Question

Assertion: Angular velocity of the second’s needle of a watch is $\dfrac{{30}}{\pi }$ rad/s.
Reason: Angular velocity $\omega = \dfrac{T}{{2\pi }}$, where T is the time period.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
C. Assertion is correct but Reason is incorrect.
D. Both Assertion and Reason are incorrect.

Answer 1

The second’s needle of the watch completes one revolution in 60 seconds. The angular velocity is the ratio of angular displacement to the time elapsed. The angular velocity of the body is inversely proportional to the time period.

Complete step by step answer:
We have given in the assertion that the angular velocity of the second’s needle is $\dfrac{{30}}{\pi }$ rad/s.
We know that the second’s needle of the watch completes one revolution in 60 seconds. We also know that,
$1\,{\text{rev}} = 2\pi \,{\text{rad}}$
Therefore, the angular velocity of the second’s needle is,
$\omega = \dfrac{{2\pi \,{\text{rad}}}}{{60\,{\text{s}}}}$
$\Rightarrow \omega = \dfrac{\pi }{{30}}\,{\text{rad/s}}$
Therefore, we can say, the assertion is the incorrect statement.

Now, we have the relationship between the angular velocity and frequency,
$\omega = 2\pi f$
Here, f is the frequency.
But, the frequency is the reciprocal of the time period. Therefore, we can write,
$\omega = \dfrac{{2\pi }}{T}$
Here, $\omega$ is the angular velocity of the second’s needle and T is the time period of revolution of the second’s needle.
We have given the reason that the angular velocity is given as, $\omega = \dfrac{T}{{2\pi }}$. But we have derived that the angular velocity is given as, $\omega = \dfrac{{2\pi }}{T}$.
Thus, we can say, the reason is also the incorrect statement.

So, the correct answer is option D.

Note: One complete rotation covers 360 degrees of the circle and one should know that $360^\circ = 2{\pi ^c}$. The S.I unit of angular velocity is rad/s and therefore, students must convert the angular velocity from rev/s to rad/s whenever necessary. Also, one should always remember, the minute hand of the watch takes 60 second to complete the complete revolution of the minute hand.