Just by using our intuition, we can begin to see how rotational quantities like $\theta $, $\omega $, and $\alpha $ are related to one another. For example, if a motorcycle wheel has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotates through many revolutions. In more technical terms, if the wheel’s angular acceleration $\alpha $ is large for a long period of time $t$, then the final angular velocity $\omega $ and angle of rotation $\theta $ are large. The wheel’s rotational motion is exactly analogous to the fact that the motorcycle’s large translational acceleration produces a large final velocity, and the distance traveled will also be large.

Kinematics is the description of motion. The kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time. Let us start by finding an equation relating $\omega $, $\alpha $, and $t$. To determine this equation, we recall a familiar kinematic equation for translational, or straight-line, motion:

Note that in rotational motion $a={a}_{\text{t}}$, and we shall use the symbol $a$ for tangential or linear acceleration from now on. As in linear kinematics, we assume $a$ is constant, which means that angular acceleration $\alpha $ is also a constant, because $a=\mathrm{r\alpha}$. Now, let us substitute $v=\mathrm{r\omega}$ and $a=\mathrm{r\alpha}$ into the linear equation above:

The radius $r$ cancels in the equation, yielding

where ${\omega}_{0}$ is the initial angular velocity. This last equation is a *kinematic relationship* among $\omega $, $\alpha $, and $t$ —that is, it describes their relationship without reference to forces or masses that may affect rotation. It is also precisely analogous in form to its translational counterpart.

Starting with the four kinematic equations we developed in One-Dimensional Kinematics, we can derive the following four rotational kinematic equations (presented together with their translational counterparts):

Rotational | Translational | |

$$\theta =\overline{\omega}t$$ | $$x=\stackrel{-}{v}t$$ | |

$$\omega ={\omega}_{0}+\mathrm{\alpha t}$$ | $$v={v}_{0}+\text{at}$$ | (constant $\alpha $, $a$) |

$$\theta ={\omega}_{0}t+\frac{1}{2}{\mathrm{\alpha t}}^{2}$$ | $$x={v}_{0}t+\frac{1}{2}{\text{at}}^{2}$$ | (constant $\alpha $, $a$) |

$${\omega}^{2}={{\omega}_{0}}^{2}+2\text{\alpha \theta}$$ | $${v}^{2}={{v}_{0}}^{2}+2\text{ax}$$ | (constant $\alpha $, $a$) |

In these equations, the subscript 0 denotes initial values (_{${\theta}_{0}$}, ${x}_{0}$, and ${t}_{0}$ are initial values), and the average angular velocity $\stackrel{-}{\omega}$ and average velocity $\stackrel{-}{v}$ are defined as follows:

The equations given above in [link] can be used to solve any rotational or translational kinematics problem in which $a$ and $\alpha $ are constant.

A deep-sea fisherman hooks a big fish that swims away from the boat pulling the fishing line from his fishing reel. The whole system is initially at rest and the fishing line unwinds from the reel at a radius of 4.50 cm from its axis of rotation. The reel is given an angular acceleration of $\text{110}\phantom{\rule{0.25em}{0ex}}{\text{rad/s}}^{2}$ for 2.00 s as seen in [link].

(a) What is the final angular velocity of the reel?

(b) At what speed is fishing line leaving the reel after 2.00 s elapses?

(c) How many revolutions does the reel make?

(d) How many meters of fishing line come off the reel in this time?

**Strategy**

In each part of this example, the strategy is the same as it was for solving problems in linear kinematics. In particular, known values are identified and a relationship is then sought that can be used to solve for the unknown.

**Solution for (a)**

Here $\alpha $ and $t$ are given and $\omega $ needs to be determined. The most straightforward equation to use is $\omega ={\omega}_{0}+\mathrm{\alpha t}$ because the unknown is already on one side and all other terms are known. That equation states that

We are also given that ${\omega}_{0}=0$ (it starts from rest), so that

**Solution for (b) **

Now that $\omega $ is known, the speed $v$ can most easily be found using the relationship

where the radius $r$ of the reel is given to be 4.50 cm; thus,

Note again that radians must always be used in any calculation relating linear and angular quantities. Also, because radians are dimensionless, we have $\text{m}\times \text{rad}=\text{m}$.

**Solution for (c)**

Here, we are asked to find the number of revolutions. Because $\text{1 rev}=\text{2\pi rad}$, we can find the number of revolutions by finding $\theta $ in radians. We are given $\alpha $ and $t$, and we know ${\omega}_{{}_{0}}$ is zero, so that $\theta $ can be obtained using $\theta ={\omega}_{0}t+\frac{1}{2}{\mathrm{\alpha t}}^{2}$.

Converting radians to revolutions gives

**Solution for (d)**

The number of meters of fishing line is $x$, which can be obtained through its relationship with $\theta $:

**Discussion**

This example illustrates that relationships among rotational quantities are highly analogous to those among linear quantities. We also see in this example how linear and rotational quantities are connected. The answers to the questions are realistic. After unwinding for two seconds, the reel is found to spin at 220 rad/s, which is 2100 rpm. (No wonder reels sometimes make high-pitched sounds.) The amount of fishing line played out is 9.90 m, about right for when the big fish bites.

Now let us consider what happens if the fisherman applies a brake to the spinning reel, achieving an angular acceleration of $\u2013\text{300}\phantom{\rule{0.25em}{0ex}}{\text{rad/s}}^{2}$. How long does it take the reel to come to a stop?

**Strategy**

We are asked to find the time $t$ for the reel to come to a stop. The initial and final conditions are different from those in the previous problem, which involved the same fishing reel. Now we see that the initial angular velocity is ${\omega}_{0}=\text{220 rad/s}$ and the final angular velocity $\omega $ is zero. The angular acceleration is given to be $\alpha =-\text{300}\phantom{\rule{0.25em}{0ex}}{\text{rad/s}}^{2}$. Examining the available equations, we see all quantities but *t* are known in $\omega ={\omega}_{0}+\mathrm{\alpha t},$ making it easiest to use this equation.

**Solution**

The equation states

We solve the equation algebraically for *t*, and then substitute the known values as usual, yielding

**Discussion**

Note that care must be taken with the signs that indicate the directions of various quantities. Also, note that the time to stop the reel is fairly small because the acceleration is rather large. Fishing lines sometimes snap because of the accelerations involved, and fishermen often let the fish swim for a while before applying brakes on the reel. A tired fish will be slower, requiring a smaller acceleration.

Large freight trains accelerate very slowly. Suppose one such train accelerates from rest, giving its 0.350-m-radius wheels an angular acceleration of $0\text{.}\text{250}\phantom{\rule{0.25em}{0ex}}{\text{rad/s}}^{2}$. After the wheels have made 200 revolutions (assume no slippage): (a) How far has the train moved down the track? (b) What are the final angular velocity of the wheels and the linear velocity of the train?

**Strategy**

In part (a), we are asked to find $x$, and in (b) we are asked to find $\omega $ and $v$. We are given the number of revolutions $\theta $, the radius of the wheels $r$, and the angular acceleration $\alpha $.

**Solution for (a)**

The distance $x$ is very easily found from the relationship between distance and rotation angle:

Solving this equation for $x$ yields

Before using this equation, we must convert the number of revolutions into radians, because we are dealing with a relationship between linear and rotational quantities:

Now we can substitute the known values into $x=\mathrm{r\theta}$ to find the distance the train moved down the track:

**Solution for (b)**

We cannot use any equation that incorporates $t$ to find $\omega $, because the equation would have at least two unknown values. The equation ${\omega}^{2}={{\omega}_{0}}^{2}+2\text{\alpha \theta}$ will work, because we know the values for all variables except $\omega $:

Taking the square root of this equation and entering the known values gives

We can find the linear velocity of the train, $v$, through its relationship to $\omega $:

**Discussion**

The distance traveled is fairly large and the final velocity is fairly slow (just under 32 km/h).

There is translational motion even for something spinning in place, as the following example illustrates. [link] shows a fly on the edge of a rotating microwave oven plate. The example below calculates the total distance it travels.

A person decides to use a microwave oven to reheat some lunch. In the process, a fly accidentally flies into the microwave and lands on the outer edge of the rotating plate and remains there. If the plate has a radius of 0.15 m and rotates at 6.0 rpm, calculate the total distance traveled by the fly during a 2.0-min cooking period. (Ignore the start-up and slow-down times.)

**Strategy**

First, find the total number of revolutions $\theta $, and then the linear distance $x$ traveled. $\theta =\overline{\omega}t$ can be used to find $\theta $ because $\stackrel{-}{\omega}$ is given to be 6.0 rpm.

**Solution**

Entering known values into $\theta =\overline{\omega}t$ gives

As always, it is necessary to convert revolutions to radians before calculating a linear quantity like $x$ from an angular quantity like $\theta $:

Now, using the relationship between $x$ and $\theta $, we can determine the distance traveled:

**Discussion**

Quite a trip (if it survives)! Note that this distance is the total distance traveled by the fly. Displacement is actually zero for complete revolutions because they bring the fly back to its original position. The distinction between total distance traveled and displacement was first noted in One-Dimensional Kinematics.

# Section Summary

- Kinematics is the description of motion.
- The kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time.
- Starting with the four kinematic equations we developed in the One-Dimensional Kinematics, we can derive the four rotational kinematic equations (presented together with their translational counterparts) seen in [link].
- In these equations, the subscript 0 denotes initial values (
_{${x}_{0}$}and_{${t}_{0}$}are initial values), and the average angular velocity $\stackrel{-}{\omega}$ and average velocity $\stackrel{-}{v}$ are defined as follows:$\overline{\omega}=\frac{{\omega}_{0}+\omega}{2}\text{and}\overline{v}=\frac{{v}_{0}+v}{2}.$

# Problems & Exercises

With the aid of a string, a gyroscope is accelerated from rest to 32 rad/s in 0.40 s.

(a) What is its angular acceleration in rad/s^{2}?

(b) How many revolutions does it go through in the process?

(a) $80\phantom{\rule{0.25em}{0ex}}{\text{rad/s}}^{2}$

(b) 1.0 rev

Suppose a piece of dust finds itself on a CD. If the spin rate of the CD is 500 rpm, and the piece of dust is 4.3 cm from the center, what is the total distance traveled by the dust in 3 minutes? (Ignore accelerations due to getting the CD rotating.)

A gyroscope slows from an initial rate of 32.0 rad/s at a rate of $0\text{.}\text{700}{\text{rad/s}}^{2}$.

(a) How long does it take to come to rest?

(b) How many revolutions does it make before stopping?

(a) 45.7 s

(b) 116 rev

During a very quick stop, a car decelerates at $7\text{.}\text{00}{\text{m/s}}^{2}$.

(a) What is the angular acceleration of its 0.280-m-radius tires, assuming they do not slip on the pavement?

(b) How many revolutions do the tires make before coming to rest, given their initial angular velocity is $\text{95}\text{.}0\text{rad/s}$?

(c) How long does the car take to stop completely?

(d) What distance does the car travel in this time?

(e) What was the car’s initial velocity?

(f) Do the values obtained seem reasonable, considering that this stop happens very quickly?

Everyday application: Suppose a yo-yo has a center shaft that has a 0.250 cm radius and that its string is being pulled.

(a) If the string is stationary and the yo-yo accelerates away from it at a rate of $1\text{.}\text{50}{\text{m/s}}^{2}$, what is the angular acceleration of the yo-yo?

(b) What is the angular velocity after 0.750 s if it starts from rest?

(c) The outside radius of the yo-yo is 3.50 cm. What is the tangential acceleration of a point on its edge?

a) $6{\text{00 rad/s}}^{2}$

b) 450 rad/s

c) 21.0 m/s

- College Physics
- Preface
- Introduction: The Nature of Science and Physics
- Kinematics
- Introduction to One-Dimensional Kinematics
- Displacement
- Vectors, Scalars, and Coordinate Systems
- Time, Velocity, and Speed
- Acceleration
- Motion Equations for Constant Acceleration in One Dimension
- Problem-Solving Basics for One-Dimensional Kinematics
- Falling Objects
- Graphical Analysis of One-Dimensional Motion

- Two-Dimensional Kinematics
- Dynamics: Force and Newton's Laws of Motion
- Introduction to Dynamics: Newton’s Laws of Motion
- Development of Force Concept
- Newton’s First Law of Motion: Inertia
- Newton’s Second Law of Motion: Concept of a System
- Newton’s Third Law of Motion: Symmetry in Forces
- Normal, Tension, and Other Examples of Forces
- Problem-Solving Strategies
- Further Applications of Newton’s Laws of Motion
- Extended Topic: The Four Basic Forces—An Introduction

- Further Applications of Newton's Laws: Friction, Drag, and Elasticity
- Uniform Circular Motion and Gravitation
- Work, Energy, and Energy Resources
- Linear Momentum and Collisions
- Statics and Torque
- Rotational Motion and Angular Momentum
- Introduction to Rotational Motion and Angular Momentum
- Angular Acceleration
- Kinematics of Rotational Motion
- Dynamics of Rotational Motion: Rotational Inertia
- Rotational Kinetic Energy: Work and Energy Revisited
- Angular Momentum and Its Conservation
- Collisions of Extended Bodies in Two Dimensions
- Gyroscopic Effects: Vector Aspects of Angular Momentum

- Fluid Statics
- Fluid Dynamics and Its Biological and Medical Applications
- Introduction to Fluid Dynamics and Its Biological and Medical Applications
- Flow Rate and Its Relation to Velocity
- Bernoulli’s Equation
- The Most General Applications of Bernoulli’s Equation
- Viscosity and Laminar Flow; Poiseuille’s Law
- The Onset of Turbulence
- Motion of an Object in a Viscous Fluid
- Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

- Temperature, Kinetic Theory, and the Gas Laws
- Heat and Heat Transfer Methods
- Thermodynamics
- Introduction to Thermodynamics
- The First Law of Thermodynamics
- The First Law of Thermodynamics and Some Simple Processes
- Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency
- Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated
- Applications of Thermodynamics: Heat Pumps and Refrigerators
- Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy
- Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

- Oscillatory Motion and Waves
- Introduction to Oscillatory Motion and Waves
- Hooke’s Law: Stress and Strain Revisited
- Period and Frequency in Oscillations
- Simple Harmonic Motion: A Special Periodic Motion
- The Simple Pendulum
- Energy and the Simple Harmonic Oscillator
- Uniform Circular Motion and Simple Harmonic Motion
- Damped Harmonic Motion
- Forced Oscillations and Resonance
- Waves
- Superposition and Interference
- Energy in Waves: Intensity

- Physics of Hearing
- Electric Charge and Electric Field
- Introduction to Electric Charge and Electric Field
- Static Electricity and Charge: Conservation of Charge
- Conductors and Insulators
- Coulomb’s Law
- Electric Field: Concept of a Field Revisited
- Electric Field Lines: Multiple Charges
- Electric Forces in Biology
- Conductors and Electric Fields in Static Equilibrium
- Applications of Electrostatics

- Electric Potential and Electric Field
- Electric Current, Resistance, and Ohm's Law
- Circuits, Bioelectricity, and DC Instruments
- Magnetism
- Introduction to Magnetism
- Magnets
- Ferromagnets and Electromagnets
- Magnetic Fields and Magnetic Field Lines
- Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
- Force on a Moving Charge in a Magnetic Field: Examples and Applications
- The Hall Effect
- Magnetic Force on a Current-Carrying Conductor
- Torque on a Current Loop: Motors and Meters
- Magnetic Fields Produced by Currents: Ampere’s Law
- Magnetic Force between Two Parallel Conductors
- More Applications of Magnetism

- Electromagnetic Induction, AC Circuits, and Electrical Technologies
- Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies
- Induced Emf and Magnetic Flux
- Faraday’s Law of Induction: Lenz’s Law
- Motional Emf
- Eddy Currents and Magnetic Damping
- Electric Generators
- Back Emf
- Transformers
- Electrical Safety: Systems and Devices
- Inductance
- RL Circuits
- Reactance, Inductive and Capacitive
- RLC Series AC Circuits

- Electromagnetic Waves
- Geometric Optics
- Vision and Optical Instruments
- Wave Optics
- Introduction to Wave Optics
- The Wave Aspect of Light: Interference
- Huygens's Principle: Diffraction
- Young’s Double Slit Experiment
- Multiple Slit Diffraction
- Single Slit Diffraction
- Limits of Resolution: The Rayleigh Criterion
- Thin Film Interference
- Polarization
- *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light

- Special Relativity
- Introduction to Quantum Physics
- Atomic Physics
- Introduction to Atomic Physics
- Discovery of the Atom
- Discovery of the Parts of the Atom: Electrons and Nuclei
- Bohr’s Theory of the Hydrogen Atom
- X Rays: Atomic Origins and Applications
- Applications of Atomic Excitations and De-Excitations
- The Wave Nature of Matter Causes Quantization
- Patterns in Spectra Reveal More Quantization
- Quantum Numbers and Rules
- The Pauli Exclusion Principle

- Radioactivity and Nuclear Physics
- Medical Applications of Nuclear Physics
- Particle Physics
- Frontiers of Physics
- Atomic Masses
- Selected Radioactive Isotopes
- Useful Information
- Glossary of Key Symbols and Notation