Linear and Angular Velocity Examples

Total Page：16

File Type：pdf, Size：1020Kb

Linear and Angular Velocity Examples Example 1 Determine the angular displacement in radians of 6.5 revolutions. Round to the nearest tenth. Each revolution equals 2 radians. For 6.5 revolutions, the number of radians is 6.5 2 or 13 . 13 radians equals about 40.8 radians. Example 2 Determine the angular velocity if 4.8 revolutions are completed in 4 seconds. Round to the nearest tenth. The angular displacement is 4.8 2 or 9.6 radians. = t 9.6 = = 9.6 , t = 4 4 7.539822369 Use a calculator. The angular velocity is about 7.5 radians per second. Example 3 AMUSEMENT PARK Jack climbs on a horse that is 12 feet from the center of a merry-go-round. 1 The merry-go-round makes 3 rotations per minute. Determine Jack’s angular velocity in radians 4 per second. Round to the nearest hundredth. 1 The merry-go-round makes 3 or 3.25 revolutions per minute. Convert revolutions per minute to radians 4 per second. 3.25 revolutions 1 minute 2 radians 0.3403392041 radian per second 1 minute 60 seconds 1 revolution Jack has an angular velocity of about 0.34 radian per second. Example 4 Determine the linear velocity of a point rotating at an angular velocity of 12 radians per second at a distance of 8 centimeters from the center of the rotating object. Round to the nearest tenth. v = r v = (8)(12 ) r = 8, = 12 v 301.5928947 Use a calculator. The linear velocity is about 301.6 centimeters per second. Example 5 Refer to the application in Example 3. Determine Jack’s linear velocity. v = r v (12)(0.34) r = 12, = 0.34 v 4.08 Use a calculator. Jack’s linear velocity is about 4.08 feet per second. Example 6 BICYCLES The tires on a bicycle have a diameter of 24 inches. If the tires are turning at a rate of 50 revolutions per minute, determine the bicycle’s speed in miles per hour (mph). If the diameter is 24 inches, the radius is 12 inches. This measure needs to be written in miles. The rate needs to be written in hours. v = r 1 ft 1 mi 50 rev 2 60 min v = 12 in. 12 in. 5280 ft 1 min 1 rev 1 h v 3.569991652 Use a calculator. The speed of the bicycle is about 3.6 miles per hour. .

Copy Link

Recommended publications

Rotational Motion (The Dynamics of a Rigid Body)
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Robert Katz Publications Research Papers in Physics and Astronomy 1-1958 Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body) Henry Semat City College of New York Robert Katz University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/physicskatz Part of the Physics Commons Semat, Henry and Katz, Robert, "Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body)" (1958). Robert Katz Publications. 141. https://digitalcommons.unl.edu/physicskatz/141 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Robert Katz Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 11 Rotational Motion (The Dynamics of a Rigid Body) 11-1 Motion about a Fixed Axis The motion of the flywheel of an engine and of a pulley on its axle are examples of an important type of motion of a rigid body, that of the motion of rotation about a fixed axis. Consider the motion of a uniform disk rotat ing about a fixed axis passing through its center of gravity C perpendicular to the face of the disk, as shown in Figure 11-1. The motion of this disk may be de scribed in terms of the motions of each of its individual particles, but a better way to describe the motion is in terms of the angle through which the disk rotates.
[Show full text]
Guide for the Use of the International System of Units (SI)
Guide for the Use of the International System of Units (SI) m kg s cd SI mol K A NIST Special Publication 811 2008 Edition Ambler Thompson and Barry N. Taylor NIST Special Publication 811 2008 Edition Guide for the Use of the International System of Units (SI) Ambler Thompson Technology Services and Barry N. Taylor Physics Laboratory National Institute of Standards and Technology Gaithersburg, MD 20899 (Supersedes NIST Special Publication 811, 1995 Edition, April 1995) March 2008 U.S. Department of Commerce Carlos M. Gutierrez, Secretary National Institute of Standards and Technology James M. Turner, Acting Director National Institute of Standards and Technology Special Publication 811, 2008 Edition (Supersedes NIST Special Publication 811, April 1995 Edition) Natl. Inst. Stand. Technol. Spec. Publ. 811, 2008 Ed., 85 pages (March 2008; 2nd printing November 2008) CODEN: NSPUE3 Note on 2nd printing: This 2nd printing dated November 2008 of NIST SP811 corrects a number of minor typographical errors present in the 1st printing dated March 2008. Guide for the Use of the International System of Units (SI) Preface The International System of Units, universally abbreviated SI (from the French Le Système International d’Unités), is the modern metric system of measurement. Long the dominant measurement system used in science, the SI is becoming the dominant measurement system used in international commerce. The Omnibus Trade and Competitiveness Act of August 1988 [Public Law (PL) 100-418] changed the name of the National Bureau of Standards (NBS) to the National Institute of Standards and Technology (NIST) and gave to NIST the added task of helping U.S.
[Show full text]
Rotational Motion of Electric Machines
Rotational Motion of Electric Machines • An electric machine rotates about a fixed axis, called the shaft, so its rotation is restricted to one angular dimension. • Relative to a given end of the machine’s shaft, the direction of counterclockwise (CCW) rotation is often assumed to be positive. • Therefore, for rotation about a fixed shaft, all the concepts are scalars. 17 Angular Position, Velocity and Acceleration • Angular position – The angle at which an object is oriented, measured from some arbitrary reference point – Unit: rad or deg – Analogy of the linear concept • Angular acceleration =d/dt of distance along a line. – The rate of change in angular • Angular velocity =d/dt velocity with respect to time – The rate of change in angular – Unit: rad/s2 position with respect to time • and >0 if the rotation is CCW – Unit: rad/s or r/min (revolutions • >0 if the absolute angular per minute or rpm for short) velocity is increasing in the CCW – Analogy of the concept of direction or decreasing in the velocity on a straight line. CW direction 18 Moment of Inertia (or Inertia) • Inertia depends on the mass and shape of the object (unit: kgm2) • A complex shape can be broken up into 2 or more of simple shapes Definition Two useful formulas mL2 m J J() RRRR22 12 3 1212 m 22 JRR()12 2 19 Torque and Change in Speed • Torque is equal to the product of the force and the perpendicular distance between the axis of rotation and the point of application of the force. T=Fr (Nm) T=0 T T=Fr • Newton’s Law of Rotation: Describes the relationship between the total torque applied to an object and its resulting angular acceleration.
[Show full text]
Physics B Topics Overview ∑
Physics 106 Lecture 1 Introduction to Rotation SJ 7th Ed.: Chap. 10.1 to 3 • Course Introduction • Course Rules & Assignment • TiTopics Overv iew • Rotation (rigid body) versus translation (point particle) • Rotation concepts and variables • Rotational kinematic quantities Angular position and displacement Angular velocity & acceleration • Rotation kinematics formulas for constant angular acceleration • Analogy with linear kinematics 1 Physics B Topics Overview PHYSICS A motion of point bodies COVERED: kinematics - translation dynamics ∑Fext = ma conservation laws: energy & momentum motion of “Rigid Bodies” (extended, finite size) PHYSICS B rotation + translation, more complex motions possible COVERS: rigid bodies: fixed size & shape, orientation matters kinematics of rotation dynamics ∑Fext = macm and ∑ Τext = Iα rotational modifications to energy conservation conservation laws: energy & angular momentum TOPICS: 3 weeks: rotation: ▪ angular versions of kinematics & second law ▪ angular momentum ▪ equilibrium 2 weeks: gravitation, oscillations, fluids 1 Angular variables: language for describing rotation Measure angles in radians simple rotation formulas Definition: 360o 180o • 2π radians = full circle 1 radian = = = 57.3o • 1 radian = angle that cuts off arc length s = radius r 2π π s arc length ≡ s = r θ (in radians) θ ≡ rad r r s θ’ θ Example: r = 10 cm, θ = 100 radians Æ s = 1000 cm = 10 m. Rigid body rotation: angular displacement and arc length Angular displacement is the angle an object (rigid body) rotates through during some time interval..... ...also the angle that a reference line fixed in a body sweeps out A rigid body rotates about some rotation axis – a line located somewhere in or near it, pointing in some y direction in space • One polar coordinate θ specifies position of the whole body about this rotation axis.
[Show full text]
TECHNICAL Nbte D-626
NASA TN D-626 LIBRARY COPY NOV 16 1960 SPACE FLIGHT LANGLEY FIELD, VIRGINIA t TECHNICAL NbTE D-626 SATELLITE ATTDUDE CONTROL USING A COMBINATION OF INERTIA WHEELS AND A BAR MAGNET By James J. Adams and Roy F. Brissenden Langley Research Center Langley Field, Va. NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON November 1960 . NATIONAL AERONAUTICS AND SPACE ADMINISTRATION TECHNICAL NOTE D-626 SATELLITE ATTITUDE CONTROL USING A COMBINATION OF INERTIA WHEELS AND A BAR MAGNET By James J. Adams and Roy F. Brissenden SUMMARY The use of inertia wheels to control the attitude of a satellite has currently aroused much interest. The stability of such a system has been studied in this investigation. A single-degree-of-freedom analysis indicates that a response with suitable dynamic characteristics and pre- cise control can be achieved by commanding the angular velocity of the inertia wheel with an error signal that is the sum of the attitude error, the attitude rate, and the integral of the attitude error. A digital computer was used to study the three-degree-of-freedom response to step displacements, and the results indicate that the cross-coupling effects of inertia coupling and precession coupling had no effect on system sta- bility. A study was also made of the use of a bar magnet to supplement the inertia wheels by providing a means of removing any momentum intro- duced into the system by disturbances such as aerodynamic torques. A study of a case with large aerodynamic torques, with a typical orbit, indicated that the magnet was a suitable device for supplying the essen- tial trimming force.
[Show full text]
Rotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn that where the force is applied and how the force is applied is just as important as how much force is applied when we want to make something rotate. This tutorial discusses the dynamics of an object rotating about a fixed axis and introduces the concepts of torque and moment of inertia. These concepts allows us to get a better understanding of why pushing a door towards its hinges is not very a very effective way to make it open, why using a longer wrench makes it easier to loosen a tight bolt, etc. This module begins by looking at the kinetic energy of rotation and by defining a quantity known as the moment of inertia which is the rotational analog of mass. Then it proceeds to discuss the quantity called torque which is the rotational analog of force and is the physical quantity that is required to changed an object's state of rotational motion. Moment of Inertia Kinetic Energy of Rotation Consider a rigid object rotating about a fixed axis at a certain angular velocity. Since every particle in the object is moving, every particle has kinetic energy. To find the total kinetic energy related to the rotation of the body, the sum of the kinetic energy of every particle due to the rotational motion is taken. The total kinetic energy can be expressed as ..
[Show full text]
Average Angular Velocity’ As the Solution This Problem and Discusses Some Applications Briefly
Average Angular Velocity Hanno Essén Department of Mechanics Royal Institute of Technology S-100 44 Stockholm, Sweden 1992, December Abstract This paper addresses the problem of the separation of rota- tional and internal motion. It introduces the concept of average angular velocity as the moment of inertia weighted average of particle angular velocities. It extends and elucidates the concept of Jellinek and Li (1989) of separation of the energy of overall rotation in an arbitrary (non-linear) N-particle system. It gen- eralizes the so called Koenig’s theorem on the two parts of the kinetic energy (center of mass plus internal) to three parts: center of mass, rotational, plus the remaining internal energy relative to an optimally translating and rotating frame. Published in: European Journal of Physics 14, pp.201-205, (1993). arXiv:physics/0401146v1 [physics.class-ph] 28 Jan 2004 1 1 Introduction The motion of a rigid body is completely characterized by its (center of mass) translational velocity and its angular velocity which describes the rotational motion. Rotational motion as a phenomenon is, however, not restricted to rigid bodies and it is then a kinematic problem to define exactly what the rotational motion of the system is. This paper introduces the new concept of ‘average angular velocity’ as the solution this problem and discusses some applications briefly. The average angular velocity concept is closely analogous to the concept of center of mass velocity. For a system of particles the center of mass velocity is simply the mass weighted average of the particle velocities. In a similar way the average angular velocity is the moment of inertia weighted average of the angular velocities of the particle position vectors.
[Show full text]
7 Rotational Motion
7 Rotational Motion © 2010 Pearson Education, Inc. Slide 7-2 © 2010 Pearson Education, Inc. Slide 7-3 Recall from Chapter 6… . Angular displacement = θ . θ= ω t . Angular Velocity = ω (Greek: Omega) . ω = 2 π f and ω = θ/ t . All points on a rotating object rotate through the same angle in the same time, and have the same frequency. Angular velocity: all points on a rotating object have the same angular velocity, ω, but different speeds, v, and v =ωr. v =ωr © 2010 Pearson Education, Inc. ω is positive if object is rotating counterclockwise. (Negative if rotation is clockwise.) . Conversion: 1 revolution = 2 π rad © 2010 Pearson Education, Inc. Checking Understanding Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The angular velocity of A is twice that of B. B. The angular velocity of A equals that of B. C. The angular velocity of A is half that of B. © 2010 Pearson Education, Inc. Slide 7-13 Answer Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The angular velocity of A is twice that of B. B. The angular velocity of A equals that of B. C. The angular velocity of A is half that of B. All points on the turntable rotate through the same angle in the same time. ω = θ/ t All points have the same period, therefore, all points have the same frequency. ω = 2 π f © 2010 Pearson Education, Inc. Slide 7-14 Checking Understanding Two coins rotate on a turntable.
[Show full text]
Circular Motion Angular Velocity
PHY131H1F - Class 8 Quiz time… – Angular Notation: it’s all Today, finishing off Chapter 4: Greek to me! d • Circular Motion dt • Rotation θ is an angle, and the S.I. unit of angle is rad. The time derivative of θ is ω. What are the S.I. units of ω ? A. m/s2 B. rad / s C. N/m D. rad E. rad /s2 Last day I asked at the end of class: Quiz time… – Angular Notation: it’s all • You are driving North Highway Greek to me! d 427, on the smoothly curving part that will join to the Westbound 401. v dt Your speedometer is constant at 115 km/hr. Your steering wheel is The time derivative of ω is α. not rotating, but it is turned to the a What are the S.I. units of α ? left to follow the curve of the A. m/s2 highway. Are you accelerating? B. rad / s • ANSWER: YES! Any change in velocity, either C. N/m magnitude or speed, implies you are accelerating. D. rad • If so, in what direction? E. rad /s2 • ANSWER: West. If your speed is constant, acceleration is always perpendicular to the velocity, toward the centre of circular path. Circular Motion r = constant Angular Velocity s and θ both change as the particle moves s = “arc length” θ = “angular position” when θ is measured in radians when ω is measured in rad/s 1 Special case of circular motion: Uniform Circular Motion A carnival has a Ferris wheel where some seats are located halfway between the center Tangential velocity is and the outside rim.
[Show full text]
Lecture 24 Angular Momentum
LECTURE 24 ANGULAR MOMENTUM Instructor: Kazumi Tolich Lecture 24 2 ¨ Reading chapter 11-6 ¤ Angular momentum n Angular momentum about an axis n Newton’s 2nd law for rotational motion Angular momentum of an rotating object 3 ¨ An object with a moment of inertia of � about an axis rotates with an angular speed of � about the same axis has an angular momentum, �, given by � = �� ¤ This is analogous to linear momentum: � = �� Angular momentum in general 4 ¨ Angular momentum of a point particle about an axis is defined by � � = �� sin � = �� sin � = �-� = ��. � �- ¤ �⃗: position vector for the particle from the axis. ¤ �: linear momentum of the particle: � = �� ⃗ ¤ � is moment arm, or sometimes called “perpendicular . Axis distance.” �. Quiz: 1 5 ¨ A particle is traveling in straight line path as shown in Case A and Case B. In which case(s) does the blue particle have non-zero angular momentum about the axis indicated by the red cross? A. Only Case A Case A B. Only Case B C. Neither D. Both Case B Quiz: 24-1 answer 6 ¨ Only Case A ¨ For a particle to have angular momentum about an axis, it does not have to be Case A moving in a circle. ¨ The particle can be moving in a straight path. Case B ¨ For it to have a non-zero angular momentum, its line of path is displaced from the axis about which the angular momentum is calculated. ¨ An object moving in a straight line that does not go through the axis of rotation has an angular position that changes with time. So, this object has an angular momentum.
[Show full text]
09. Central Force Motion II Gerhard Müller University of Rhode Island, [email protected] Creative Commons License
CORE Metadata, citation and similar papers at core.ac.uk Provided by DigitalCommons@URI University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 09. Central Force Motion II Gerhard Müller University of Rhode Island, [email protected] Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. Follow this and additional works at: http://digitalcommons.uri.edu/classical_dynamics Abstract Part nine of course materials for Classical Dynamics (Physics 520), taught by Gerhard Müller at the University of Rhode Island. Entries listed in the table of contents, but not shown in the document, exist only in handwritten form. Documents will be updated periodically as more entries become presentable. Recommended Citation Müller, Gerhard, "09. Central Force Motion II" (2015). Classical Dynamics. Paper 13. http://digitalcommons.uri.edu/classical_dynamics/13 This Course Material is brought to you for free and open access by the Physics Course Materials at DigitalCommons@URI. It has been accepted for inclusion in Classical Dynamics by an authorized administrator of DigitalCommons@URI. For more information, please contact [email protected]. Contents of this Document [mtc9] 9. Central Force Motion II • Kepler’s laws of planetary motion [msl22] • Orbits of Kepler problem [msl23] • Motion in time on elliptic orbit [mln19] • Cometary motion on parabolic orbit [mex44] • Cometary motion on hyperbolic orbit [mex234] • Close encounter of the ﬁrst kind [mex145]
[Show full text]
Rotational Motion Angular Position Simple Vs. Complex Objects Speed
3/11/16 Rotational Motion Angular Position In physics we distinguish two types of motion for objects: • Translational Motion (change of location): • We will measure Whole object moves through space. angular position in radians: • Rotational Motion - object turns around an axis (axle); axis does not move. (Wheels) • Counterclockwise (CCW): positive We can also describe objects that do both at rotation the same time! • Clockwise (CW): negative rotation Linear Distance d vs. Angular Simple vs. Complex Objects distance Δθ Model motion with just Model motion with For a point at Position position and Rotation radius R on the wheel, R d = RΔθ ONLY works if Δθ is in radians! Speed in Rotational Motion Analogues Between Linear and Rotational Motion (see pg. 303) • Rotational Speed ω: rad per second • Tangential speed vt: distance per second • Two objects can have the same rotational speed, but different tangential speeds! 1 3/11/16 Relationship Between Angular and Linear Example: Gears Quantities • Displacements • Every point on the • Two wheels are rotating object has the connected by a chain s = θr same angular motion • Speeds that doesn’t slip. • Every point on the v r • Which wheel (if either) t = ω rotating object does • Accelerations not have the same has the higher linear motion rotational speed? a = αr t • Which wheel (if either) has the higher tangential speed for a point on its rim? Angular Acceleration Directions Corgi on a Carousel • If the angular acceleration and the angular • What is the carousel’s angular speed? velocity are in the same direction, the • What is the corgi’s angular speed? angular speed will increase with time.
[Show full text]