![]() ![]() Point mass M at a distance r from the axis of rotation.Ī point mass does not have a moment of inertia around its own axis, but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved. In general, the moment of inertia is a tensor, see below. This article mainly considers symmetric mass distributions, with constant density throughout the object, and the axis of rotation is taken to be through the center of mass unless otherwise specified.įollowing are scalar moments of inertia. When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems. In general, it may not be straightforward to symbolically express the moment of inertia of shapes with more complicated mass distributions and lacking symmetry. Typically this occurs when the mass density is constant, but in some cases the density can vary throughout the object as well. The mass moment of inertia is often also known as the rotational inertia, and sometimes as the angular mass.įor simple objects with geometric symmetry, one can often determine the moment of inertia in an exact closed-form expression. It should not be confused with the second moment of area, which is used in beam calculations. The moments of inertia of a mass have units of dimension ML 2( × 2). Lots of examples.Moment of inertia, denoted by I, measures the extent to which an object resists rotational acceleration about a particular axis, it is the rotational analogue to mass (which determines an object's resistance to linear acceleration). The best way to learn how to do this is by example. What can I say about the perpendicular axis theorem other than it's interesting. What if an object isn't being rotated about the axis used to calculate the moment of inertia? Apply the parallel axis theorem. Where α is a simple rational number like 1 for a hoop, ½ for a cylinder, or ⅖ for a sphere. When you are done with all of this, you oftentimes end up with a nice little formula that looks something like this… These methods can be used to find the moment of inertia of things like spheres, hollow spheres, thin spherical shells and other more exotic shapes like cones, buckets, and eggs - basically, anything that might roll and that has a fairly simple mathematical description. Or this for stacked disks and washers I = Something like for nested, cylindrical shells… I = When shapes get more complicated, but are still somewhat simple geometrically, break them up into pieces that resemble shapes that have already been worked on and add up these known moments of inertia to get the total.įor slightly more complicated round shapes, you may have to revert to an integral that I'm not sure how to write. This method can be applied to disks, pipes, tubes, cylinders, pencils, paper rolls and maybe even tree branches, vases, and actual leeks (if they have a simple mathematical description). The volume of each infinitesimal layer is then…įor many cylindrical objects, you basically start with something like this… I = Imagine a leek.Įach layer of the leek has a circumference 2π r, thickness dr, and height h. The other easy volume element to work with is the infinitesimal tube. Note that although the strict mathematical description requires a triple integral, for many simple shapes the actual number of integrals worked out through brute force analysis may be less. ![]() This is the way to find the moment of inertia for cubes, boxes, plates, tiles, rods and other rectangular stuff. When an object is essentially rectangular, you get a set up something like this… I = The volume of each infinitesimal piece is… ![]() The pieces are dx wide, dy high, and dz deep. The infinitesimal box is probably the easiest conceptually. In practice, this may take one of two forms (but it is not limited to these two forms). The infinitesimal quantity dV is a teeny tiny piece of the whole body. In practice, for objects with uniform density ( ρ = m/ V) you do something like this… I =įor objects with nonuniform density, replace density with a density function, ρ( r). You add up (integrate) all the moments of inertia contributed by the teeny, tiny masses ( dm) located at whatever distance ( r) from the axis they happen to lie. It works like mass in this respect as long as you're adding moments that are measured about the same axis.įor an extended body, replace the summation with an integral and the mass with an infinitesimal mass. Say it, kilogram meter squared and don't say it some other way by accident.įor a collection of objects, just add the moments. It's a scalar quantity (like its translational cousin, mass), but has unusual looking units. Logic behind the moment of inertia: Why do we need this? ![]()
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