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2: Moments of Inertia

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    In this chapter we shall consider how to calculate the (second) moment of inertia for different sizes and shapes of body, as well as certain associated theorems. But the question should be asked: "What is the purpose of calculating the squares of the distances of lots of particles from an axis, multiplying these squares by the mass of each, and adding them all together?

    • 2.1: Definition of Moment of Inertia
      This page covers the second moment of mass, or moment of inertia, defined as \( \sum m_{i} r_{i}^{2} \). It emphasizes its importance in mechanics, illustrating the connection between mass and inertia, particularly in the context of force and acceleration. The chapter aims to explore methods to calculate moment of inertia for different shapes and discusses relevant theorems, encouraging readers to reflect on its practical applications beyond simple calculations.
    • 2.2: Meaning of Rotational Inertia
      If a force acts of a body, the body will accelerate. The ratio of the applied force to the resulting acceleration is the inertia (or mass) of the body.
    • 2.3: Moments of Inertia of Some Simple Shapes
      "For how many different shapes of body must I commit to memory the formulas for their moments of inertia?" I would be tempted to say: "None". However, if any are to be committed to memory, I would suggest that the list to be memorized should be limited to those few bodies that are likely to be encountered very often (particularly if they can be used to determine quickly the moments of inertia of other bodies) and for which it is easier to remember the formulas than to derive them.
    • 2.4: Radius of Gyration
      The second moment of inertia of any body can be written in the form mk², where k is the radius of gyration. If all the mass of a body were concentrated at its radius of gyration, its moment of inertia would remain the same.
    • 2.5: Plane Laminas and Mass Points distributed in a Plane
      This page covers the moment of inertia for point masses relative to a movable axis, highlighting that it is minimized when the axis is at the center of mass. It introduces the Parallel Axes Theorem, which connects moments of inertia about parallel axes, and the Perpendicular Axes Theorem for two-dimensional distributions. Additionally, it includes examples demonstrating the application of these theorems to calculate moments of inertia for different shapes.
    • 2.6: Three-dimensional Solid Figures. Spheres, Cylinders, Cones.
      This page covers the calculation of moments of inertia for various shapes, including spheres, cylinders, triangular prisms, and cones. It derives the second moment of inertia for solid spheres and hemispheres as \(\frac{2}{5}ma^2\), explores hollow spheres with a polynomial equation for a parameter \(x\), and demonstrates how geometric dimensions and mass distribution affect rotational inertia for solid cylinders, triangular prisms, and cones.
    • 2.7: Three-dimensional Hollow Figures. Spheres, Cylinders, Cones
      This page covers the calculations of moments of inertia for hollow and solid geometric shapes, including spheres, cylinders, and cones. It derives the moment of inertia for a hollow spherical shell and relates it to solid spheres. The page also discusses integration methods for nonuniform solid spheres and offers formulas for hollow cylinders and cones, highlighting the importance of mass distribution in these calculations.
    • 2.8: Torus
      The rotational inertias of solid and hollow toruses (large radius a , small radius b ) are given below for reference and without derivation. They can be derived by integral calculus, and their derivation is recommended as a challenge to the reader.
    • 2.9: Linear Triatomic Molecule
      This page explains how to calculate a molecule's rotational inertia, requiring interatomic distance measurements. It details the rotational inertia equation and illustrates it with the OCS molecule. The text recommends using an isotopic variant for precise calculations of interatomic spacings and discusses two methods for solving the equations, emphasizing the importance of numerical precision in calculations.
    • 2.10: Pendulums
      We are familiar with the equation of motion for a mass vibrating at the end of a spring of force constant - this is simple harmonic motion. The mechanics of the torsion pendulum is similar.
    • 2.11: Plane Laminas. Product Moment. Translation of Axes (Parallel Axes Theorem)
      This page covers the moments of inertia for point masses in a plane, focusing on second moments \( A \), \( B \), and \( C \), along with their relation \( C = A + B \). It introduces the product moment of inertia \( H \) and its calculation for a right triangular lamina, while inviting readers to explore its physical significance and derive results for other laminas through exercises.
    • 2.12: Rotation of Axes
      This page explores the relationship between two inclined coordinate systems regarding the moments of inertia of a plane lamina. It introduces rotation matrices for coordinate transformation and calculates moments of inertia for both systems. Using trigonometric identities, the page expresses the new system's moments in terms of the original ones.
    • 2.13: Momental Ellipse
      This page explains the radius of gyration and its connection to the moment of inertia in a plane lamina, using a vector \( \bf P \) to illustrate mass distribution. It demonstrates that the moment of inertia varies by axis while the endpoint of \( \bf P \) creates a momental ellipse aligned with the lamina's principal axes. It notes that a regular polygon maintains a constant moment of inertia across axes through its centroid, resulting in circular momental ellipses in this special case.
    • 2.14: Eigenvectors and Eigenvalues
      This page introduces eigenvectors and eigenvalues, emphasizing their application in moments of inertia for solid bodies. It covers the mathematical operations to derive these properties through matrix-vector multiplication and the characteristic equation method, detailing the significance of this approach for both 2x2 and 3x3 matrices, particularly for numerical solutions of the cubic characteristic equation.
    • 2.15: Solid Body
      This page explores the calculations of moments and products of inertia for point masses in three-dimensional space, focusing on axes at the center of mass. It defines six moments (A, B, C) and three products of inertia (F, G, H) and introduces the parallel axes theorem for determining these measures relative to a new point, P. The content assumes the center of mass as the origin for simplification in calculations.
    • 2.16: Rotation of Axes - Three Dimensions
      If it is possible to find a set of axes with respect to which the product moments F, G and H are all zero, these axes are called the principal axes of the body, and the moments of inertia with respect to these axes are the principal moments of inertia.
    • 2.17: Solid Body Rotation and the Inertia Tensor
      It is intended that this chapter should be limited to the calculation of the moments of inertia of bodies of various shapes, and not with the huge subject of the rotational dynamics of solid bodies, which requires a chapter on its own. In this section I mention merely for interest two small topics involving the principal axes.
    • 2.18: Determination of the Principal Axes
      The Principals Axes are the three mutually perpendicular axes in a body about which the moment of inertia is maximized.
    • 2.19: Moment of Inertia with Respect to a Point
      By “moment of inertia” we have hitherto meant the second moment of mass with respect to an axis. We were easily able to identify it with the rotational inertia with respect to the axis, namely the ratio of an applied torque to the resulting angular acceleration.
    • 2.20: Ellipses and Ellipsoids
      This page covers principal moments of inertia for various shapes, including elliptical laminae, rings, and triaxial ellipsoids, with specific methods for calculations and distinctions between thicknesses in triaxial shells. It also introduces functions for surface area and moment of inertia based on the parameter \( \chi \), detailing changes from discs to spheres and hollow spheres.
    • 2.21: Tetrahedra
      The solid regular tetrahedron and the methane molecule are both spherical tops, and the moment of inertia is the same about any axis through the center of mass.

    Thumbnail: Moment of Inertia of a thin rod about an axis perpendicular to the length of the rod and passing through its center. (Public Domain; Krishnavedala).

    This page titled 2: Moments of Inertia is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.