91Ó°ÊÓ

The moment of inertia is a central concept when analyzing rotational motion. It signifies an object's resistance to changes in its rotational state. For a simple object like a disk, the moment of inertia (\( I \)) can be calculated using the formula: \( I = \frac{1}{2} m r^2 \), where \( m \) is the mass and \( r \) is the radius of the disk.
In the given exercise, each disk (large and small) has its moment of inertia. The large disk's moment of inertia was initially calculated as \( I_{\text{large}} = 45mr^2 \). The small disk's initial position gives it a moment of inertia of \( I_{\text{small initial}} = \frac{1}{2}mr^2 \).
Once the smaller disk slides to the edge, its inertia changes to \( I_{\text{small final}} = \frac{9}{2}mr^2 \) due to the increased radius. These changes in inertia reflect shifts in how mass distribution affects the system's ability to rotate.

Angular velocity describes how fast an object rotates. It's measured in radians per second (\( \text{rad/s} \)). Initially, both disks in the exercise rotate together at an angular velocity of \( 20 \text{ rad/s} \).
When the smaller disk moves outward, conservation of angular momentum comes into play. This principle suggests that unless external torques act upon it, the angular momentum (\( L \)) remains constant. Therefore, even as the moment of inertia changes, the formula \( L_i = L_f \) (initial angular momentum equals final angular momentum) helps us find the new angular velocity.
We use the relationship: \[ \omega_f = \frac{I_{\text{total initial}} \cdot \omega_i}{I_{\text{total final}}} \]This gives a new angular velocity of \( \frac{910}{49.5} \text{ rad/s} \), showing how redistributing mass affects rotational speed.

The kinetic energy in rotational systems is vital for understanding dynamics. It's given by the formula for rotational kinetic energy:\[ K = \frac{1}{2} I \omega^2 \]Initially, the system's kinetic energy when both disks rotate together with an angular velocity of \( 20 \text{ rad/s} \) is computed as \( 9100mr^2 \).
After the smaller disk slides to the edge, the new kinetic energy is influenced by both the system's final angular velocity and its updated moment of inertia. This new kinetic energy reflects the system's altered energy state due to its mass redistribution, though exact calculations in later steps ensure precision.
Finally, the ratio of the kinetic energies (\( \frac{K}{K_0} \)) provides insights into how much energy remains after such transformations in the system. It confirms the conservation laws at play and highlights energy changes that occur when redistributing mass within a rotational setup.