Watch
Watching this resources will notify you when proposed changes or new versions are created so you can keep track of improvements that have been made.
Favorite
Favoriting this resource allows you to save it in the “My Resources” tab of your account. There, you can easily access this resource later when you’re ready to customize it or assign it to your students.
Relationship Between Linear and Rotational Quantitues
The description of motion could be sometimes easier with angular quantities such as angular velocity, rotational inertia, torque, etc.
Learning Objective

Derive uniform circular motion from linear equations
Key Points
 As we use mass, linear momentum, translational kinetic energy, and Newton's 2nd law to describe linear motion, we can describe a general rotational motion using corresponding scalar/vector/tensor quantities.
 Angular and linear velocity have the following relationship:
$\bf{v = \omega \times r}$ .  As we use the equation of motion
$F = ma$ to describe a linear motion, we can use its counterpart$\bf{\tau} = \frac{d\bf{L}}{dt} = \bf{r} \times \bf{F}$ , to describe angular motion. The descriptions are equivalent, and the choice can be made purely for the convenience of use.
Terms

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

uniform circular motion
Movement around a circular path with constant speed.

torque
A rotational or twisting effect of a force; (SI unit newtonmeter or Nm; imperial unit footpound or ftlb)
Full Text
Defining Circular Motion
The description of circular motion is described better in terms of angular quantity than its linear counter part. The reasons are easy to understand. For example, consider the case of uniform circular motion. Here, the velocity of particle is changing  though the motion is "uniform". The two concepts do not go together. The general connotation of the term "uniform" indicates "constant", but the velocity is actually changing all the time.
A Rotating Body
Each particle constituting the body executes a uniform circular motion about the fixed axis. For the description of the motion, angular quantities are the better choice.
When we describe the uniform circular motion in terms of angular velocity, there is no contradiction. The velocity (i.e. angular velocity) is indeed constant. This is the first advantage of describing uniform circular motion in terms of angular velocity.
Second advantage is that angular velocity conveys the physical sense of the rotation of the particle as against linear velocity, which indicates translational motion. Alternatively, angular description emphasizes the distinction between two types of motion (translational and rotational).
Relationship Between Linear and Angular Speed
For simplicity, let's consider a uniform circular motion. For the length of the arc subtending angle " at the origin and "r" is the radius of the circle containing the position of the particle, we have
Differentiating with respect to time, we have
Because
Rotational Kinematic Equations
With the relationship of the linear and angular speed/acceleration, we can derive the following four rotational kinematic equations for constant
Mass, Momentum, Energy, and Newton's Second Law
As we use mass, linear momentum, translational kinetic energy, and Newton's 2nd law to describe linear motion, we can describe a general rotational motion using corresponding scalar/vector/tensor quantities:
 Mass/Rotational inertia:
 Linenar/angular momentum:
 Force/Torque:
 Kinetic energy:
For example, just as we use the equation of motion
Assign just this concept or entire chapters to your class for free.
Key Term Reference
 Law
 Appears in these related concepts: Physics and Other Fields, Photon Interactions and Pair Production, and Models, Theories, and Laws
 Length
 Appears in these related concepts: Length, Atomic Structure, and Introduction to The Sampling Theorem
 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, FreeFalling Objects, and The Second Law: Force and Acceleration
 angular
 Appears in these related concepts: Wavelength, Freqency in Relation to Speed, Bohr Orbits, and B.1 Chapter 1
 angular acceleration
 Appears in these related concepts: Relationship Between Torque and Angular Acceleration, Angular Acceleration, Alpha, and Conservation of Angular Momentum
 angular motion
 Appears in this related concept: Torque
 angular velocity
 Appears in these related concepts: Angular Quantities as Vectors, Angular vs. Linear Quantities, and Kepler's Second Law
 circular motion
 Appears in these related concepts: Water Waves, Sinusoidal Nature of Simple Harmonic Motion, and B.3 Chapter 3
 energy
 Appears in these related concepts: Energy Transportation, Surface Tension, and Introduction to Work and Energy
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 inertia
 Appears in these related concepts: Rotational Kinetic Energy: Work, Energy, and Power, Mass, and The First Law: Inertia
 kinematic
 Appears in these related concepts: Motion with Constant Acceleration, Temperature, and Constant Acceleration
 kinematics
 Appears in these related concepts: Applications, Constant Angular Acceleration, and Defining Kinematics
 kinetic
 Appears in these related concepts: Friction: Static, The Kinetic Molecular Theory of Matter, and Postmodernist Sculpture
 kinetic energy
 Appears in these related concepts: Escape Speed, The Ideal Gas Equation, and Friction: Kinetic
 linear velocity
 Appears in this related concept: Rotational Angle and Angular Velocity
 mass
 Appears in these related concepts: Mass Spectrometer, Mass, and Pop Art
 momentum
 Appears in these related concepts: The Uncertainty Principle, Differentiation and Rates of Change in the Natural and Social Sciences, and Elastic Collisions in One Dimension
 motion
 Appears in these related concepts: Motion Diagrams, TwoComponent Forces, and Moving Source
 origin
 Appears in these related concepts: Adding and Subtracting Vectors Graphically, Types of Muscle Tissue, and ThreeDimensional Coordinate Systems
 position
 Appears in these related concepts: Damped Harmonic Motion, Longitudinal Waves, and Graphical Interpretation
 rotation
 Appears in these related concepts: Rotational Collisions, Lever Systems, and Center of Mass and Translational Motion
 velocity
 Appears in these related concepts: Rolling Without Slipping, Arc Length and Speed, and Centripetial Acceleration
Sources
Boundless vets and curates highquality, openly licensed content from around the Internet. This particular resource used the following sources:
Cite This Source
Source: Boundless. “Relationship Between Linear and Rotational Quantitues.” Boundless Physics. Boundless, 13 Apr. 2016. Retrieved 01 May. 2016 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/rotationalkinematicsangularmomentumandenergy9/linearandrotationalquantities89/relationshipbetweenlinearandrotationalquantitues3337735/