An oxygen molecule \(\left(\mathrm{O}_{2}\right)\) rotates in the \(x y\) -plane about the \(z\) -axis. The axis of rotation passes through the center of the molecule, perpendicular to its length. The mass of each oxygen atom is \(2.66 \cdot 10^{-26} \mathrm{~kg},\) and the average separation between the two atoms is \(d=1.21 \cdot 10^{-10} \mathrm{~m}\) a) Calculate the moment of inertia of the molecule about the \(z\) -axis. b) If the angular speed of the molecule about the \(z\) -axis is \(4.60 \cdot 10^{12} \mathrm{rad} / \mathrm{s},\) what is its rotational kinetic energy?

Understanding the moment of inertia is crucial when dealing with rotational motion. It's the rotational equivalent of mass for linear motion. This property indicates how much torque is required to change the object’s rotational speed. For a single point mass, it depends only on the mass of that point and the square of its perpendicular distance from the axis of rotation. The formula is as simple as I = md^2, with I representing the moment of inertia, m the mass, and d the distance from the axis.

In more complex systems, like the oxygen molecule in our problem, composed of two point masses, each mass contributes to the total moment of inertia. We calculate each point mass's moment of inertia and sum them. Recognizing that both atoms are equally distanced from the rotation axis simplifies our work: we double the moment of inertia of one atom. Therefore, the formula becomes I_total = 2m(d/2)^2, enabling us to calculate the molecule's moment of inertia accurately.

Angular velocity is a measure of how fast an object rotates or revolves relative to another point, in this case, the axis of rotation. The faster the object spins, the greater its angular velocity, commonly denoted by the Greek letter \(\omega\). The unit of angular velocity is radians per second (rad/s). Radian is a measure of angle, equivalent to the angle at the center of a circle that subtends an arc equal in length to the radius.

To comprehend angular velocity’s impact, consider holding a string with a weight at one end and spinning it. If you speed up the spinning, the force you feel pulling on the string increases due to the higher angular velocity. In our molecule example, the angular velocity given is exceptionally high, which implies a very rapid spinning motion around the axis.

Rotational motion is movement around an axis or point. It's foundational in understanding how various objects move in our universe—from planets orbiting stars to the electron clouds within an atom. In our daily lives, we see rotational motion in wheels, fans, and even merry-go-rounds. What makes rotational motion interesting and sometimes complex is the interplay between the moment of inertia and the angular velocity.

The rotational kinetic energy is a form of energy an object possesses due to its rotation, and it's influenced by both the object's moment of inertia and its angular velocity. The formula K_{rot} = \(\frac{1}{2}I\omega^2\) expresses this relationship mathematically. It shows that if either the moment of inertia or the angular velocity increases, the rotational kinetic energy jumps up even more because of the squared term for \(\omega\). In practice, this means that when our oxygen molecule speeds up, its energy increases rapidly, highlighting the significance of controlling rotational speeds in machinery and technology to manage energy usage and safety.