Question

the hour hand of a standard clock moves with an angular velocity of \\(4\pi\\) radians per day. how many radians does the hour hand rotate through in one hour?

\\(\pi/30\\) radians
\\(\pi/12\\) radians
\\(\pi/6\\) radians
\\(\pi\\) radians

Explanation:

Identify the given values

We are given the angular velocity of the hour hand of a standard clock:

$$\omega = 4\pi \text{ radians per day}$$

Identify the target unknown

We need to find the angle in radians that the hour hand rotates through in exactly one hour:

$$\theta = \text{angle in radians for } t = 1 \text{ hour}$$

Establish the conversion factor

To convert the rate from "per day" to "per hour", we use the standard time conversion:

$$1 \text{ day} = 24 \text{ hours}$$

Calculate the rotation per hour

We divide the total daily angular rotation by the number of hours in a day:

$$\theta = \frac{4\pi \text{ radians}}{24 \text{ hours}}$$

Simplifying the fraction:

$$\theta = \frac{\pi}{6} \text{ radians per hour}$$

Match with the given options

The calculated value is \(\pi/6\) radians, which matches the third option.

Answer:

• \(\pi/30\) radians
• \(\pi/12\) radians
• \(\pi/6\) radians (Correct answer)
• \(\pi\) radians