91Ó°ÊÓ

The moment of inertia is essentially a measure of how much an object resists changes to its rotation. Imagine trying to spin a spinning top or a frying pan. The top, being more compact, is easier to spin than the pan because it has a smaller moment of inertia. This property depends on the mass of the object and how this mass is distributed with respect to the axis of rotation. The farther the mass is from the axis, the greater the moment of inertia.
In the case of the student on the stool, when their arms are outstretched, the masses in their hands are further from the axis of rotation, increasing the moment of inertia. The formula to calculate it in such scenarios is:

• The moment of inertia, denoted by \( I \), is calculated as \( I = I_0 + 2 imes m imes r^2 \), where:
• \( I_0 \) is the initial moment of inertia without the masses,
• \( m \) is the mass of the objects held in the student's arms,
• \( r \) is the distance of the masses from the axis of rotation.

Understanding how the distribution of mass affects the moment of inertia helps explain why rotating bodies behave the way they do when that distribution changes.

Angular speed tells us how fast something is spinning. It is like the rotational version of linear speed but instead of measuring how fast something moves from one place to another, it measures how quickly it spins through an angle. It is measured in revolutions per second (rev/s) or in radians per second (rad/s).
For the student on the stool, the initial angular speed is 2.95 rev/s. This translates to how quickly they are rotating at the start. When the masses are pulled inwards, the student spins faster, increasing the angular speed to 3.54 rev/s.
Why does this change occur? The principle of conservation of angular momentum answers this. Since no external torques are involved, the initial angular momentum \( L_i = I_i \cdot \omega_i \) will equal the final angular momentum \( L_f = I_f \cdot \omega_f \). Here, \( \omega_i \) and \( \omega_f \) are the initial and final angular speeds, respectively. This relationship shows that as the moment of inertia decreases by pulling in the arms, the angular speed must increase to keep the angular momentum constant.

Kinetic energy in rotational motion works similarly to linear kinetic energy but with a rotational twist. It represents the energy possessed due to a body's rotation. Calculated using the formula \( KE = \frac{1}{2} I \omega^2 \), it determines how much energy is involved in the rotating motion.
Initially, using an initial moment of inertia \( I_i \) and angular speed \( \omega_i \), the kinetic energy can be calculated for the student's system. When the arms are outstretched, the kinetic energy is computed using these initial values:

• Initial kinetic energy \( KE_i \approx 109.64 \text{ J} \)
• Final kinetic energy once the arms are drawn in \( KE_f \approx 126.68 \text{ J} \)

Notice the increase in kinetic energy as the angular speed increases. This change illustrates how pulling the masses inward, changing the moment of inertia, transfers energy into speeding up the rotation, leading to increased kinetic energy in the system.