Rotational Inertia
Rotational inertia is the tendency of a rotating object to remain rotating unless a torque is applied to it.
Learning Objective

Explain the relationship between the force, mass, radius, and angular acceleration
Key Points
 The farther the force is applied from the pivot, the greater the angular acceleration.
 Angular acceleration is inversely proportional to mass.
 The equation τ = m(r^2)α is the rotational analog of Newton's second law (F=ma), where torque is analogous to force, angular acceleration is analogous to translational acceleration, and mr2 is analogous to mass (or inertia).
Terms

rotational inertia
The tendency of a rotating object to remain rotating unless a torque is applied to it.

torque
A rotational or twisting effect of a force; (SI unit newtonmeter or Nm; imperial unit footpound or ftlb)
Full Text
If you have ever spun a bike wheel or pushed a merrygoround, you have experienced the force needed to change angular velocity. Our intuition is reliable in predicting many of the factors that are involved. For example, we know that a door opens slowly if we push too close to its hinges. Furthermore, we know that the more massive the door, the more slowly it opens. The first example implies that the farther the force is applied from the pivot, the greater the angular acceleration; another implication is that angular acceleration is inversely proportional to mass. These relationships should seem very similar to the familiar relationships among force, mass, and acceleration embodied in Newton's second law of motion. There are, in fact, precise rotational analogs to both force and mass.
Rotational inertia, as illustrated in , is the resistance of objects to changes in their rotation. In other words, a rotating object will stay rotating and a nonrotating object will stay nonrotating unless acted on by a torque. This should remind you of Newton's First Law.
Rotational Inertia
Force is required to spin the bike wheel. The greater the force, the greater the angular acceleration produced. The more massive the wheel, the smaller the angular acceleration. If you push on a spoke closer to the axle, the angular acceleration will be smaller.
To develop the precise relationship among force, mass, radius, and angular acceleration, consider what happens if we exert a force F on a point mass m that is at a distance r from a pivot point. Because the force is perpendicular to r, an acceleration
Recall that torque is the turning effectiveness of a force. In this case, because F is perpendicular to r, torque is simply τ=Fr. So, if we multiply both sides of the equation above by r, we get torque on the lefthand side. That is, rF = mr^{2}α, or
τ = mr^{2}α.
This equation is the rotational analog of Newton's second law (F=ma), where torque is analogous to force, angular acceleration is analogous to translational acceleration, and mr^{2} is analogous to mass (or inertia). The quantity mr^{2} is called the rotational inertia or moment of inertia of a point mass m a distance r from the center of rotation.
Different shapes of objects have different rotational inertia which depend on the distribution of their mass.
Key Term Reference
 Law
 Appears in these related concepts: Physics and Other Fields, Damped Harmonic Motion, and Models, Theories, and Laws
 Newton's Second Law
 Appears in these related concepts: Momentum, Force, and Newton's Second Law, Centripetal Force, and Matter Exists in Space and Time
 acceleration
 Appears in these related concepts: Position, Displacement, Velocity, and Acceleration as Vectors, Scientific Applications of Quadratic Functions, and Centripetial Acceleration
 angular
 Appears in these related concepts: Wavelength, Freqency in Relation to Speed, Bohr Orbits, and Constant Angular Acceleration
 angular acceleration
 Appears in these related concepts: Relationship Between Torque and Angular Acceleration, Angular Acceleration, Alpha, and Conservation of Angular Momentum
 angular velocity
 Appears in these related concepts: Angular Quantities as Vectors, Angular vs. Linear Quantities, and Angular Velocity, Omega
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 force
 Appears in these related concepts: Force of Muscle Contraction, Force, and First Condition
 inertia
 Appears in these related concepts: Rotational Kinetic Energy: Work, Energy, and Power, Mass, and The First Law: Inertia
 mass
 Appears in these related concepts: Mass Spectrometer, Pop Art, and Mass
 moment of inertia
 Appears in these related concepts: Applications of Multiple Integrals, Internal Energy of an Ideal Gas, and ProblemSolving Techniques
 motion
 Appears in these related concepts: Motion Diagrams, TwoComponent Forces, and Moving Source
 perpendicular
 Appears in these related concepts: The Cross Product, Tangent Vectors and Normal Vectors, and Circular Motion
 point mass
 Appears in these related concepts: Moment of Inertia, Elastic Collisions in Multiple Dimensions, and Weight of the Earth
 resistance
 Appears in these related concepts: Resistors in Parallel, Resisitors in Series, and Introduction to Blood Flow, Pressure, and Resistance
 rotation
 Appears in these related concepts: Rotational Collisions, Lever Systems, and Transformations of Functions
 velocity
 Appears in these related concepts: Velocity of Blood Flow, RootMeanSquare Speed, and Rolling Without Slipping
Sources
Boundless vets and curates highquality, openly licensed content from around the Internet. This particular resource used the following sources: