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Rotational inertia, also known as the moment of inertia, is a key concept when dealing with rotating objects. It is essentially the rotational equivalent of mass in linear motion. Just like mass resists changes in velocity, rotational inertia resists changes in angular speed. This property depends on how the mass of an object is distributed in relation to its axis of rotation.

For example, the farther the mass is from the central axis, the higher the rotational inertia. In the exercise we’re looking at, the man initially has his arms outstretched with bricks. This leads to a rotational inertia of 6.0 kg·m². However, when he pulls the bricks inwards, closer to his body, the rotational inertia decreases to 2.0 kg·m². By understanding how to manipulate rotational inertia, you can change how an object behaves while rotating.

In practical terms, athletes like figure skaters use this principle to spin faster by pulling their limbs closer to their body, effectively reducing their rotational inertia and thus spinning at higher speeds without additional force.

Angular speed denotes how fast an object is rotating around a particular axis. We often measure it in revolutions per second (rev/s) or radians per second (rad/s). The principle of angular momentum conservation lets us mathematically relate rotational inertia and angular speed.

In the provided exercise, the man's angular momentum is conserved. Initially, he spins at 1.2 rev/s with a rotational inertia of 6.0 kg·m². By changing this inertia to 2.0 kg·m², we apply the conservation of angular momentum: the product of the initial rotational inertia and angular speed must equal the product of the new rotational inertia and the new angular speed.

Using this principle, we calculate the resulting angular speed to be 3.6 rev/s after the change. So, even without any external force, the man's angular speed increases due to the manipulation of his body's rotational inertia.

Rotational kinetic energy is the energy an object possesses due to its rotation, closely related to its mass distribution (rotational inertia) and its angular speed. The formula for rotational kinetic energy is given by \[ KE = \frac{1}{2} I \omega^2 \] where \( I \) is the rotational inertia, and \( \omega \) is the angular speed.

In our exercise, we first calculate the initial kinetic energy using the initial rotational inertia and angular speed: \[ KE_i = \frac{1}{2} \times 6.0 \, \text{kg} \cdot \text{m}^2 \times (2\pi \times 1.2)^2 \]
Then, we find the final kinetic energy with the reduced rotational inertia and increased angular speed:\[ KE_f = \frac{1}{2} \times 2.0 \, \text{kg} \cdot \text{m}^2 \times (2\pi \times 3.6)^2 \]
Calculating these, we can determine the ratio of the new kinetic energy to the original kinetic energy by dividing \( KE_f \) by \( KE_i \). The interesting revelation here is that the extra energy comes from the internal work done by the man—illustrating the basic principles of physics where energy transformation within a closed system can result in increased motion.