Moment of inertia

Moment of inertia is a fundamental property in rotational mechanics that applies to any object that can be rotated. It is also sometimes known as mass moment of inertia or rotational inertia. Specifically, moment of inertia is a measure of an object's resistance to a change in its rotational state. Put more simply, moment of inertia is a determination of how difficult it is to change the rotational velocity of an object about its rotational axis. As a physics concept, moment of inertia is one of the core principles of dynamics, a branch of mechanical science concerned with the motion of bodies under the action of various forces. In rotational dynamics, which deals with the motion of bodies in rotation, moment of inertia plays essentially the same role that mass plays in linear dynamics, which deals with the motion of bodies in a straight line. This means that moment of inertia is an important concept in determining the relationship between such factors as angular momentum and angular velocity, and torque and angular acceleration.

Background

Moment of inertia is a concept that is broadly tied to a branch of physics called mechanics. Concerned primarily with the study of motion, this branch is further subdivided into two separate fields: statics and dynamics. Statics is the study of bodies in equilibrium, or a state of balance between opposing forces. Dynamics, meanwhile, is the study of bodies in motion. Within the discipline of dynamics, there are two additional subfields: kinematics and kinetics. While kinematics is concerned strictly with the geometry and time evolution of motion, kinetics is concerned with the relationship between motion and the various forces involved. It is with kinetics that moment of inertia is most closely associated.

The earliest significant studies in dynamics were first carried out by Italian scientist Galileo Galilei in the seventeenth century. Known mainly for his pioneering and often controversial contributions to astronomy and physics, Galileo conducted a series of experiments that ultimately led to the first major breakthroughs in the field of dynamics. In addition to discovering the laws of motion that govern projectiles and object in free fall, Galileo successfully outlined the law of inertia. Even more impressive was the fact that he did all this at a time when key branches of mathematics like differential and integral calculus were nonexistent and none of the modern instruments that allow for the exact measurement of time had yet been invented.

Later in the seventeenth century, English mathematician, astronomer, and physicist Isaac Newton further developed the scientific basis of dynamics when he created what are today known as Newton's laws of motion. These three laws represented a major scientific advance in that they provided a clear explanation of the experimental evidence previously gathered by scientists such as Galileo and showed that it was possible to make precise, accurate predictions about the behavior of objects in motion. Moreover, Newton's laws of motion served as the fundamental basis for the development of classical mechanics. As it pertains to moment of inertia, Newton's second law of motion, which states that the force acting on an object is equal to the mass of that object times its acceleration, is the most relevant of the three.

Overview

Moment of inertia is a measure of an object's resistance to changes in its rate of rotation. To fully understand what moment of inertia is and how it works, it is first necessary to know a number of key terms, including angular momentum, angular velocity, axis of rotation, torque, and angular acceleration. Angular momentum is the quantity of rotation of a body and is the product of its moment of inertia and its angular velocity. Angular velocity is the rate of change in angular position in a rotating body. Axis of rotation is the straight line through all fixed points of a rotating body around which all other points of the body circle. Torque is a measure of the force that can cause an object to rotate about an axis. Angular acceleration is the rate of change in angular velocity. All of these factors play into moment of inertia in different ways.

The second key to understanding moment of inertia is recognizing the nature of its relationship to Newton's second law of motion. As written, Newton's second law applies specifically to linear dynamics. When applied to rotational dynamics, the moment of inertia takes the place of mass in the original law. As a result, the angular equivalent to Newton's second law states that the rate of change of angular momentum of a system is equal to the torque (or moment of inertia) applied to that system. In practice, this simply means that the farther away a mass moves from the axis of rotation, the more difficult it becomes to change the system's rotational velocity. This is true because momentum and the momentum vector, which is a quantity that is described by magnitude and direction, are both dependent on a mass's distance from the axis of rotation. As a mass moves away from the axis of rotation, its speed and momentum increase and its momentum vector begins changing more rapidly. This, in turn, makes it more difficult to turn the mass.

An object's moment of inertia is also affected by its size and shape. Where size is concerned, the bigger and heavier an object is, the higher its moment of inertia will be. At the same time, awkwardly shaped objects are also likely to have a higher moment of inertia. In either case, a higher moment of inertia means that larger or more awkwardly shaped objects will be more resistant to changes in their rate of rotation. Put another way, it can be accurately said that the more mass an object has farther away from the axis around which it rotates, the more work it will take to turn the object. As an example, if one were to attach a pair of weights at the center of a stick and an identical pair of weights at the end of the same type of stick and attempted to rotate each stick from the center, it would be easier to rotate the first stick. This is because a greater portion of the mass of the second stick is located farther away from its axis of rotation, which means that its moment of inertia is higher.

Bibliography

Gross, Dietmar, and Werner Hauger. Engineering Mechanics 3: Dynamics. Springer, 2011.

Holzner, Steven. "How to Calculate the Momentum of Inertia for Different Shapes and Solids." Dummies.com, www.dummies.com/education/science/physics/how-to-calculate-the-momentum-of-inertia-for-different-shapes-and-solids/. Accessed 26 June 2017.

Holzner, Steven, and Daniel Funch Wohns. "Moments of Inertia in Physics." Dummies.com, www.dummies.com/education/science/physics/moments-of-inertia-in-physics/. Accessed 26 June 2017.

"Moment of Inertia." Isaac Physics, isaacphysics.org/concepts/cp‗moment‗inertia. Accessed 26 June 2017.

"Moment of Inertia." Shmoop, www.shmoop.com/rotation/inertia.html. Accessed 26 June 2017.

Rhett, Allain. "Let's Explore the Physics of Rotational Motion with a Fidget Spinner." Wired, 23 May 2017, www.wired.com/2017/05/physics-of-a-fidget-spinner/. Accessed 26 June 2017.

"Rotational Inertia." Khan Academy, www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/a/rotational-inertia. Accessed 26 June 2017.

"What Is Moment of Inertia?" The Constructor, theconstructor.org/structural-engg/moment-of-inertia-calculation-formula/2825/. Accessed 26 June 2017.