To increase the angular velocity $\omega$ of the discus while keeping everything else constant, we need to consider the relationship between angular velocity and the system's key parameters. The primary relationship for angular velocity is tied to the moment of inertia $I$ and the rotational kinetic energy, which is given by:

$E_{\text{rot}} = \frac{1}{2} I \omega^2$

Here are some factors that could increase $\omega$:

1. Reduce the moment of inertia: The moment of inertia $I$ depends on the mass distribution relative to the axis of rotation. Specifically, for a point mass or system rotating at a radius $r$, the moment of inertia is:

$I = m r^2$

Where $m$ is the mass and $r$ is the radius. Decreasing the radius $r$ would decrease the moment of inertia, and because angular momentum $L = I \omega$ is conserved (assuming no external torque), a decrease in $I$ would lead to an increase in $\omega$. Thus, reducing the radius of the discus' rotation increases the angular velocity the most.

2. Increase the rotational kinetic energy: If the athlete exerts more force or energy to the system, the total energy increases. This would directly increase the angular velocity as well.

3. Reduce friction: If there’s any friction in the system (for example, air resistance or friction between the athlete’s grip and the discus), reducing it would allow for more efficient transfer of energy into increasing $\omega$.

Conclusion:

The most effective way to increase the angular velocity is to reduce the radius $r$, since angular momentum is conserved and $\omega$ will increase as $r$ decreases.

Would you like further details or have any questions?

Here are 5 follow-up questions for you:

1. How does moment of inertia affect rotational motion in other systems?
2. What is the relationship between linear velocity and angular velocity?
3. How does the conservation of angular momentum work in a closed system?
4. What other examples demonstrate changes in angular velocity due to changes in radius?
5. How does torque affect angular velocity?

Tip: The relationship between angular velocity and radius is inversely proportional, meaning that small changes in radius can have a large impact on $\omega$.