The length of the seconds hand of a watch is 10 mm. What is the change in the angular speed of the watch after 15 seconds

Correct Answer: Option Zero

To find the change in the angular speed of the seconds hand of the watch after 15 seconds, we need to analyze how angular speed is defined and whether it varies over time.

1. Understanding Angular Speed:
Angular speed ($\omega$) is a measure of how quickly an object rotates or revolves relative to another point. For a rotating object, it is defined as the angle rotated per unit time:

$\omega =\frac{\Delta \theta }{\Delta t}$

2. Calculation for a Seconds Hand:
The seconds hand of a watch completes one full revolution (which is $2\pi$ radians) in exactly 60 seconds. Since a watch functions at a constant, uniform rate, the seconds hand moves in uniform circular motion.
Its angular speed is:

$\omega =\frac{2\pi \text{rad}}{60\text{s}}=\frac{\pi }{30}\text{rad/s}$

3. Change in Angular Speed:
Since the rotation of the seconds hand is uniform, its angular speed remains constant at all times.
This means the initial angular speed (${\omega }_{\text{initial}}$) at $t=0$ and the final angular speed (${\omega }_{\text{final}}$) at $t=15\text{s}$ are exactly the same:

${\omega }_{\text{initial}}={\omega }_{\text{final}}=\frac{\pi }{30}\text{rad/s}$

Therefore, the change in angular speed ($\Delta \omega$) is:

$\Delta \omega ={\omega }_{\text{final}}-{\omega }_{\text{initial}}=0$

Thus, the change in the angular speed of the watch after 15 seconds is Zero.