Question

example 6: a 15 - inch diameter tire on a car makes 9.3 revolutions per second. a. find the angular speed of the tire in rad/sec b. find the linear speed of the car in in/sec

Explanation:

Step1: Recall angular - speed formula

The formula for angular speed $\omega$ is $\omega = 2\pi n$, where $n$ is the number of revolutions per second.

Step2: Substitute the given value of $n$

Given $n = 9.3$ revolutions per second. So, $\omega=2\pi\times9.3 = 18.6\pi$ rad/sec.

Step3: Recall linear - speed formula

The formula for linear speed $v$ is $v = r\omega$, where $r$ is the radius of the tire and $\omega$ is the angular speed. The radius $r=\frac{d}{2}$, and given $d = 15$ inches, so $r=\frac{15}{2}$ inches.

Step4: Calculate the linear speed

We know $\omega = 18.6\pi$ rad/sec and $r=\frac{15}{2}$ inches. Then $v=r\omega=\frac{15}{2}\times18.6\pi$.
$v=\frac{15\times18.6\pi}{2}=140.25\pi\approx440.6$ in/sec.

Answer:

a. The angular speed of the tire is $18.6\pi$ rad/sec.
b. The linear speed of the car is approximately $440.6$ in/sec.