Before we learn to calculate the moment of inertia, let us first understand what the word means and its basic definition. Moment of inertia or rotational momentum which is more commonly called angular momentum or gravitational force is an amount that exists at any given time that acts to balance the amount of force acting on a system. This concept can be illustrated using the example of the earth rotating on its axis. The moment of inertia represents the power or strength with which the earth spins on its axis due to the gravity.
Moment of Inertia can be calculated for nearly any dynamic system by taking the difference between the torque applied to the system and the force that it would counter if it were not rotating. The result is the momentum change, which is the product of the force and the change in velocity. It is usually denoted by some symbols like MP0(inired), MP1 (determined), RV (reacted) or VS (vibration). When an observer measures the change in velocity of a system, they are measuring the Moment of Inertia by expressing it in terms of the change in velocity on one of their reference axes.
For systems with no reference axis, like a spring, an observer measures the change in momentum due to the gravitational force acting on the mass. For a system with a single reference axis, such as a rod, an observer measures the change in momentum due to the magnetism of that particular rod. The results are the values on the left and right panels of the plots on the left and right of the figure. The lower left panel shows the first moment and the lower right panel shows the second moment. There is another data plot that shows the time evolution of the system. This last data plot shows the torque, which changes as the system is subject to the changing gravitational field.
A second type of reference axis used to determine the Moment of Inertia is the System of Moments (SOM). A SOM includes all components of the kinetic energy of a system. The components can be divided into two groups: first is the force component and the second is the potential energy. The force component, which include the work done by the part attached to the component that is moving, and the potential energy, which include any change in the gravitational field, are derived from the equation
A third way of calculating the Moment of Inertia is to use a tensometric function. It is necessary to define a tensometric function first. Then all the component of momentum changes of the object can be evaluated by the tensometric function, where t is the derivative of the object’s velocity with respect to the x axis, and n is the angular momentum of the body.
It is easy to calculate the moments of inertia by using the law of integrals. For an object to be considered an integral member of an integrals series, it must have the same value for all its components at all times. For the purpose of computing the Moment of Inertia of an object, the distance of the center of mass from some reference rod is the distance from the center of mass of the rotating rod. It is also necessary to adjust the distance of the center of mass to the rotational axis for the proper computation of the moment.