Each bit of length (dx) has mass (dm) and r=|x|. To solve the problem, we usually put m in terms of another variable which we can iterate over more easily.įor example, consider the moment of inertia of a rod of length L around its center with total mass of L. We still sum up the function value (r^2), but this time we multiply it by the strip mass (dm). When you see an expression like dm, we are iterating over masses instead. Note that the area of each strip is approximated as the function value (x^2) times the strip length (dx). Integral calculus deals with finding the sum of infinitesimal parts of the. When you see an expression like $I = \int x^2 dx$ we effectively iterating over a range on the x axis and adding up the area of an infinite amount of infinitesimally small strips. Along with differentiation, integration is the fundamental object of calculus. Seeing an expression like $I = \int r^2 dm$ is certainly confusing the first time you see it. So I would solve this, using my experience with calculus (which encompasses a read through the Sparks Notes packet) as $ I = m r^2 $.īut obviously, this is wrong? $r$ is not a constant! How do I deal with it? Do I need to replace $r$ with an expression that varies with $m$? But how could $r$ possibly vary with $m$? Isn't it more likely the other way around? But how can $m$ vary with $r$? It's all rather confusing me.Ĭould someone help me figure out what to do with all these substitutions for, example, figuring out the Moment of Inertia of a hoop with no thickness and width $w$, with the axis of rotation running through its center orthogonal to its plane? This really doesn't make any sense to me.you have two independent variables? I am only used to having one independent variable and one constant. The equivalent rotational equation is $\tau = I \alpha$, where $\tau$ is rotational force, $\alpha$ is rotational acceleration, and $I$ is rotational inertia.įor a point about an axis, $I$ is $m r^2$, where $r$ is the distance from the point to the axis of rotation.įor a continuous body, this is an integral - $I = \int r^2 \,dm$. It's basically the equivalent of mass in Netwon's $F = m a$ in linear motion. For example, when replacing car tires, it's often necessary to balance the wheels by attaching small lead weights to ensure the coincidence of the rotation axis with the principal axis of inertia and to eliminate vibration.I just came back from my Introduction to Rotational Kinematics class, and one of the important concepts they described was Rotational Inertia, or Moment of Inertia. In this section we develop computational techniques for finding the center of mass and moments of inertia of several types of physical objects, using double integrals for a lamina (flat plate) and triple integrals for a three-dimensional object with variable density. Therefore, when designing such devices it is necessary the axis of rotation to be coinciding with one of the principal axes of inertia. If a body rotates about an axis which does not coincide with a principal axis of inertia, it will experience vibrations at the high rotation speeds. This tensor is symmetric and, hence, it can be transformed to a diagonal view by choosing the appropriate coordinate axes \(Ox', Oy', Oz'.\) The values of the diagonal elements (after transforming the tensor to a diagonal form) are called the main moments of inertia, and the indicated directions of the axes are called the eigenvalues or the principal axes of inertia of the body.
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