To solve this problem, we can use the equations of rotational motion with constant angular acceleration. Here’s the approach step-by-step.

Given:

• Initial angular velocity ($\omega_0$) = 0 (since the disk starts from rest)
• Let $n_1 = 10$ revolutions required to reach angular velocity $\omega$.
• Final angular velocity in the first phase ($\omega$) is reached after 10 revolutions.
• We need to find the additional revolutions $n_2$ required to reach an angular velocity of $\sqrt{2} \omega$.

Step 1: Relate Revolutions to Angular Velocity

For rotational motion, we can use the following kinematic equation: $\omega^2 = \omega_0^2 + 2 \alpha \theta$ where:

• $\omega$ is the final angular velocity,
• $\omega_0$ is the initial angular velocity,
• $\alpha$ is the angular acceleration, and
• $\theta$ is the angular displacement in radians.

Since $\omega_0 = 0$, this simplifies to: $\omega^2 = 2 \alpha \theta$

Step 2: Find $\alpha$ in Terms of $\omega$

For the first 10 revolutions, we have: $\omega^2 = 2 \alpha (10 \times 2\pi) = 20 \alpha \pi$ So, $\alpha = \frac{\omega^2}{20 \pi}$

Step 3: Calculate Revolutions for $\sqrt{2} \omega$

Now we need the total revolutions to reach $\sqrt{2} \omega$ as the final angular velocity. Using the same kinematic equation: $(\sqrt{2} \omega)^2 = 2 \alpha \theta$ $2 \omega^2 = 2 \alpha \theta$

Substitute $\alpha = \frac{\omega^2}{20 \pi}$: $2 \omega^2 = 2 \left(\frac{\omega^2}{20 \pi}\right) \theta$ $\theta = 20 \pi \text{ revolutions}$

Step 4: Find Additional Revolutions

The additional revolutions $n_2$ required are: $n_2 = \theta - 10 = 20 - 10 = 10 \text{ revolutions}$

Thus, 10 additional revolutions are required.

Would you like further clarification on any part of this solution?

Related Questions:

1. What if the disk took 15 revolutions to reach angular velocity $\omega$ instead?
2. How does this calculation change if the angular acceleration is not constant?
3. How many total revolutions would be needed to reach $2 \omega$ instead of $\sqrt{2} \omega$?
4. Can this approach be used for linear motion problems as well? How?
5. How does the angular displacement affect the time taken if the initial velocity is not zero?

Tip:

Always remember to convert revolutions to radians when dealing with rotational kinematic equations, as radians are the standard unit for angular displacement in physics.