To find the angular velocity of the ring, we can use the formula:

$\text{Angular Velocity} (\omega) = \frac{\theta}{t}$

where:

• $\theta$ is the angular displacement in radians (for one full revolution, $\theta = 2\pi$ radians),
• $t$ is the time taken to complete one full revolution.

Given:

• $t = 230,000$ hours (the time for one full revolution),
• $\theta = 2\pi$ radians (since one full revolution corresponds to $2\pi$ radians).

Now, plug in the values:

$\omega = \frac{2\pi \text{ radians}}{230,000 \text{ hours}}$

Calculating this:

$\omega = \frac{2\pi}{230,000} \text{ radians/hour}$

$\omega \approx \frac{6.2832}{230,000} \text{ radians/hour}$

$\omega \approx 2.73 \times 10^{-5} \text{ radians/hour}$

So, the angular velocity of the ring is approximately $2.73 \times 10^{-5}$ radians per hour.

Would you like more details or have any questions?

Here are 5 relative questions for you:

1. How do you convert angular velocity from radians per hour to degrees per hour?
2. What is the significance of angular velocity in planetary rings?
3. How would the angular velocity change if the ring's radius increased but the period remained the same?
4. What are some practical applications of angular velocity in astronomy?
5. How would you calculate the linear velocity of a point on the ring's circumference?

Tip: Always check the units of your measurements to ensure consistency when performing calculations.