Question

The oxygen molecule has a total mass of \(5.30 \times 10^{-26} \mathrm{~kg}\) and a rotational inertia of \(1.94 \times 10^{-46} \mathrm{~kg} \cdot \mathrm{m}^{2}\) about an axis through the center perpendicular to the line joining the atoms. Suppose that such a molecule in a gas has a mean speed of \(500 \mathrm{~m} / \mathrm{s}\) and that its rotational kinetic energy is two- thirds of its translational kinetic energy. Find its average angular velocity.

Short answer

The average angular velocity can be found by first finding the translational and rotational kinetic energy, then plugging these values into the equation for the kinetic energy of rotation and solving for the angular velocity.

Step-by-step solution

Calculate the translational kinetic energy

First, we need to calculate the translational kinetic energy using the given values. The formula for the kinetic energy of translation is \(KE_{translational}=\frac{1}{2}mv^2\). Substituting the values, we get: \(KE_{translational}=\frac{1}{2}(5.30 \times 10^{-26}\) kg \(\times (500\) m/s\()\)^2.

Find the rotational kinetic energy

The problem tells us that the rotational kinetic energy is two-thirds of the translational kinetic energy. We can find it by: \(KE_{rotational}=\frac{2}{3}KE_{translational}\)

Calculate the average angular velocity

Now using the formula of the kinetic energy for rotation, which is \(KE_{rotational}=\frac{1}{2}Iw^2\), we can solve for the average angular velocity \(w\). The rotational inertia \(I\) is given as \(1.94 \times 10^{-46}\) kg m\(^2\). Rearranging the equation for \(w\) and plugging in the known values, we can find the average angular velocity.

Key concepts

Translational Kinetic Energy

Translational kinetic energy refers to the energy possessed by an object in motion related to the translation (or straight-line movement) of its center of mass. For any given object with mass m moving at a speed v, this energy can be calculated using the equation:
\( KE_{translational} = \frac{1}{2}mv^2 \).
A real-world example of translational kinetic energy could be a soccer ball flying toward the goal post after being kicked. The equation captures how the energy scales with both mass and the square of the object's speed, indicating that even a small increase in speed results in a significantly larger increase in translational kinetic energy.
In the context of the textbook exercise, the translational kinetic energy of the oxygen molecule is critical because it sets the baseline for calculating the molecule's rotational kinetic energy, which is said to be two-thirds of this value. This relationship between translational and rotational kinetic energies highlights a key interplay between different forms of motion in physics. Understanding translation is essential as it's often the first step in complex motion analysis.

Angular Velocity

Angular velocity, often represented by the Greek letter ω (omega), is the rate at which an object rotates or revolves around an axis. It's the rotational equivalent of linear velocity and is usually measured in radians per second (rad/s). For instance, when looking at the spinning Earth, its angular velocity would be the rate at which it completes a rotation on its axis.
Calculating the angular velocity is vital in many branches of physics and engineering, especially when dealing with rotating systems. The relationship between angular velocity and translational kinetic energy is illustrated in the exercise where angular velocity must be derived from the given kinetic energies.
The equation used to describe rotational kinetic energy is \( KE_{rotational} = \frac{1}{2}I\omega^2 \), where I is the rotational inertia. By knowing the rotational kinetic energy, one can rearrange this formula to solve for ω, which is precisely what the exercise requires. Understanding angular velocity is fundamental for students as it appears in a wide variety of contexts, from simple merry-go-rounds to the complex rotations of molecules in gases.

Rotational Inertia

Rotational inertia (also known as the moment of inertia) quantifies an object's resistance to changes in its rotation. It is influenced by both the mass of the object and the distribution of that mass relative to the axis of rotation. The formula general for a collection of particles is
\( I = \sum m_ir^2_i \), where m_i is the mass of a particle in the object and r_i is the distance of that particle from the axis of rotation.
Just like mass is a measure of how much an object resists being moved (translational motion), rotational inertia is a measure of how much an object resists being spun or rotated. Something with high rotational inertia, like a discus, requires a larger torque for it to achieve the same angular acceleration as something with a lower rotational inertia, such as a frisbee.
The rotational inertia is crucial for solving the textbook problem because it directly affects the rotational kinetic energy and, therefore, the object's angular velocity. Students must appreciate the concept of rotational inertia to understand how it affects rotational motion and the energy associated with it.