Torque and Rotational Inertia Torque
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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 ................................................................................................................................................ -
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. -
The Stress Tensor
An Internet Book on Fluid Dynamics The Stress Tensor The general state of stress in any homogeneous continuum, whether fluid or solid, consists of a stress acting perpendicular to any plane and two orthogonal shear stresses acting tangential to that plane. Thus, Figure 1: Differential element indicating the nine stresses. as depicted in Figure 1, there will be a similar set of three stresses acting on each of the three perpendicular planes in a three-dimensional continuum for a total of nine stresses which are most conveniently denoted by the stress tensor, σij, defined as the stress in the j direction acting on a plane normal to the i direction. Equivalently we can think of σij as the 3 × 3 matrix ⎡ ⎤ σxx σxy σxz ⎣ ⎦ σyx σyy σyz (Bha1) σzx σzy σzz in which the diagonal terms, σxx, σyy and σzz, are the normal stresses which would be equal to −p in any fluid at rest (note the change in the sign convention in which tensile normal stresses are positive whereas a positive pressure is compressive). The off-digonal terms, σij with i = j, are all shear stresses which would, of course, be zero in a fluid at rest and are proportional to the viscosity in a fluid in motion. In fact, instead of nine independent stresses there are only six because, as we shall see, the stress tensor is symmetric, specifically σij = σji. In other words the shear stress acting in the i direction on a face perpendicular to the j direction is equal to the shear stress acting in the j direction on a face perpendicular to the i direction. -
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. -
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. -
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. -
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. -
Unit 1: Motion
Macomb Intermediate School District High School Science Power Standards Document Physics The Michigan High School Science Content Expectations establish what every student is expected to know and be able to do by the end of high school. They also outline the parameters for receiving high school credit as dictated by state law. To aid teachers and administrators in meeting these expectations the Macomb ISD has undertaken the task of identifying those content expectations which can be considered power standards. The critical characteristics1 for selecting a power standard are: • Endurance – knowledge and skills of value beyond a single test date. • Leverage - knowledge and skills that will be of value in multiple disciplines. • Readiness - knowledge and skills necessary for the next level of learning. The selection of power standards is not intended to relieve teachers of the responsibility for teaching all content expectations. Rather, it gives the school district a common focus and acts as a safety net of standards that all students must learn prior to leaving their current level. The following document utilizes the unit design including the big ideas and real world contexts, as developed in the science companion documents for the Michigan High School Science Content Expectations. 1 Dr. Douglas Reeves, Center for Performance Assessment Unit 1: Motion Big Ideas The motion of an object may be described using a) motion diagrams, b) data, c) graphs, and d) mathematical functions. Conceptual Understandings A comparison can be made of the motion of a person attempting to walk at a constant velocity down a sidewalk to the motion of a person attempting to walk in a straight line with a constant acceleration. -
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 -
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. -
Exploring Robotics Joel Kammet Supplemental Notes on Gear Ratios
CORC 3303 – Exploring Robotics Joel Kammet Supplemental notes on gear ratios, torque and speed Vocabulary SI (Système International d'Unités) – the metric system force torque axis moment arm acceleration gear ratio newton – Si unit of force meter – SI unit of distance newton-meter – SI unit of torque Torque Torque is a measure of the tendency of a force to rotate an object about some axis. A torque is meaningful only in relation to a particular axis, so we speak of the torque about the motor shaft, the torque about the axle, and so on. In order to produce torque, the force must act at some distance from the axis or pivot point. For example, a force applied at the end of a wrench handle to turn a nut around a screw located in the jaw at the other end of the wrench produces a torque about the screw. Similarly, a force applied at the circumference of a gear attached to an axle produces a torque about the axle. The perpendicular distance d from the line of force to the axis is called the moment arm. In the following diagram, the circle represents a gear of radius d. The dot in the center represents the axle (A). A force F is applied at the edge of the gear, tangentially. F d A Diagram 1 In this example, the radius of the gear is the moment arm. The force is acting along a tangent to the gear, so it is perpendicular to the radius. The amount of torque at A about the gear axle is defined as = F×d 1 We use the Greek letter Tau ( ) to represent torque. -
Newton's Second
Newton's Second Law INTRODUCTION Sir Isaac Newton1 put forth many important ideas in his famous book The Principia. His three laws of motion are the best known of these. The first law seems to be at odds with our everyday experience. Newton's first law states that any object at rest that is not acted upon by outside forces will remain at rest, and that any object in motion not acted upon by outside forces will continue its motion in a straight line at a constant velocity. If we roll a ball across the floor, we know that it will eventually come to a stop, seemingly contradicting the First Law. Our experience seems to agree with Aristotle's2 idea, that the \impetus"3 given to the ball is used up as it rolls. But Aristotle was wrong, as is our first impression of the ball's motion. The key is that the ball does experience an outside force, i.e., friction, as it rolls across the floor. This force causes the ball to decelerate (that is, it has a \negative" acceleration). According to Newton's second law an object will accelerate in the direction of the net force. Since the force of friction is opposite to the direction of travel, this acceleration causes the object to slow its forward motion, and eventually stop. The purpose of this laboratory exercise is to verify Newton's second law. DISCUSSION OF PRINCIPLES Newton's second law in vector form is X F~ = m~a or F~net = m~a (1) This force causes the ball rolling on the floor to decelerate (that is, it has a \negative" accelera- tion).