What is the ratio of the angular speed of the second hand to the angular speed of the hour hand on a clock?

The second hand completes one full revolution ($2\pi$ radians) in 60 seconds.
The time period for the second hand is $T_s = 60$ seconds.
The angular speed $\omega_s$ is given by:

$$\omega_s = \frac{2\pi}{T_s} = \frac{2\pi}{60} \text{ rad/s}$$

The hour hand completes one full revolution ($2\pi$ radians) in 12 hours.
First, convert 12 hours to seconds:

$$T_h = 12 \text{ hours} \times 60 \text{ minutes/hour} \times 60 \text{ seconds/minute}$$ $$T_h = 12 \times 3600 \text{ seconds} = 43200 \text{ seconds}$$

The angular speed $\omega_h$ is given by:

$$\omega_h = \frac{2\pi}{T_h} = \frac{2\pi}{43200} \text{ rad/s}$$

$$\frac{\omega_s}{\omega_h} = \frac{\frac{2\pi}{60}}{\frac{2\pi}{43200}}$$ $$\frac{\omega_s}{\omega_h} = \frac{2\pi}{60} \times \frac{43200}{2\pi}$$ $$\frac{\omega_s}{\omega_h} = \frac{43200}{60}$$ $$\frac{\omega_s}{\omega_h} = \frac{4320}{6}$$ $$\frac{\omega_s}{\omega_h} = 720$$

The ratio is $720:1$.

The final answer is $\text{(c) } 720:1$.