Consider a smoothly running analog wall clock. What is the exact ratio of the angular velocity of it

Question

Consider a smoothly running analog wall clock. What is the exact ratio of the angular velocity of its minute hand to the angular velocity of its hour hand?

Professor PC Thomas Classes · Answer

(d) : The angular velocity of any rotating object is given by $\omega = \frac{2\pi}{T}$, where $T$ is the time period of one complete revolution. For the minute hand, it completes one revolution in $1 	ext{ hour}$, so $\omega_{min} = \frac{2\pi}{1 	ext{ hr}}$. For the hour hand, it completes one revolution in $12 	ext{ hours}$, so $\omega_{hr} = \frac{2\pi}{12 	ext{ hr}}$. Taking the ratio: $\frac{\omega_{min}}{\omega_{hr}} = \frac{2\pi / 1}{2\pi / 12} = 12$. The ratio is $12:1$.