Question

example 6: the second - hand of a clock is 10.2 centimeters long. find the following: a. find the angular speed in radians per second of the tip of the second hand as it passes around the clock face. write the exact answer (do not round). b. find the linear speed in centimeters per second of the tip of this second hand as it passes around the clock face. write the exact answer and then round to two decimal places.

Explanation:

Step1: Recall angular - speed formula for a clock

The second - hand of a clock makes a full revolution ($2\pi$ radians) in 60 seconds. The formula for angular speed $\omega$ is $\omega=\frac{\theta}{t}$, where $\theta$ is the angle of rotation and $t$ is the time. For the second - hand, $\theta = 2\pi$ radians and $t = 60$ s. So, $\omega=\frac{2\pi}{60}=\frac{\pi}{30}$ rad/s.

Step2: Recall linear - speed formula

The formula for linear speed $v$ is $v = r\omega$, where $r$ is the radius and $\omega$ is the angular speed. Given that $r = 10.2$ cm and $\omega=\frac{\pi}{30}$ rad/s. Then $v=(10.2)\times\frac{\pi}{30}=\frac{10.2\pi}{30}=\frac{1.7\pi}{5}$ cm/s.

Step3: Calculate the decimal value of linear speed

$v=\frac{1.7\pi}{5}\approx\frac{1.7\times3.14}{5}=\frac{5.338}{5}=1.0676\approx1.07$ cm/s.

Answer:

a. $\omega=\frac{\pi}{30}$ rad/s
b. Exact value of $v$: $v = \frac{1.7\pi}{5}$ cm/s, Rounded value of $v$: $v\approx1.07$ cm/s