91Ó°ÊÓ

The principle of conservation of angular momentum is pivotal in understanding rotational dynamics. Angular momentum is defined as the product of the moment of inertia and angular velocity, expressed mathematically as \( L = I \omega \). For a point mass \( m \) rotating at a distance \( r \) from an axis, the moment of inertia can be represented as \( I = mr^2 \). Thus, the angular momentum for the block revolving in a circle is \( L = mr^2 \omega \).
In our exercise, since the surface is frictionless and no external torques are applied, the system's angular momentum is conserved. This means that the initial angular momentum \( L_i = mr_i^2 \omega_i \) before the cord is pulled remains equal to the final angular momentum \( L_f = mr_f^2 \omega_f \).
By employing conservation laws, we determine that when the radius decreases, the angular speed must increase to maintain the same angular momentum, leading to the equation \( \omega_f = \left(\frac{r_i^2}{r_f^2}\right) \omega_i \). Understanding this conservation principle is crucial to solving the problem of changing circle radii.

Kinetic energy in rotational motion is similar to linear motion, but here it depends on both the mass distribution and the square of the angular velocity. Mathematically, it's expressed as \( K = \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia. For a mass orbiting a circle with radius \( r \), the kinetic energy becomes \( K = \frac{1}{2} m r^2 \omega^2 \).
In our situation, initially, the block has an angular speed of \( 1.75 \, \text{rad/s} \) at a radius of \( 0.300 \, \text{m} \). Substituting these values into the equation gives the initial kinetic energy \( K_i = 0.00344 \, \text{J} \).
After pulling the cord, the radius reduces to \( 0.150 \, \text{m} \) and the angular speed increases to \( 7.00 \, \text{rad/s} \), resulting in a new kinetic energy \( K_f = 0.013125 \, \text{J} \). The increased energy stems from the system's rotational adjustment to preserve angular momentum, illustrating energy's dynamic relationship with rotational motion.

The work-energy principle connects the net work done on an object to its change in kinetic energy. In cases where friction is absent, like in this exercise, all the work done translates directly into changing the system's kinetic energy.
Here, as the cord is pulled and the block's motion changes, the work done on the block is quantifiable using the calculated change in kinetic energy. Initially, the kinetic energy is \( 0.00344 \, \text{J} \) and increases to \( 0.013125 \, \text{J} \). The work done, calculated as \( \Delta K = K_f - K_i \), results in \( 0.009685 \, \text{J} \).
This means all the energy resulting from the work done manifests as kinetic energy, showcasing the work-energy principle's role in rotational dynamics. This principle helps us understand how work done through force application can change an object's energy state without considering energy loss due to friction or other forces.