The term mass times radius squared is called the moment of inertia or rotational inertia. If there is more than one particle involved, the moment of inertia is equal to the sum of the inertias of all the particles. Inertia is similar to mass in translational motion. The units for the moment of inertia are kilogram-meters squared. There are complex formulations for inertia that depend on whether or not the object is a cylinder or a hoop. The inertia for a hoop is the total mass times the radius of the hoop squared. This means is that the net torque is equal to the moment of inertia times the angular acceleration. This means that the ability to accelerate a mass is related to its mass and the square of the distance from the center of gravity. In a solid disc, the inertia is half the mass times the radius squared. In looking at torque and angular acceleration as it applies to inertia, you need to make the following relationship, which is “the angular acceleration is equal to the total torque divided by the moment of inertia”. This would mean that the heavier the object and the square of its distance from the center are inversely proportional to the degree of the acceleration of the object when a force is applied. In this case, the force applied would be the torque. In such cases, linear force is related to torque and inertia is related to mass but you need to know that these things are not the same. Torque depends on three factors: the direction of the force, the magnitude of the force, and the point of application. Inertia is related to the mass and the square of the radius. In both cases, the radius plays a big role in the determination of the angular acceleration.
WORK OF ROTATION Work must be done in order to rotate different objects, such as a merry-go-round and grindstones. The net work would be the net force times the arc length traveled of the disc. Substituting the various parameters, you get the net work equaling the net torque multiplied by the angle in radians. Figure 54 shows the work of rotating a disc:
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