Question

a thread is being pulled off a spool at the rate of 67.3 cm per sec. find the radius of the spool if it makes 116 revolutions per min.
r≈ cm
(round to two decimal places as needed.)

Explanation:

Step1: Convert revolutions per minute to radians per second

First, convert 116 revolutions per minute to radians per second. One - revolution is $2\pi$ radians and there are 60 seconds in a minute. So the angular velocity $\omega$ is $\omega=116\times2\pi\div60=\frac{116\times2\pi}{60}=\frac{116\pi}{30}=\frac{58\pi}{15}$ radians per second.

Step2: Use the linear - velocity formula

The linear - velocity formula is $v = r\omega$, where $v$ is the linear velocity, $r$ is the radius, and $\omega$ is the angular velocity. We know that $v = 67.3$ cm/s and $\omega=\frac{58\pi}{15}$ rad/s. Rearranging the formula for $r$, we get $r=\frac{v}{\omega}$.

Step3: Calculate the radius

Substitute $v = 67.3$ and $\omega=\frac{58\pi}{15}$ into the formula $r=\frac{v}{\omega}$:
\[

$$\begin{align*} r&=\frac{67.3}{\frac{58\pi}{15}}\\ &=67.3\times\frac{15}{58\pi}\\ &=\frac{1009.5}{58\pi}\\ &\approx\frac{1009.5}{58\times3.14159}\\ &=\frac{1009.5}{182.21222}\\ &\approx5.54 \end{align*}$$

\]

Answer:

$5.54$