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2: Moments of Inertia

Last updated

Aug 8, 2022
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- 1.S: Centers of Mass (Summary)
- 2.1: Definition of Moment 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
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
2.6: Three-dimensional Solid Figures. Spheres, Cylinders, Cones.
2.7: Three-dimensional Hollow Figures. Spheres, Cylinders, Cones
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
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)
2.12: Rotation of Axes
2.13: Momental Ellipse
2.14: Eigenvectors and Eigenvalues
2.15: Solid Body
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
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).

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