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The Nature of Science and Physics

Kinematics

Introduction to One-Dimensional KinematicsDisplacementVectors, Scalars, and Coordinate SystemsTime, Velocity, and SpeedVideo: One-Dimensional KinematicsAccelerationMotion Equations for Constant Acceleration in One DimensionProblem-Solving Basics for One-Dimensional KinematicsFalling ObjectsGraphical Analysis of One-Dimensional Motion

Two-Dimensional Kinematics

Dynamics: Force and Newton's Laws of Motion

Introduction to Dynamics: Newton's Laws of MotionDevelopment of Force ConceptNewton's First Law of Motion: InertiaNewton's Second Law of Motion: Concept of a SystemNewton's Third Law of Motion: Symmetry in ForcesVideo: Newton's LawsNormal, Tension, and Other Examples of ForcesProblem-Solving StrategiesFurther Applications of Newton's Laws of MotionExtended Topic: The Four Basic Forces—An Introduction

Further Applications of Newton's Laws: Friction, Drag, and Elasticity

Uniform Circular Motion and Gravitation

Introduction to Uniform Circular Motion and GravitationRotation Angle and Angular VelocityCentripetal AccelerationCentripetal ForceFictitious Forces and Non-inertial Frames: The Coriolis ForceNewton's Universal Law of GravitationVideo: GravitationSatellites and Kepler's Laws: An Argument for Simplicity

Work, Energy, and Energy Resources

Introduction to Work, Energy, and Energy ResourcesWork: The Scientific DefinitionKinetic Energy and the Work-Energy TheoremGravitational Potential EnergyVideo: Potential and Kinetic EnergyConservative Forces and Potential EnergyNonconservative ForcesConservation of EnergyPowerWork, Energy, and Power in HumansWorld Energy Use

Linear Momentum and Collisions

Rotational Motion and Angular Momentum

Introduction to Rotational Motion and Angular MomentumAngular AccelerationKinematics of Rotational MotionVideo: Rotational MotionDynamics of Rotational Motion: Rotational InertiaRotational Kinetic Energy: Work and Energy RevisitedAngular Momentum and Its ConservationVideo: Angular MomentumCollisions of Extended Bodies in Two DimensionsGyroscopic Effects: Vector Aspects of Angular Momentum

Statics and Torque

Fluid Statics

Introduction to Fluid StaticsWhat Is a Fluid?DensityPressureVariation of Pressure with Depth in a FluidPascal's PrincipleGauge Pressure, Absolute Pressure, and Pressure MeasurementArchimedes' PrincipleVideo: BuoyancyCohesion and Adhesion in Liquids: Surface Tension and Capillary ActionPressures in the Body

Fluid Dynamics and Its Biological and Medical Applications

Introduction to Fluid Dynamics and Biological and Medical ApplicationsFlow Rate and Its Relation to VelocityBernoulli's EquationVideo: Fluid FlowThe Most General Applications of Bernoulli's EquationViscosity and Laminar Flow; Poiseuille's LawThe Onset of TurbulenceMotion of an Object in a Viscous FluidMolecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

Temperature, Kinetic Theory, and the Gas Laws

Heat and Heat Transfer Methods

Thermodynamics

Introduction to ThermodynamicsThe First Law of ThermodynamicsThe First Law of Thermodynamics and Some Simple ProcessesIntroduction to the Second Law of Thermodynamics: Heat Engines and Their EfficiencyCarnot's Perfect Heat Engine: The Second Law of Thermodynamics RestatedApplications of Thermodynamics: Heat Pumps and RefrigeratorsEntropy and the Second Law of Thermodynamics: Disorder and the Unavailability of EnergyStatistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

Oscillatory Motion and Waves

Introduction to Oscillatory Motion and WavesHooke's Law: Stress and Strain RevisitedPeriod and Frequency in OscillationsSimple Harmonic Motion: A Special Periodic MotionVideo: Harmonic MotionThe Simple PendulumEnergy and the Simple Harmonic OscillatorUniform Circular Motion and Simple Harmonic MotionDamped Harmonic MotionForced Oscillations and ResonanceWavesSuperposition and InterferenceEnergy in Waves: Intensity

Physics of Hearing

- Define arc length, rotation angle, radius of curvature and angular velocity.
- Calculate the angular velocity of a car wheel spin.

$\displaystyle\Delta\theta=\frac{\Delta{s}}{r}\\$

The

We know that for one complete revolution, the arc length is the circumference of a circle of radius

$\displaystyle\Delta\theta=\frac{2\pi{r}}{r}=2\pi\\$

.
This result is the basis for defining the units used to measure rotation angles, ΔA comparison of some useful angles expressed in both degrees and radians is shown in Table 1.

Table 1. Comparison of Angular Units | |
---|---|

Degree Measures | Radian Measure |

30º | $\displaystyle\frac{\pi}{6}\\$ |

60º | $\displaystyle\frac{\pi}{3}\\$ |

90º | $\displaystyle\frac{\pi}{2}\\$ |

120º | $\displaystyle\frac{2\pi}{3}\\$ |

135º | $\displaystyle\frac{3\pi}{4}\\$ |

180º | π |

If Δ

$1\text{ rad}=\frac{360^{\circ}}{2\pi}\approx57.3^{\circ}\\$

.
$\omega=\frac{\Delta\theta}{\Delta{t}}\\$

, where an angular rotation ΔAngular velocity

$v=\frac{\Delta{s}}{\Delta{t}}\\$

.From

$\Delta\theta=\frac{\Delta{s}}{r}\\$

we see that Δ$v=\frac{r\Delta\theta}{\Delta{t}}=r\omega\\$

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

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

.
The first relationship in $v=r\omega\text{ or }\omega\frac{v}{r}\\$

states that the linear velocity $v=r\omega\text{ or }\omega\frac{v}{r}\\$

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

to calculate the angular velocity.$\omega\frac{v}{r}\\$

.Substituting the knowns,

$\omega=\frac{15.0 \text{ m/s}}{0.300\text{ m}}=50.0\text{ rad/s}\\$

.
$\omega=\frac{15.0\text{ m/s}}{1.20\text{ m}}=12.5\text{ rad/s}\\$

.- Uniform circular motion is motion in a circle at constant speed. The rotation angle $\Delta\theta\\$is defined as the ratio of the arc length to the radius of curvature:$\Delta\theta=\frac{\Delta{s}}{r}\\$, where arc length Δ
*s*is distance traveled along a circular path and*r*is the radius of curvature of the circular path. The quantity$\Delta\theta\\$is measured in units of radians (rad), for which$2\pi\text{rad}=360^{\circ}= 1\text{ revolution}\\$. - The conversion between radians and degrees is $1\text{ rad}=57.3^{\circ}\\$.
- Angular velocity ω is the rate of change of an angle, $\omega=\frac{\Delta\theta}{\Delta{t}}\\$, where a rotation$\Delta\theta\\$takes place in a time$\Delta{t}\\$. The units of angular velocity are radians per second (rad/s). Linear velocity
*v*and angular velocity ω are related by$v=\mathrm{r\omega }\text{ or }\omega =\frac{v}{r}\text{.}$

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

- 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?
- 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?
- (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 $6.4\times{10}^6\text{ m}\\$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?
- 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?
**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 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.**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.

$\Delta\theta=\frac{\Delta{s}}{r}\\$

3. 5 × 10

5. 117 rad/s

7. 76.2 rad/s; 728 rpm

8. (a) 33.3 rad/s; (b) 500 N; (c) 40.8 m