Question

If the length of the second of the wall clock is 3 cm, the angular velocity and the linear velocity of the tip is equal to:
(A) 0.201 $\dfrac{{rad}}{s}$ , 0.314 $\dfrac{m}{s}$
(B) 0.254 $\dfrac{{rad}}{s}$ , 0.314 $\dfrac{m}{s}$
(C) 0.104 $\dfrac{{rad}}{s}$ , 0.631 $\dfrac{m}{s}$
(D) 0.104 $\dfrac{{rad}}{s}$ , 0.0031 $\dfrac{m}{s}$

Answer 1

Hint: We first need to calculate the angular displacement of the second hand of the clock in 1s. then we will use this to find the angular velocity of the clock. As we know that the linear velocity is the cross product of the radius of the circle and its angular velocity, so we can find linear velocity from this relation.

Complete step by step solution:
 The angular velocity of a body is given as the angular displacement per unit time. a second-hand takes the 60s to complete 1 circle around the clock. This 1 one complete circle is represented as $2\pi $ . So the angular velocity of the tip of the clock will be:
 $\omega = \dfrac{{2\pi }}{{60}} = \dfrac{\pi }{{30}} = \dfrac{{3.14}}{{60}} = 0.104$
Now, we know the angular velocity of the body. To find the linear velocity, we use:
$v = rx\omega $
$ \Rightarrow v = r\omega \sin \theta $
$ \Rightarrow v = 0.03 \times 0.104\sin (90)$
$ \Rightarrow v = 0.031$
Therefore, the option with the correct answer is option D. 0.104 $\dfrac{{rad}}{s}$ , 0.0031 $\dfrac{m}{s}$

Note:
In a uniform circular motion, the angular velocity does not change its magnitude because it is dependent on the angular displacement of the body. However, the linear velocity is dependent on the radius of the body and every point which lies on the radius of the circle will have a different angular velocity.