91Ó°ÊÓ

The moment of inertia is a critical parameter in rotational dynamics. It measures how mass is distributed relative to the axis of rotation. For a rigid body, it is expressed as the sum of the products of the mass of each particle and the square of its distance from the rotation axis. Mathematically, it can be represented as: \[ I = \frac{1}{2} m R^2 \] for a disk rotating about its center and for point masses at a distance R, \[ I = m R^2 \]. In our exercise, initially, the disk and the cockroach together, have a combined moment of inertia of \[ I_1 = (5m + m) R^2 = 6m R^2 \]. When the cockroach moves halfway, new moment of inertia becomes \[ I_2 = 5m R^2 + \frac{1}{4} m R^2 = 5.25 m R^2 \]. This reduction in moment of inertia signifies that the mass is moving closer to the rotation axis, impacting the system's rotational characteristics.

Rotational kinematics describe the motion of rotating objects, including how their angular velocity changes over time. When you're dealing with rotating bodies, angular velocity (\( \omega \)) is a fundamental concept. It represents how fast an object rotates or spins around an axis. In this problem, the conservation of angular momentum is vital. Since no external torque acts on the cockroach-disk system, angular momentum remains constant: \[ I_1 \omega_1 = I_2 \omega_2 \]. Given the moments of inertia before and after the cockroach moves, we use them to find the final angular velocity: \[ \omega_2 = \omega_1 \frac{I_1}{I_2} = \omega_1 \frac{6}{5.25} = 1.1429 \omega_1 \]. This shows an increase in angular velocity, even though the cockroach moved toward the center, demonstrating how distributing mass affects rotational speed.

Kinetic energy in rotational motion is the energy due to the rotation of an object and is given by: \[ K = \frac{1}{2}I \omega^2 \]. Initially, the kinetic energy of the system is calculated using its initial moment of inertia and angular velocity: \[ K_1 = \frac{1}{2}I_1 \omega_1^2 = 3mR^2 \omega_1^2 \]. After the cockroach moves, the new kinetic energy is: \[ K_2 = \frac{1}{2}I_2 \omega_2^2 = 3.43mR^2 \omega_1^2 \]. When comparing the two kinetic energies, the ratio is: \[ \frac{K_2}{ K_1} = \frac{3.43mR^2 \omega_1^2}{3mR^2 \omega_1^2} = 1.143 \]. This increase in kinetic energy is due to the cockroach moving closer to the center, decreasing the moment of inertia and causing the system to rotate faster. Even in the absence of external torque, this change in mass distribution causes a higher rotational kinetic energy.