In Kinematics, we studied motion along a straight line and introduced such concepts as displacement, velocity, and acceleration. Two-Dimensional Kinematics dealt with motion in two dimensions. Projectile motion is a special case of two-dimensional kinematics in which the object is projected into the air, while being subject to the gravitational force, and lands a distance away. In this chapter, we consider situations where the object does not land but moves in a curve. We begin the study of uniform circular motion by defining two angular quantities needed to describe rotational motion.

# Rotation Angle

When objects rotate about some axis—for example, when the CD (compact disc) in [link] rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle $\text{\Delta}\theta $ to be the ratio of the arc length to the radius of curvature:

The arc length$\phantom{\rule{0.25em}{0ex}}\text{\Delta}s$ is the distance traveled along a circular path as shown in [link] Note that $r$ is the radius of curvature of the circular path.

We know that for one complete revolution, the arc length is the circumference of a circle of radius $r$. The circumference of a circle is $\mathrm{2\pi}r$. Thus for one complete revolution the rotation angle is

This result is the basis for defining the units used to measure rotation angles, $\text{\Delta}\theta $ to be radians (rad), defined so that

A comparison of some useful angles expressed in both degrees and radians is shown in [link].

Degree Measures | Radian Measure |

$$\text{30\xba}$$ | $$\frac{\pi}{6}$$ |

$$\text{60\xba}$$ | $$\frac{\pi}{3}$$ |

$$\text{90\xba}$$ | $$\frac{\pi}{2}$$ |

$$\text{120\xba}$$ | $$\frac{\mathrm{2\pi}}{3}$$ |

$$\text{135\xba}$$ | $$\frac{\mathrm{3\pi}}{4}$$ |

$$\text{180\xba}$$ | $$\pi $$ |

If $\text{\Delta}\theta =2\pi $ rad, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are $\text{360\xba}$ in a circle or one revolution, the relationship between radians and degrees is thus

so that

# Angular Velocity

How fast is an object rotating? We define angular velocity $\omega $ as the rate of change of an angle. In symbols, this is

where an angular rotation $\text{\Delta}\theta $ takes place in a time $\text{\Delta}t$. The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad/s).

Angular velocity $\omega $ is analogous to linear velocity $v$. To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length $\text{\Delta}s$ in a time $\text{\Delta}t$, and so it has a linear velocity

From $\text{\Delta}\theta =\frac{\text{\Delta}s}{r}$ we see that $\text{\Delta}s=r\text{\Delta}\theta $. Substituting this into the expression for $v$ gives

We write this relationship in two different ways and gain two different insights:

The first relationship in $v=\mathrm{r\omega}\text{or}\omega =\frac{v}{r}$ states that the linear velocity $v$ is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest $r$), as you might expect. We can also call this linear speed $v$ of a point on the rim the *tangential speed*. The second relationship in $v=\mathrm{r\omega}\text{or}\omega =\frac{v}{r}$ can be illustrated by considering the tire of a moving car. Note that the speed of a point on the rim of the tire is the same as the speed $v$ of the car. See [link]. So the faster the car moves, the faster the tire spins—large $v$ means a large $\omega $, because $v=\mathrm{r\omega}$. Similarly, a larger-radius tire rotating at the same angular velocity ($\omega $) will produce a greater linear speed ($v$) for the car.

Calculate the angular velocity of a 0.300 m radius car tire when the car travels at $\text{15}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{m/s}$ (about $\text{54}\phantom{\rule{0.25em}{0ex}}\text{km/h}$). See [link].

**Strategy**

Because the linear speed of the tire rim is the same as the speed of the car, we have $v=\text{15.0 m/s}.$ The radius of the tire is given to be $r=\text{0.300 m}.$ Knowing $v$ and $r$, we can use the second relationship in $v=\mathrm{r\omega},\omega =\frac{v}{r}$ to calculate the angular velocity.

**Solution**

To calculate the angular velocity, we will use the following relationship:

Substituting the knowns,

**Discussion**

When we cancel units in the above calculation, we get 50.0/s. But the angular velocity must have units of rad/s. Because radians are actually unitless (radians are defined as a ratio of distance), we can simply insert them into the answer for the angular velocity. Also note that if an earth mover with much larger tires, say 1.20 m in radius, were moving at the same speed of 15.0 m/s, its tires would rotate more slowly. They would have an angular velocity

Both $\omega $ and $v$ have directions (hence they are angular and linear *velocities*, respectively). Angular velocity has only two directions with respect to the axis of rotation—it is either clockwise or counterclockwise. Linear velocity is tangent to the path, as illustrated in [link].

# Section Summary

- Uniform circular motion is motion in a circle at constant speed. The rotation angle $\text{\Delta}\theta $ is defined as the ratio of the arc length to the radius of curvature:
$\text{\Delta}\theta =\frac{\text{\Delta}s}{r}\text{,}$
where arc length $\text{\Delta}s$ is distance traveled along a circular path and $r$ is the radius of curvature of the circular path. The quantity $\text{\Delta}\theta $ is measured in units of radians (rad), for which

$\mathrm{2\pi}\phantom{\rule{0.25em}{0ex}}\text{rad}=\text{360\xba}\text{=}1\text{revolution.}$ - The conversion between radians and degrees is $1\phantom{\rule{0.25em}{0ex}}\text{rad}=\text{57}\text{.}3\text{\xba}$.
- Angular velocity $\omega $ is the rate of change of an angle,
$\omega =\frac{\text{\Delta}\theta}{\text{\Delta}t}\text{,}$
where a rotation $\text{\Delta}\theta $ takes place in a time $\text{\Delta}t$. The units of angular velocity are radians per second (rad/s). Linear velocity $v$ and angular velocity $\omega $ are related by

$v=\mathrm{r\omega}\text{or}\omega =\frac{v}{r}\text{.}$

# Conceptual Questions

There is an analogy between rotational and linear physical quantities. What rotational quantities are analogous to distance and velocity?

# Problem Exercises

Semi-trailer trucks have an odometer on one hub of a trailer wheel. The hub is weighted so that it does not rotate, but it contains gears to count the number of wheel revolutions—it then calculates the distance traveled. If the wheel has a 1.15 m diameter and goes through 200,000 rotations, how many kilometers should the odometer read?

723 km

Microwave ovens rotate at a rate of about 6 rev/min. What is this in revolutions per second? What is the angular velocity in radians per second?

An automobile with 0.260 m radius tires travels 80,000 km before wearing them out. How many revolutions do the tires make, neglecting any backing up and any change in radius due to wear?

$5\times {\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{rotations}$

(a) What is the period of rotation of Earth in seconds? (b) What is the angular velocity of Earth? (c) Given that Earth has a radius of $6\text{.}4\times {\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{m}$ at its equator, what is the linear velocity at Earth’s surface?

A baseball pitcher brings his arm forward during a pitch, rotating the forearm about the elbow. If the velocity of the ball in the pitcher’s hand is 35.0 m/s and the ball is 0.300 m from the elbow joint, what is the angular velocity of the forearm?

117 rad/s

In lacrosse, a ball is thrown from a net on the end of a stick by rotating the stick and forearm about the elbow. If the angular velocity of the ball about the elbow joint is 30.0 rad/s and the ball is 1.30 m from the elbow joint, what is the velocity of the ball?

A truck with 0.420 m radius tires travels at 32.0 m/s. What is the angular velocity of the rotating tires in radians per second? What is this in rev/min?

76.2 rad/s

728 rpm

**Integrated Concepts** When kicking a football, the kicker rotates his leg about the hip joint.

(a) If the velocity of the tip of the kicker’s shoe is 35.0 m/s and the hip joint is 1.05 m from the tip of the shoe, what is the shoe tip’s angular velocity?

(b) The shoe is in contact with the initially nearly stationary 0.500 kg football for 20.0 ms. What average force is exerted on the football to give it a velocity of 20.0 m/s?

(c) Find the maximum range of the football, neglecting air resistance.

(a) 33.3 rad/s

(b) 500 N

(c) 40.8 m

**Construct Your Own Problem**

Consider an amusement park ride in which participants are rotated about a vertical axis in a cylinder with vertical walls. Once the angular velocity reaches its full value, the floor drops away and friction between the walls and the riders prevents them from sliding down. Construct a problem in which you calculate the necessary angular velocity that assures the riders will not slide down the wall. Include a free body diagram of a single rider. Among the variables to consider are the radius of the cylinder and the coefficients of friction between the riders’ clothing and the wall.

- 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
- Introduction to Work, Energy, and Energy Resources
- Work: The Scientific Definition
- Kinetic Energy and the Work-Energy Theorem
- Gravitational Potential Energy
- Conservative Forces and Potential Energy
- Nonconservative Forces
- Conservation of Energy
- Power
- Work, Energy, and Power in Humans
- World Energy Use

- 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
- Introduction to Fluid Statics
- What Is a Fluid?
- Density
- Pressure
- Variation of Pressure with Depth in a Fluid
- Pascal’s Principle
- Gauge Pressure, Absolute Pressure, and Pressure Measurement
- Archimedes’ Principle
- Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action
- Pressures in the Body

- 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
- Introduction to Electric Potential and Electric Energy
- Electric Potential Energy: Potential Difference
- Electric Potential in a Uniform Electric Field
- Electrical Potential Due to a Point Charge
- Equipotential Lines
- Capacitors and Dielectrics
- Capacitors in Series and Parallel
- Energy Stored in Capacitors

- 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