Question

Calculate the angular speed and linear speed of the tip of a second hand of the clock if the second hand is 5cm long.

Answer 1

Hint: Let us consider the clock as a circular path and take the length of the second hand of the clock as the radius of the circular path. Angular speed is calculated as angular displacement per unit time and linear speed is calculated as total distance travelled per unit time.
Formula Used: $\text{Angular Speed}\left(\omega \right)\text{ = }{\displaystyle \frac{\text{Angular Displacement}\left(\theta \right)}{\text{time}\left(\text{t}\right)}}$
$\text{Linear Speed}\left(v\right)\text{ = Radius}\left(r\right)\times \text{Angular Velocity}\left(\omega \right)$.

Complete step by step answer:
Angular Speed is the rate at which an object changes its angle in radians, in a given period of time.
In complete revolution, Second hand of the clock covers an angular displacement of ${360}^{\circ }$ or $2\pi$ radian in 60 seconds.
So, we have: Angular Displacement$\left(\theta \right)$=$2\pi$ radian and time (t) = 60 seconds……….. (i)
Thus, $\text{Angular Speed}\left(\omega \right)\text{ = }{\displaystyle \frac{\text{Angular Displacement}\left(\theta \right)}{\text{time}\left(\text{t}\right)}}$
Now, we put the value of angular displacement and time from (i) in above equation, we have:
$$\begin{gathered} \therefore \text{Angular Speed}\left(\omega \right)\text{ = }{\displaystyle \frac{2\pi }{60 \\ }} \end{gathered}$$
Putting the value of$\pi =3.14$, we have:
$$\begin{gathered} \text{Angular Speed}\left(\omega \right)\text{ = }{\displaystyle \frac{2\times 3.14}{60}}={\displaystyle \frac{6.28}{60}}=0.1046 \\ \therefore \text{Angular Speed = 0}\text{.1046 rad/sec} \end{gathered}$$
Linear speed is defined as the rate of change of distance covered in unit time.
The relation between linear speed and angular speed is:
$\text{Linear Speed}\left(v\right)\text{ = Radius}\left(r\right)\times \text{Angular Velocity}\left(\omega \right)$……………… (ii)
Here, Radius is the length of a second hand.

Given, length of second hand = 5 cm
So, Radius (r) = 5 cm
As, we know that 1 cm=0.01 m, so we convert the unit of radius in metre, then we have:
 $\therefore \text{Radius(r) = }5\times 0.01=0.05m$
From above, the value of angular speed is 0.1046 rad/sec.
Now, we put the value of radius and angular speed in equation (ii),then we have:
$$\begin{gathered} \text{Linear Speed (v)}=0.05\times 0.1046 \\ \therefore \text{Linear Speed (v)}=0.00523\text{ m/sec} \end{gathered}$$
Thus, we have calculated the value of Angular Speed = 0.1046 rad/sec and
Linear speed = 0.00523m/sec.

Note: SI unit of angular speed is rad/sec. But, in many physical applications, angular speed is given as revolution per min, abbreviated as rpm. To convert angular speed from rpm to rad/sec, we have the following formula: $v\left(rad/\sec \right)={\displaystyle \frac{v\left(rpm\right)\times 60}{2\pi }}$