D

Step 1: The second's hand of a clock makes one complete revolution in one minute (60 seconds).
Step 2: The angular velocity ($\omega$) in radians per second can be given by the formula: $\omega = \frac{\theta}{t}$, where $\theta$ is the angle covered in radians and $t$ is the time in seconds.
Step 3: For one complete revolution, $\theta = 2\pi$ radians, and $t = 60$ seconds.
Step 4: Substitute these values into the formula:
\(\omega = \frac{2\pi}{60} = \frac{\pi}{30}\).
Step 5: Therefore, the angular velocity of the second's needle is \(\frac{\pi}{30}\). Hence, the correct option is D.