An oxygen molecule (O2) rotates in the x y -plane about the z -axis. The axis of rotation passes

Name: Problem 71, Chapter 10: University Physics with Modern Physics
Uploaded: 2021-07-29T15:31:02-08:00
Duration: 3 min 23 s
Channel: Keshav Singh

step 1 For this problem on the topic of rotation, we are told that an oxygen molecule is rotating in th

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?

   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?

University Physics with Modern Physics

Wolfgang Bauer, Gary… 2nd Edition

Chapter 10, Problem 71

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Solved by Expert Keshav Singh

07/29/2021

Step 1

In this case, the molecule consists of two oxygen atoms, each of mass $m = 2.66 \times 10^{-26} \mathrm{kg}$, and the distance between them is $d = 1.21 \times 10^{-10} \mathrm{m}$. Since the axis of rotation passes through the center of the molecule, the distance Show more…

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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?

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Key Concepts

Rotational Kinetic Energy

Rotational kinetic energy is the energy possessed by a rotating body due to its angular motion. It is determined by the moment of inertia and the square of the angular speed, typically expressed as one-half times the moment of inertia times the square of the angular velocity. This concept is key to understanding energy distribution and transfer in rotational dynamics.

Moment of Inertia

The moment of inertia is a measure of how mass is distributed about an axis of rotation. It quantifies the resistance of a body to changes in its rotational motion and is calculated as the sum of each mass element multiplied by the square of its distance from the axis. This concept is fundamental in analyzing the dynamics of rotating systems in classical mechanics.

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