Question

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?

Short answer

Question: Calculate the moment of inertia and rotational kinetic energy of an oxygen molecule (\(O_2\)), given the following information: - Mass of one oxygen atom: \(2.66 \cdot 10^{-26}\, \mathrm{kg}\) - Distance between oxygen atoms: \(1.21 \cdot 10^{-10}\, \mathrm{m}\) - Angular speed of the molecule: \(4.60 \cdot 10^{12}\, \mathrm{rad} / \mathrm{s}\) Answer: The moment of inertia of the oxygen molecule is approximately \(4.89 \times 10^{-46}\, \mathrm{kg\cdot m^2}\), and its rotational kinetic energy is approximately \(5.15 \times 10^{-21}\, \mathrm{J}\).

Step-by-step solution

Calculate the moment of inertia

To calculate the moment of inertia of the molecule about the z-axis, we will consider each oxygen atom as a point mass. The moment of inertia for a point mass is given by: \(I = m r^2\) where \(I\) is the moment of inertia, \(m\) is the mass of the particle, and \(r\) is the distance from the axis of rotation. Since the z-axis passes through the center of the molecule and both atoms have the same mass, the distance of each atom from the axis of rotation is half the separation between the atoms, which is \(\frac{d}{2}\). So the moment of inertia of one oxygen atom is: \(I_{1} = m\left(\frac{d}{2}\right)^2\) The moment of inertia of the second oxygen atom is the same as that of the first atom: \(I_{2} = I_{1}\) The total moment of inertia of the molecule is the sum of the moments of inertia of the two atoms: \(I_{total} = I_{1} + I_{2}\) Substituting the values for mass and distance, we get: \(I_{total} = 2\cdot(2.66 \cdot 10^{-26} \mathrm{~kg})\left(\frac{1.21 \cdot 10^{-10} \mathrm{~m}}{2}\right)^2\)

Calculate the moment of inertia

Now we will calculate the moment of inertia using the formula derived in step 1: \(I_{total} = 2\cdot(2.66 \cdot 10^{-26} \mathrm{~kg})\left(\frac{1.21 \cdot 10^{-10} \mathrm{~m}}{2}\right)^2 = 4.89 \times 10^{-46}\, \mathrm{kg\cdot m^2}\)

Find the rotational kinetic energy

Now that we have found the moment of inertia, we can calculate the rotational kinetic energy of the molecule using the formula: \(K_{rot} = \frac{1}{2} I \omega^2\) where \(K_{rot}\) is the rotational kinetic energy, \(I\) is the moment of inertia, and \(\omega\) is the angular speed. Given that the angular speed is \(4.60 \cdot 10^{12} \mathrm{rad} / \mathrm{s}\), we plug in the values and calculate the rotational kinetic energy: \(K_{rot} = \frac{1}{2} (4.89 \times 10^{-46}\, \mathrm{kg\cdot m^2}) (4.60 \cdot 10^{12} \mathrm{rad} / \mathrm{s})^2 = 5.15 \times 10^{-21}\, \mathrm{J}\) So, the rotational kinetic energy of the molecule is approximately \(5.15 \times 10^{-21}\, \mathrm{J}\).

Key concepts

Moment of Inertia

Understanding the moment of inertia is crucial when studying rotational motion. It's analogous to mass in linear motion, but instead of relating to how much matter there is, it involves how that matter is distributed with respect to the axis of rotation.

For a point mass, the moment of inertia is calculated by the formula
\(I = m r^2\), where \(m\) stands for mass and \(r\) is the radius or distance from the rotation axis. Therefore, the further the mass is from the axis, the more it affects the rotation. This concept explains why ice skaters spin faster when they tuck in their arms – they are reducing their moment of inertia.

In the provided exercise, each oxygen atom is regarded as a point mass, with their combined moment of inertia being the sum of each individual atom's contribution. Since the atoms are equidistant from the axis, their moments of inertia are uniform. The total moment of inertia of the molecule is then a simple matter of adding the two together. This calculation is essential because it sets the stage for finding out other properties of rotational motion, such as kinetic energy.

Angular Speed

Angular speed is a measure of how fast an object rotates or revolves relative to another point – usually the center point of a circle. It is the angle an object rotates through in a certain amount of time and is often measured in radians per second (rad/s).

Just as linear speed tells us how quickly something travels along a straight path, angular speed indicates the rapidity of rotation. A higher angular speed means an object is spinning faster. It's expressed as \(\frac{radians}{time}\), where 'radians' is the standard unit of angular measurements and 'time' typically in seconds.

In our exercise, the angular speed given is exceptionally high, showing that the oxygen molecule spins at an extremely rapid rate. This speed is integral to determining the rotational kinetic energy, which involves not just how much mass is rotating but how quickly it's spinning. Increasing the angular speed will proportionally increase the rotational kinetic energy, assuming that the distribution of mass stays the same.

Rotational Dynamics

Rotational dynamics revolve around the forces and moments that cause an object to rotate. The principles governing rotational motion are similar to Newton's laws of motion for linear dynamics but adapted for rotating bodies. It involves torque, angular acceleration, and the already mentioned moment of inertia.

If we apply a force to an object at a point away from its center of mass, we cause it to rotate around an axis. This turning effect is torque. In rotational dynamics, we study the relationship between applied torque and the resulting angular acceleration, which depends on the object's moment of inertia.

The exercise brought into focus the final kinetic aspect of rotational dynamics: the rotational kinetic energy. It's the energy an object possesses because of its rotation, and it's given by \(K_{rot} = \frac{1}{2} I \omega^2\), where \(I\) is the moment of inertia and \(\omega\) the angular speed. The formula shows that for a given moment of inertia, more energy is stored in faster spins. Thus, a spinning top, a twirling ice skater, and even the oxygen molecule from our textbook problem have this type of energy until an outside force acts on them to slow them down.