To solve this problem, we need to apply the principle of conservation of angular momentum. Since there are no external torques acting on the system, the total angular momentum before and after the two disks stick together must be the same.

Step 1: Calculate the initial angular momentum.

Let the moment of inertia of one disk be $I$ and the angular velocity of the first disk be $\omega_1 = 17.4 \, \text{rad/s}$.

For a uniform cylindrical disk, the moment of inertia about its central axis is:

$I = \frac{1}{2} m r^2$

where $m$ is the mass of the disk and $r$ is its radius.

Before the second disk is dropped, the angular momentum of the system is just the angular momentum of the rotating disk:

$L_{\text{initial}} = I \omega_1 = \frac{1}{2} m r^2 \cdot 17.4$

Step 2: Calculate the final angular momentum.

When the second disk is dropped on top of the first disk, the system now consists of two disks, each with the same moment of inertia $I$. Since the two disks are identical, the total moment of inertia of the system after they stick together is:

$I_{\text{final}} = 2I = 2 \cdot \frac{1}{2} m r^2 = m r^2$

Let the common angular velocity of the system after the collision be $\omega_f$. According to the conservation of angular momentum:

$L_{\text{initial}} = L_{\text{final}}$

This gives us the equation:

$I \omega_1 = (2I) \omega_f$

Substitute $I = \frac{1}{2} m r^2$ into the equation:

$\frac{1}{2} m r^2 \cdot 17.4 = m r^2 \cdot \omega_f$

Step 3: Solve for the final angular velocity.

Canceling $m r^2$ from both sides:

$\frac{1}{2} \cdot 17.4 = \omega_f$

$\omega_f = \frac{17.4}{2} = 8.70 \, \text{rad/s}$

Final Answer:

The common angular velocity of the two disks after they stick together is:

$\omega_f = 8.70 \, \text{rad/s}$

Would you like more details or have any questions about the steps?

Here are 5 related questions for practice:

1. How would the result change if the second disk had a different mass?
2. What would happen if the disks did not stick together?
3. How would you calculate the rotational kinetic energy before and after the collision?
4. What if the second disk was rotating before being dropped on the first one?
5. How does the radius of the disks affect the final angular velocity?

Tip: Conservation laws, such as conservation of angular momentum, are powerful tools in solving collision and rotational dynamics problems.