Question

1. the pulley shown has a radius of 13.58 cm. suppose it takes 12 sec for 63 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 roller is cm per second. (type an integer or decimal rounded to the nearest tenth as needed.) (b) the angular speed of the roller is radian(s) per second. (type an integer or decimal rounded to the nearest hundredth as needed. use the answer from part (a) to find this answer.)

Explanation:

Step1: Calculate the linear speed of the belt

The belt travels \(63\) cm in \(12\) seconds. The formula for linear speed \(v\) is \(v=\frac{s}{t}\), where \(s\) is the distance and \(t\) is the time.
\(v = \frac{63}{12}=5.25\) cm per second.

Step2: Calculate the angular speed of the pulley

The formula relating linear speed \(v\), radius \(r\), and angular speed \(\omega\) is \(v = r\omega\). Given \(r = 13.58\) cm and \(v=5.25\) cm/s. We can solve for \(\omega\) by \(\omega=\frac{v}{r}\).
\(\omega=\frac{5.25}{13.58}\approx0.39\) radians per second.

Answer:

(a) \(5.3\) cm per second (rounded to the nearest tenth)
(b) \(0.39\) radians per second (rounded to the nearest hundredth)