Question

linear velocity and angular velocity
a tire with radius 9 inches is spinning at 80 revolutions
per minute.
a) find the angular speed of the tire in radians per
second.
$omega=\frac{\theta}{t}$
b) find the speed in inches per minute and miles per
hour.
$v = romega$
$v=\frac{r\theta}{t}$
a hamster running in a wheel of radius 13 cm spins the wheel at 14 revolutions per
minute.
a) what is the angular velocity of the wheel? (in radians/sec)
b) at what linear velocity is the hamster running? (in cm/sec)

Explanation:

Step1: Calculate angular velocity

Angular velocity \(\omega=\frac{\theta}{t}\). One revolution is \(2\pi\) radians.
The wheel spins at \(14\) revolutions per minute.
First, convert minutes to seconds: \(1\) minute \( = 60\) seconds.
\(\theta = 14\times2\pi\) radians, \(t = 60\) seconds.
\(\omega=\frac{14\times2\pi}{60}=\frac{14\pi}{30}=\frac{7\pi}{15}\approx1.47\) radians/sec.

Step2: Calculate linear velocity

Linear velocity \(v = r\omega\). Given \(r = 13\) cm and \(\omega=\frac{7\pi}{15}\) radians/sec.
\(v=13\times\frac{7\pi}{15}=\frac{91\pi}{15}\approx19.0\) cm/sec.

Answer:

a) \(\frac{7\pi}{15}\approx1.47\) radians/sec
b) \(\frac{91\pi}{15}\approx19.0\) cm/sec