathrm{\alpha }=0.75\mathrm{rad}/{\mathrm{s}}^2 \end{gathered}$$