Question

Find ratio of angular speeds of second hand and minute hand?

Answer 1

The second hand of a clock completes one full rotation in 60 seconds, while the minute hand completes one full rotation in 60 minutes, or 3600 seconds. To find the ratio of the angular speeds of the second hand and the minute hand, we use the following formula:

\[ \frac{\text{Angular speed of second hand}}{\text{Angular speed of minute hand}} = \frac{2\pi \text{ radians}}{60 \text{ seconds}} \div \frac{2\pi \text{ radians}}{3600 \text{ seconds}} \]

Now, simplifying this expression step-by-step:

\[ \frac{\text{Angular speed of second hand}}{\text{Angular speed of minute hand}} = \frac{\frac{2\pi}{60}}{\frac{2\pi}{3600}} \]

Multiplying by the reciprocal of the second term:

\[ \frac{\text{Angular speed of second hand}}{\text{Angular speed of minute hand}} = \frac{2\pi}{60} \times \frac{3600}{2\pi} \]

Notice that the factors of \(2\pi\) cancel each other out:

\[ \frac{\text{Angular speed of second hand}}{\text{Angular speed of minute hand}} = \frac{3600}{60} = 60 \]

Therefore, the ratio of the angular speeds of the second hand and minute hand is 60:1. This means that the second hand rotates 60 times faster than the minute hand.