Question

a thread is being pulled off a spool at the rate of 64.4 cm per sec. find the radius of the spool if it makes 158 revolutions per min. (round to two decimal places as needed.) r ≈ □ cm

Explanation:

Step1: Convert revolutions per minute to revolutions per second

The spool makes 158 revolutions per minute. To convert to revolutions per second, divide by 60. So, the number of revolutions per second $n=\frac{158}{60}\text{ rev/s}$.

Step2: Find the angular - velocity $\omega$

One revolution is $2\pi$ radians. So the angular - velocity $\omega = 2\pi n$. Substituting $n=\frac{158}{60}\text{ rev/s}$, we get $\omega=2\pi\times\frac{158}{60}\text{ rad/s}$.

Step3: Use the relationship between linear - velocity $v$, angular - velocity $\omega$ and radius $r$

The relationship between linear - velocity $v$, angular - velocity $\omega$ and radius $r$ is $v = r\omega$. We know that $v = 64.4\text{ cm/s}$. Rearranging for $r$, we have $r=\frac{v}{\omega}$. Substituting $v = 64.4\text{ cm/s}$ and $\omega=2\pi\times\frac{158}{60}\text{ rad/s}$ into the formula:
\[

$$\begin{align*} r&=\frac{64.4}{2\pi\times\frac{158}{60}}\\ &=\frac{64.4\times60}{2\pi\times158}\\ &=\frac{3864}{2\pi\times158}\\ &=\frac{3864}{992.08}\\ &\approx3.89\text{ cm} \end{align*}$$

\]

Answer:

$3.89$