Question

the pulley shown has a radius of 17.96 cm. suppose it takes 15 sec for 64 cm of belt to go around the pulley. (a) find the linear speed of the belt in centimeters per second. (b) find the angular speed of the pulley in radians per second. (a) the linear speed of the belt is cm per second. (type an integer or decimal rounded to the nearest tenth as needed.)

Explanation:

Step1: Recall linear - speed formula

The formula for linear speed $v$ is $v=\frac{s}{t}$, where $s$ is the distance traveled and $t$ is the time taken.
Given $s = 64$ cm and $t=15$ s.
$v=\frac{64}{15}\text{ cm/s}$

Step2: Calculate linear speed

$v=\frac{64}{15}\approx4.3$ cm/s

Step3: Recall relationship between linear and angular speed

The relationship between linear speed $v$, angular speed $\omega$ and radius $r$ is $v = r\omega$. We know $v\approx4.3$ cm/s and $r = 17.96$ cm. So, $\omega=\frac{v}{r}$.

Step4: Calculate angular speed

$\omega=\frac{\frac{64}{15}}{17.96}=\frac{64}{15\times17.96}\approx\frac{64}{269.4}\approx0.2$ rad/s

Answer:

(a) $4.3$
(b) $0.2$