6.5: Rotational Inertia
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- Nov 1, 2024
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Rotational Kinetic Energy
Recall that kinetic energy is described by the mass of the object and its speed. For rotating bodies there must be an analog to kinetic energy as well. We already have a relationship between linear and angular speed, which we can use to redefine kinetic energy for rotational motion. Let us simplify our wheel example by observing a few point masses rotating on a massless rod as shown in Figure 7.4.1 below. The pivot shown in the figure defines a fixed point about which the object rotates. In some cases it is an obvious choice, such as in this example since we are assuming rod is help in place at a pilot. In other examples a rotating object might not be fixed at a particular point, but will still rotate about the center-of-mass of the object, which we will discuss later.
Figure 7.4.1: Rotational Kinetic Energy
In this scenario the total kinetic energy of the rotating object is the sum of the kinetic energies of the four masses shown. For mass 1,
The expression is similar for all the other masses, using the appropriate distances from the pivot and recognizing that all masses will rotate with the same angular velocity,
In general, for a set of N masses, the kinetic energy of a rotating object about a fixed pivot becomes:
Rotational Inertia
Let us now use the result in Equation
where I, is the rotational inertia of a object consisting of point masses:
The SI units of rotational inertia are
In general, most objects are not made-up of point masses but are continuous mass. Equation
We will not go into deriving rotational inertia for different objects, but the table below gives the rotational inertia of several simple geometric shapes, as calculated in the limit of infinitesimal increments of mass using this equation. All of there geometric shapes have uniform mass.
Object | Rotational Inertia | Illustration |
Point mass |
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Thin ring of mass |
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Thin rod of mass |
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Thin rod of mass |
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Disk of mass |
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Sphere of mass |
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Thin hollow spherical shell of mass |
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As seen from the formulas in the table, objects with the same mass can have very different rotational inertias, depending on how the mass is distributed with respect to the axis of rotation. For example, when the mass of a sphere is concentrated at its radius (as for the spherical shell), its rotational inertia is greater than for the sphere of the same radius and mass but with the mass uniformly distributed from the center (solid sphere). This is consistent with Equation
The rotational inertia of a composite object is the sum of the rotational inertias of each component, all calculated about the same axis.
So for a ring and a disk stacked upon each other and rotating about the symmetry axis of both, the rotational inertia is: