Work of Rotation

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:

Figure 54.

Remember that work is analogous to energy, and this is no different in rotational work. The kinetic energy of rotation equals half of the inertia multiplied by the angular velocity squared. This means that energy can be sorted in a flywheel and used to create kinetic energy that can move things. The rotational kinetic energy of a disc or other spinning object is related to the work done by the torque applied to the object.

Another interesting question to ask yourself in understanding rotational energies is why do things roll downhill at different rates, even if they have the same mass? At the beginning of the project, the potential energy due to gravity is the starting point of the object’s energy. As it rolls downhill, all of its potential energy is converted into kinetic energy. The kinetic energy is partly translational (related to its linear motion) and rotational (or related to its rotational motion). Remember that energy is conserved. The more energy put into rotation, the less energy goes into translation or linear motion. This means that the greater the rotational energy, the slower the item will roll.

A can with no liquid in it will slide down a hill and won’t roll much so it slides faster. A can with thick liquid will have more rotation and will have lesser speed of translation or a lesser speed. The can with no liquid has the same mass but has less inertia than a can