To solve this problem, let's go through the steps systematically.

1. Given Information:

   • The disk starts from rest, so the initial angular velocity $\omega_0 = 0$.
   • It takes 10 revolutions to reach an angular velocity $\omega$ at constant angular acceleration $\alpha$.

2. Relevant Equations: For an object under constant angular acceleration, the relationship between angular displacement $\theta$, initial angular velocity $\omega_0$, final angular velocity $\omega$, and angular acceleration $\alpha$ is given by: $\omega^2 = \omega_0^2 + 2 \alpha \theta$

3. Step 1: Determine $\alpha$ in Terms of $\omega$:

   • Convert 10 revolutions into radians. Since one revolution is $2 \pi$ radians, 10 revolutions is: $\theta = 10 \times 2\pi = 20\pi \text{ radians}$
   • Using the equation $\omega^2 = \omega_0^2 + 2 \alpha \theta$: $\omega^2 = 0 + 2 \alpha (20\pi)$ $\alpha = \frac{\omega^2}{40\pi}$

4. Step 2: Determine the Angular Displacement to Reach $\sqrt{2} \omega$:

   • The new angular velocity is $\sqrt{2} \omega$.
   • Using the same formula: $(\sqrt{2} \omega)^2 = 0 + 2 \alpha \theta_{\text{new}}$
   • Expanding $(\sqrt{2} \omega)^2$: $2 \omega^2 = 2 \alpha \theta_{\text{new}}$
   • Substitute $\alpha = \frac{\omega^2}{40\pi}$ from earlier: $2 \omega^2 = 2 \left( \frac{\omega^2}{40\pi} \right) \theta_{\text{new}}$ $\theta_{\text{new}} = 40 \pi$

5. Step 3: Calculate Additional Revolutions:

   • The total displacement to reach $\sqrt{2} \omega$ is $40 \pi$ radians.
   • The additional revolutions required are: $\theta_{\text{additional}} = \theta_{\text{new}} - \theta = 40 \pi - 20 \pi = 20 \pi \text{ radians}$
   • Converting $20 \pi$ radians back to revolutions: $\text{Additional revolutions} = \frac{20 \pi}{2 \pi} = 10$

Answer: The disk requires an additional 10 revolutions to reach an angular velocity of $\sqrt{2} \omega$.

Would you like further details on any step, or have additional questions?

Follow-up Questions:

1. What is the relationship between angular acceleration and angular velocity?
2. How would the solution change if the initial angular velocity were not zero?
3. What would happen to the number of additional revolutions if the angular acceleration were doubled?
4. How can this concept apply to rotational motion in mechanical systems?
5. What are the physical units associated with each variable in this problem?

Tip: Always convert units consistently when dealing with angular quantities in rotational motion problems.