DOCSLIB.ORG

Chapter 11 Rotational Dynamics and Static Equilibrium

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 11 Rotational Dynamics and Static Equilibrium Chapter 11 Rotational Dynamics and Static Equilibrium 1 Copyright Weining Man Units of Chapter 11 • Torque • Torque and Angular Acceleration • Zero Torque and Static Equilibrium • Center of Mass and Balance • Dynamic Applications of Torque • Angular Momentum 2 Copyright Weining Man Units of Chapter 11 Optional Not required • Conservation of Angular Momentum • Rotational Work and Power • The Vector Nature of Rotational Motion 3 Copyright Weining Man 11-1 Torque From experience, we know that the same force will be much more effective at rotating an object such as a nut or a door if our hand is not too close to the axis. This is why we have long-handled wrenches, and why doorknobs are not next to hinges. 4 Copyright Weining Man 11-1 Torque We define a quantity called torque: The torque increases as the force increases, and also as the distance increases. r is moment arm length 5 Copyright Weining Man 11-1 Torque Only the tangential component of force causes a torque: 6 Copyright Weining Man 11-1 Torque This leads to a more general definition of torque: is called moment arm length, it’s the perpendicular distance between the rotation center to the force line.7 Copyright Weining Man r is the moment arm length. To use longer r, can saves force. Examples: Level, bottle opener, dolly, wrench, screw driver…. 8 Copyright Weining Man 11-1 Torque If the torque causes a counterclockwise angular acceleration, it is positive; if it causes a clockwise angular acceleration, it is negative. 9 Copyright Weining Man 11-3 Zero Torque and Static Equilibrium Static equilibrium occurs when an object is at rest – neither rotating nor translating. Total torque is zero respect to any rotational axis. 10 Copyright Weining Man 11-3 Zero Torque and Static Equilibrium If the net torque is zero, it doesn’t matter which axis we consider rotation to be around; we are free to choose the one that makes our calculations easiest. 11 Copyright Weining Man |τccw| = |τcw| When you consider SIZE only, use the absolute value. CCW torque=CW torque When you consider total torque = 0, Ccw torque are positive Cw torque are negative. 12 Copyright Weining Man 13 Copyright Weining Man 1, label forces, 2, choose rotational axis 3, write out torque created by each forces. Use the correct moment arm length for each force. If a force is going through the rotational axis, its torque=0 4, use equation total torque = 0, To solve bicep force, 5, Finally use equation total forces =0, to solve the unknown bone force. Will the bicep force be more or less than Mg+mg? Will the bone force point upward or downward? What if the bone becomes a soft rope? Equations: 14 Copyright Weining Man 11-4 Center of Mass and Balance If an extended object is to be balanced, it must be supported through its center of mass. 15 Copyright Weining Man 11-4 Center of Mass and Balance This fact can be used to find the center of mass of an object – suspend it from different axes and trace a vertical line. The center of mass is where the lines meet. 16 Copyright Weining Man 17 Copyright Weining Man 18 Copyright Weining Man 19 Copyright Weining Man 20 Copyright Weining Man 11-2 Torque and Angular Acceleration Newton’s second law: If we consider a mass m rotating around an axis a distance r away, we can reformat Newton’s second law to read: Or equivalently, 21 Copyright Weining Man 11-2 Torque and Angular Acceleration Once again, we have analogies between linear and angular motion: =∑mr2 = at/r = Fr s θ =s /r v ω=vt /r p=mv L =I ω KE=1/2 mv2 KE =1/2 I ω2 22 Copyright Weining Man Example: open your arm and let it fall without applying any muscle force. What will be the angular acceleration for your arm due to its gravity at that moment? What will be the linear acceleration of your finger tip at that moment? Will your finger tip fall faster, or slower than free fall? put a small object at your finger tip to compare. Will your upper arm fall faster or slower than free fall? The length of the whole arm = L; Total arm mass = m Choose the shoulder to be the rotation axis, Total torque exerted on the arm: L Assume the whole arm as a long rod of block, When rotating around its end(the shoulder) Arm’s rotational inertia: mg At that moment,the angular acceleration : Fingertip linear acceleration: Arm center linear acceleration: 23 Fingertip fall (rotates) faster than free fall. Upper arm falls slower than free fall. Copyright Weining Man 11-5 Dynamic Applications of Torque When dealing with systems that have both rotating parts and translating parts, we must be careful to account for all forces and torques correctly. 24 Copyright Weining Man 11-6 Angular Momentum Using a bit of algebra, we find for a particle moving in a circle of radius r, 25 Copyright Weining Man 11-7 Conservation of Angular Momentum If the net external torque on a system is zero, the angular momentum is conserved. The most interesting consequences occur in systems that are able to change shape: If < Ii So that, ωf > ωi As the moment of inertia decreases, the angular speed increases, so the angular momentum does not change. 26 Copyright Weining Man Summary of Chapter 11 • A force applied so as to cause an angular acceleration is said to exert a torque. • Torque due to a tangential force: • Torque in general: r is the moment arm length. Longer r, saves force. (but longer r means to apply force across longer distance. It saves forces, but nothing can save work. • Newton’s second law for rotation: Angular acceleration = total torque / rotational inertia 27 Copyright Weining Man Once again, we have analogies between linear and angular motion: =∑mr2 = at/r = Fr s θ =s /r v ω=vt /r p=mv L =I ω KE=1/2 mv2 KE =1/2 I ω2 28 Copyright Weining Man Summary of Chapter 11 • In order for an object to be in static equilibrium, the total force and the total torque acting on the object must be zero. (most important) • An object balances when it is supported at its center of mass. • In systems with both rotational and linear motion, Newton’s second law must be applied separately to each. • Angular momentum: •In systems with no external torque, total angular momentum is conserved. 29 Copyright Weining Man 11-7 Conservation of Angular Momentum Optional Planet’s elliptical orbit: Rotating around the sun Gravity of the force going through the rotational center (the sun) hence doesn’t create torque on the planet to change its angular momentum. Iω will be constant, hence mr2ω will be constant. After the same amount of time t, the area swept by the line between a planet and the sun Area=r2∆θ=r2ωt, will be the same. 30 Copyright Weining Man 11-7 Conservation of Angular Momentum Optional Angular momentum is also conserved in rotational collisions: 31 Copyright Weining Man 11-6 Angular Momentum Optional Looking at the rate at which angular momentum changes, 32 Copyright Weining Man 11-8 Rotational Work and Power Optional Not Require A torque acting through an angular displacement does work, just as a force acting through a distance does. The work-energy theorem applies as usual. 33 Copyright Weining Man 11-8 Rotational Work and Power Optional Not Require Power is the rate at which work is done, for rotational motion as well as for translational motion. Again, note the analogy to the linear form: 34 Copyright Weining Man 11-9 The Vector Nature of Rotational Motion, Optional Not Require The direction of the angular velocity vector is along the axis of rotation. A right-hand rule gives the sign. 35 Copyright Weining Man 11-9 The Vector Nature of Rotational Motion, Optional Not Require Conservation of angular momentum means that the total angular momentum around any axis must be constant. This is why gyroscopes are so stable. 36 Copyright Weining Man.
Load more

Recommended publications
  • The Experimental Determination of the Moment of Inertia of a Model Airplane Michael Koken [email protected]
    The University of Akron IdeaExchange@UAkron The Dr. Gary B. and Pamela S. Williams Honors Honors Research Projects College Fall 2017 The Experimental Determination of the Moment of Inertia of a Model Airplane Michael Koken [email protected] Please take a moment to share how this work helps you through this survey. Your feedback will be important as we plan further development of our repository. Follow this and additional works at: http://ideaexchange.uakron.edu/honors_research_projects Part of the Aerospace Engineering Commons, Aviation Commons, Civil and Environmental Engineering Commons, Mechanical Engineering Commons, and the Physics Commons Recommended Citation Koken, Michael, "The Experimental Determination of the Moment of Inertia of a Model Airplane" (2017). Honors Research Projects. 585. http://ideaexchange.uakron.edu/honors_research_projects/585 This Honors Research Project is brought to you for free and open access by The Dr. Gary B. and Pamela S. Williams Honors College at IdeaExchange@UAkron, the institutional repository of The nivU ersity of Akron in Akron, Ohio, USA. It has been accepted for inclusion in Honors Research Projects by an authorized administrator of IdeaExchange@UAkron. For more information, please contact [email protected], [email protected]. 2017 THE EXPERIMENTAL DETERMINATION OF A MODEL AIRPLANE KOKEN, MICHAEL THE UNIVERSITY OF AKRON Honors Project TABLE OF CONTENTS List of Tables ................................................................................................................................................
    [Show full text]
  • 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]
  • Pierre Simon Laplace - Biography Paper
    Pierre Simon Laplace - Biography Paper MATH 4010 Melissa R. Moore University of Colorado- Denver April 1, 2008 2 Many people contributed to the scientific fields of mathematics, physics, chemistry and astronomy. According to Gillispie (1997), Pierre Simon Laplace was the most influential scientist in all history, (p.vii). Laplace helped form the modern scientific disciplines. His techniques are used diligently by engineers, mathematicians and physicists. His diverse collection of work ranged in all fields but began in mathematics. Laplace was born in Normandy in 1749. His father, Pierre, was a syndic of a parish and his mother, Marie-Anne, was from a family of farmers. Many accounts refer to Laplace as a peasant. While it was not exactly known the profession of his Uncle Louis, priest, mathematician or teacher, speculations implied he was an educated man. In 1756, Laplace enrolled at the Beaumont-en-Auge, a secondary school run the Benedictine order. He studied there until the age of sixteen. The nest step of education led to either the army or the church. His father intended him for ecclesiastical vocation, according to Gillispie (1997 p.3). In 1766, Laplace went to the University of Caen to begin preparation for a career in the church, according to Katz (1998 p.609). Cristophe Gadbled and Pierre Le Canu taught Laplace mathematics, which in turn showed him his talents. In 1768, Laplace left for Paris to pursue mathematics further. Le Canu gave Laplace a letter of recommendation to d’Alembert, according to Gillispie (1997 p.3). Allegedly d’Alembert gave Laplace a problem which he solved immediately.
    [Show full text]
  • Magnetism, Angular Momentum, and Spin
    Chapter 19 Magnetism, Angular Momentum, and Spin P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin In 1820 Hans Christian Ørsted discovered that electric current produces a magnetic field that deflects compass needle from magnetic north, establishing first direct connection between fields of electricity and magnetism. P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Biot-Savart Law Jean-Baptiste Biot and Félix Savart worked out that magnetic field, B⃗, produced at distance r away from section of wire of length dl carrying steady current I is 휇 I d⃗l × ⃗r dB⃗ = 0 Biot-Savart law 4휋 r3 Direction of magnetic field vector is given by “right-hand” rule: if you point thumb of your right hand along direction of current then your fingers will curl in direction of magnetic field. current P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Microscopic Origins of Magnetism Shortly after Biot and Savart, Ampére suggested that magnetism in matter arises from a multitude of ring currents circulating at atomic and molecular scale. André-Marie Ampére 1775 - 1836 P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Magnetic dipole moment from current loop Current flowing in flat loop of wire with area A will generate magnetic field magnetic which, at distance much larger than radius, r, appears identical to field dipole produced by point magnetic dipole with strength of radius 휇 = | ⃗휇| = I ⋅ A current Example What is magnetic dipole moment induced by e* in circular orbit of radius r with linear velocity v? * 휋 Solution: For e with linear velocity of v the time for one orbit is torbit = 2 r_v.
    [Show full text]
  • M1=100 Kg Adult, M2=10 Kg Baby. the Seesaw Starts from Rest. Which Direction Will It Rotates?
    m1 m2 m1=100 kg adult, m2=10 kg baby. The seesaw starts from rest. Which direction will it rotates? (a) Counter-Clockwise (b) Clockwise ()(c) NttiNo rotation (d) Not enough information Effect of a Constant Net Torque 2.3 A constant non-zero net torque is exerted on a wheel. Which of the following quantities must be changing? 1. angular position 2. angular velocity 3. angular acceleration 4. moment of inertia 5. kinetic energy 6. the mass center location A. 1, 2, 3 B. 4, 5, 6 C. 1,2, 5 D. 1, 2, 3, 4 E. 2, 3, 5 1 Example: second law for rotation PP10601-50: A torque of 32.0 N·m on a certain wheel causes an angular acceleration of 25.0 rad/s2. What is the wheel's rotational inertia? Second Law example: α for an unbalanced bar Bar is massless and originally horizontal Rotation axis at fulcrum point L1 N L2 Î N has zero torque +y Find angular acceleration of bar and the linear m1gmfulcrum 2g acceleration of m1 just after you let go τnet Constraints: Use: τnet = Itotα ⇒ α = Itot 2 2 Using specific numbers: where: Itot = I1 + I2 = m1L1 + m2L2 Let m1 = m2= m L =20 cm, L = 80 cm τnet = ∑ τo,i = + m1gL1 − m2gL2 1 2 θ gL1 − gL2 g(0.2 - 0.8) What happened to sin( ) in moment arm? α = 2 2 = 2 2 L1 + L2 0.2 + 0.8 2 net = − 8.65 rad/s Clockwise torque m gL − m gL a ==+ -α L 1.7 m/s2 α = 1 1 2 2 11 2 2 Accelerates UP m1L1 + m2L2 total I about pivot What if bar is not horizontal? 2 See Saw 3.1.
    [Show full text]
  • Influence of Angular Velocity of Pedaling on the Accuracy of The
    Research article 2018-04-10 - Rev08 Influence of Angular Velocity of Pedaling on the Accuracy of the Measurement of Cyclist Power Abstract Almost all cycling power meters currently available on the The miscalculation may be—even significantly—greater than we market are positioned on rotating parts of the bicycle (pedals, found in our study, for the following reasons: crank arms, spider, bottom bracket/hub) and, regardless of • the test was limited to only 5 cyclists: there is no technical and construction differences, all calculate power on doubt other cyclists may have styles of pedaling with the basis of two physical quantities: torque and angular velocity greater variations of angular velocity; (or rotational speed – cadence). Both these measures vary only 2 indoor trainer models were considered: other during the 360 degrees of each revolution. • models may produce greater errors; The torque / force value is usually measured many times during slopes greater than 5% (the only value tested) may each rotation, while the angular velocity variation is commonly • lead to less uniform rotations and consequently neglected, considering only its average value for each greater errors. revolution (cadence). It should be noted that the error observed in this analysis This, however, introduces an unpredictable error into the power occurs because to measure power the power meter considers calculation. To use the average value of angular velocity means the average angular velocity of each rotation. In power meters to consider each pedal revolution as perfectly smooth and that use this type of calculation, this error must therefore be uniform: but this type of pedal revolution does not exist in added to the accuracy stated by the manufacturer.
    [Show full text]
  • Basement Flood Mitigation
    1 Mitigation refers to measures taken now to reduce losses in the future. How can homeowners and renters protect themselves and their property from a devastating loss? 2 There are a range of possible causes for basement flooding and some potential remedies. Many of these low-cost options can be factored into a family’s budget and accomplished over the several months that precede storm season. 3 There are four ways water gets into your basement: Through the drainage system, known as the sump. Backing up through the sewer lines under the house. Seeping through cracks in the walls and floor. Through windows and doors, called overland flooding. 4 Gutters can play a huge role in keeping basements dry and foundations stable. Water damage caused by clogged gutters can be severe. Install gutters and downspouts. Repair them as the need arises. Keep them free of debris. 5 Channel and disperse water away from the home by lengthening the run of downspouts with rigid or flexible extensions. Prevent interior intrusion through windows and replace weather stripping as needed. 6 Many varieties of sturdy window well covers are available, simple to install and hinged for easy access. Wells should be constructed with gravel bottoms to promote drainage. Remove organic growth to permit sunlight and ventilation. 7 Berms and barriers can help water slope away from the home. The berm’s slope should be about 1 inch per foot and extend for at least 10 feet. It is important to note permits are required any time a homeowner alters the elevation of the property.
    [Show full text]
  • Simple Harmonic Motion
    [SHIVOK SP211] October 30, 2015 CH 15 Simple Harmonic Motion I. Oscillatory motion A. Motion which is periodic in time, that is, motion that repeats itself in time. B. Examples: 1. Power line oscillates when the wind blows past it 2. Earthquake oscillations move buildings C. Sometimes the oscillations are so severe, that the system exhibiting oscillations break apart. 1. Tacoma Narrows Bridge Collapse "Gallopin' Gertie" a) http://www.youtube.com/watch?v=j‐zczJXSxnw II. Simple Harmonic Motion A. http://www.youtube.com/watch?v=__2YND93ofE Watch the video in your spare time. This professor is my teaching Idol. B. In the figure below snapshots of a simple oscillatory system is shown. A particle repeatedly moves back and forth about the point x=0. Page 1 [SHIVOK SP211] October 30, 2015 C. The time taken for one complete oscillation is the period, T. In the time of one T, the system travels from x=+x , to –x , and then back to m m its original position x . m D. The velocity vector arrows are scaled to indicate the magnitude of the speed of the system at different times. At x=±x , the velocity is m zero. E. Frequency of oscillation is the number of oscillations that are completed in each second. 1. The symbol for frequency is f, and the SI unit is the hertz (abbreviated as Hz). 2. It follows that F. Any motion that repeats itself is periodic or harmonic. G. If the motion is a sinusoidal function of time, it is called simple harmonic motion (SHM).
    [Show full text]
  • Hub City Powertorque® Shaft Mount Reducers
    Hub City PowerTorque® Shaft Mount Reducers PowerTorque® Features and Description .................................................. G-2 PowerTorque Nomenclature ............................................................................................ G-4 Selection Instructions ................................................................................ G-5 Selection By Horsepower .......................................................................... G-7 Mechanical Ratings .................................................................................... G-12 ® Shaft Mount Reducers Dimensions ................................................................................................ G-14 Accessories ................................................................................................ G-15 Screw Conveyor Accessories ..................................................................... G-22 G For Additional Models of Shaft Mount Reducers See Hub City Engineering Manual Sections F & J DOWNLOAD AVAILABLE CAD MODELS AT: WWW.HUBCITYINC.COM Certified prints are available upon request EMAIL: [email protected] • www.hubcityinc.com G-1 Hub City PowerTorque® Shaft Mount Reducers Ten models available from 1/4 HP through 200 HP capacity Manufacturing Quality Manufactured to the highest quality 98.5% standards in the industry, assembled Efficiency using precision manufactured components made from top quality per Gear Stage! materials Designed for the toughest applications in the industry Housings High strength ductile
    [Show full text]
  • Chapter 3 Motion in Two and Three Dimensions
    Chapter 3 Motion in Two and Three Dimensions 3.1 The Important Stuff 3.1.1 Position In three dimensions, the location of a particle is specified by its location vector, r: r = xi + yj + zk (3.1) If during a time interval ∆t the position vector of the particle changes from r1 to r2, the displacement ∆r for that time interval is ∆r = r1 − r2 (3.2) = (x2 − x1)i +(y2 − y1)j +(z2 − z1)k (3.3) 3.1.2 Velocity If a particle moves through a displacement ∆r in a time interval ∆t then its average velocity for that interval is ∆r ∆x ∆y ∆z v = = i + j + k (3.4) ∆t ∆t ∆t ∆t As before, a more interesting quantity is the instantaneous velocity v, which is the limit of the average velocity when we shrink the time interval ∆t to zero. It is the time derivative of the position vector r: dr v = (3.5) dt d = (xi + yj + zk) (3.6) dt dx dy dz = i + j + k (3.7) dt dt dt can be written: v = vxi + vyj + vzk (3.8) 51 52 CHAPTER 3. MOTION IN TWO AND THREE DIMENSIONS where dx dy dz v = v = v = (3.9) x dt y dt z dt The instantaneous velocity v of a particle is always tangent to the path of the particle. 3.1.3 Acceleration If a particle’s velocity changes by ∆v in a time period ∆t, the average acceleration a for that period is ∆v ∆v ∆v ∆v a = = x i + y j + z k (3.10) ∆t ∆t ∆t ∆t but a much more interesting quantity is the result of shrinking the period ∆t to zero, which gives us the instantaneous acceleration, a.
    [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]
  • Finite Element Model Calibration of an Instrumented Thirteen-Story Steel Moment Frame Building in South San Fernando Valley, California
    Finite Element Model Calibration of An Instrumented Thirteen-story Steel Moment Frame Building in South San Fernando Valley, California By Erol Kalkan Disclamer The finite element model incuding its executable file are provided by the copyright holder "as is" and any express or implied warranties, including, but not limited to, the implied warranties of merchantability and fitness for a particular purpose are disclaimed. In no event shall the copyright owner be liable for any direct, indirect, incidental, special, exemplary, or consequential damages (including, but not limited to, procurement of substitute goods or services; loss of use, data, or profits; or business interruption) however caused and on any theory of liability, whether in contract, strict liability, or tort (including negligence or otherwise) arising in any way out of the use of this software, even if advised of the possibility of such damage. Acknowledgments Ground motions were recorded at a station owned and maintained by the California Geological Survey (CGS). Data can be downloaded from CESMD Virtual Data Center at: http://www.strongmotioncenter.org/cgi-bin/CESMD/StaEvent.pl?stacode=CE24567. Contents Introduction ..................................................................................................................................................... 1 OpenSEES Model ........................................................................................................................................... 3 Calibration of OpenSEES Model to Observed Response
    [Show full text]