The general form of the moment of inertia involves an integral. The moment of inertia of any extended object is built up from that basic definition. So, for instance, the center of mass of a uniform rod that extends along the x axis from \(x=0\) to \(x=L\) is at (L/2, 0). The moment of inertia of a point mass with respect to an axis is defined as the product of the mass times the distance from the axis squared. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The center of mass of a uniform rod is at the center of the rod. A uniform thin rod is one for which the linear mass density \(\mu\), the mass-per-length of the rod, has one and the same value at all points on the rod. The simplest case involves a uniform thin rod. In the simplest case, the calculation of the position of the center of mass is trivial. of individual point masses at each different radius for this the formula. The ideal thin rod, however, is a good approximation to the physical thin rod as long as the diameter of the rod is small compared to its length.) moment of inertia is calculated as the sum of each particles mass times the. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis).
A physical thin rod must have some nonzero diameter. It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation.
The easiest rigid body for which to calculate the center of mass is the thin rod because it extends in only one dimension. Quite often, when the finding of the position of the center of mass of a distribution of particles is called for, the distribution of particles is the set of particles making up a rigid body. The center of mass is found to be midway between the two particles, right where common sense tells us it has to be.
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