The other formulas provided are usually more useful and represent the most common situations that physicists run into. This formula is the most "brute force" approach to calculating the moment of inertia. A new axis of rotation ends up with a different formula, even if the physical shape of the object remains the same. The consequence of this formula is that the same object gets a different moment of inertia value, depending on how it is rotating. Figure 10-31b gives the rotational inertia I of the disk about the axis as a. You do this for all of the particles that make up the rotating object and then add those values together, and that gives the moment of inertia. 34 Figure 10-30 gives angular speed versus time for a thin rod that. The mr2 is sometimes referred to the moment of inertia or. I (1/2)M(R 1 2 + R 2 2) Note: If you took this formula and set R 1 R 2 R (or, more appropriately, took the mathematical limit as R 1 and R 2 approach a common radius R. T m × × r × r or another way of stating the torque equation becomes: T (mr2) ×.
Basically, for any rotating object, the moment of inertia can be calculated by taking the distance of each particle from the axis of rotation ( r in the equation), squaring that value (that's the r 2 term), and multiplying it times the mass of that particle. We were discussing the concept of Torsion or twisting moment, Power transmitted by a circular solid shaft and power transmitted by a circular hollow shaft. A hollow cylinder with rotating on an axis that goes through the center of the cylinder, with mass M, internal radius R 1, and external radius R 2, has a moment of inertia determined by the formula. The general formula represents the most basic conceptual understanding of the moment of inertia. Choose Cartesian coordinates, with the origin at the center of mass of the rod, which is midway between the endpoints since the rod is uniform. A sketch of the rod, volume element, and axis is shown in Figure 16.9.
The general formula for deriving the moment of inertia. the moment of inertia about an axis perpendicular to the rod that passes through the center of mass of the rod.