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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

Determine 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

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

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
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.
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
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Key Term Reference
 Law
 Appears in this related concepts: Physics and Other Fields, Photon Interactions and Pair Production, and Models, Theories, and Laws
 Newton's Second Law
 Appears in this related concepts: Momentum, Force, and Newton's Second Law, Centripetal Force, and Matter Exists in Space and Time
 acceleration
 Appears in this related concepts: Motion with Constant Acceleration, Centripetial Acceleration, and Applications and ProblemSolving
 angular
 Appears in this related concepts: Wavelength, Freqency in Relation to Speed, Driven Oscillations and Resonance, and Bohr Orbits
 angular acceleration
 Appears in this 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 this related concepts: Angular vs. Linear Quantities, Angular Velocity, Omega, and Kepler's Second Law
 circular motion
 Appears in this related concepts: Water Waves, Sinusoidal Nature of Simple Harmonic Motion, and XRay Diffraction
 energy
 Appears in this related concepts: Surface Tension, Introduction to Work and Energy, and The Role of Energy and Metabolism
 equation
 Appears in this related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 inertia
 Appears in this related concepts: Moment of Inertia, The Impact of Culture on an Organization, and The First Law: Inertia
 kinematic
 Appears in this related concepts: Applications, Temperature, and Constant Acceleration
 kinematics
 Appears in this related concepts: Constant Angular Acceleration, Defining Kinematics, and ProblemSolving Techniques
 kinetic
 Appears in this related concepts: Friction: Static, The Kinetic Molecular Theory of Matter, and Sculpture
 kinetic energy
 Appears in this related concepts: Glancing Collisions, Solid Solubility and Temperature, and Escape Speed
 mass
 Appears in this related concepts: Physical and Chemical Properties of Matter, Mass Spectrometer, and Mass
 momentum
 Appears in this related concepts: The Uncertainty Principle, Rotational Collisions, and Differentiation and Rates of Change in the Natural and Social Sciences
 motion
 Appears in this related concepts: Motion Diagrams, TwoComponent Forces, and Moving Source
 origin
 Appears in this related concepts: Adding and Subtracting Vectors Graphically, Overview of Muscle Functions, and ThreeDimensional Coordinate Systems
 position
 Appears in this related concepts: Damped Harmonic Motion, Longitudinal Waves, and Graphical Interpretation
 rotation
 Appears in this related concepts: Synovial Joint Movements, Lever Systems, and Center of Mass and Translational Motion
 velocity
 Appears in this related concepts: Rolling Without Slipping, Arc Length and Speed, and Tangent and Velocity Problems
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Source: Boundless. “Relationship Between Linear and Rotational Quantitues.” Boundless Physics. Boundless, 03 Jul. 2014. Retrieved 24 Mar. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/rotationalkinematicsangularmomentumandenergy9/linearandrotationalquantities89/relationshipbetweenlinearandrotationalquantitues3337735/