DOCSLIB.ORG

Construction and Characterization of a Single Stage Dual Diaphragm Gas Gun Nathaniel Steven Helminiak Marquette University

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Construction and Characterization of a Single Stage Dual Diaphragm Gas Gun Nathaniel Steven Helminiak Marquette University Marquette University e-Publications@Marquette Master's Theses (2009 -) Dissertations, Theses, and Professional Projects Construction and Characterization of a Single Stage Dual Diaphragm Gas Gun Nathaniel Steven Helminiak Marquette University Recommended Citation Helminiak, Nathaniel Steven, "Construction and Characterization of a Single Stage Dual Diaphragm Gas Gun" (2017). Master's Theses (2009 -). 441. http://epublications.marquette.edu/theses_open/441 - 1 - CONSTRUCTION AND CHARACTERIZATION OF A SINGLE STAGE DUAL DIAPHRAGM GAS GUN by Nathaniel Steven Helminiak, B.S., E.I.T. A Thesis submitted to the Faculty of the Graduate School, Marquette University, in Partial Fulfillment of the Requirements for the Degree of Master of Science Milwaukee, Wisconsin December 2017 ABSTRACT CONSTRUCTION AND CHARACTERIZATION OF A SINGLE STAGE DUAL DIAPHRAGM GAS GUN Nathaniel Steven Helminiak, B.S., E.I.T. Marquette University, 2017 In the interest of studying the propagation of shock waves, this work sets out to design, construct, and characterize a pneumatic accelerator that performs high-velocity flyer plate impact tests. A single stage gas gun with a dual diaphragm breach allows for a non-volatile, reliable experimental testing platform for shock phenomena. This remotely operated gas gun utilizes compressed nitrogen to launch projectiles down a 14 foot long, 2 inch diameter bore barrel, which subsequently impacts a target material of interest. A dual diaphragm firing mechanism allows the 4.5 liter breech to reach a total pressure differential of 10ksi before accelerating projectiles to velocities as high as 1,000 m/s (1570-2240 mph). The projectile’s velocity is measured using a series of break pin circuits. The target response can be measured with Photon Doppler Velocimetry (PDV) and/or stress gauge system. A vacuum system eliminates the need for pressure relief in front of the projectile, while additionally allowing the system to remain closed over the entire firing cycle. Characterization of the system will allow for projectile speed to be estimated prior to launching based on initial breach pressure. i ACKNOWLEDGMENTS I submit this thesis to the public for review with the utmost thanks to God, my family, mentors, friends, and sponsors for their help and support throughout the process of generating this work. I am truly grateful for the unconditional love of my family, as there is no way that I would have made it this far without their support. Further, Dr. John Borg, my advisor for the majority of my higher academic career has provided an education that extends beyond pure scholasticism and has informed some of the greatest choices in my life to date. I can think of few thesis topics quite as extraordinary as the construction and characterization of a gas gun. My thanks are also extended to the members of my committee: Dr. Casey Allen, Dr. Anthony Bowman, Dr. Johnathan Fleischmann, and Dr. Hyunjae Park, for their mentorship and commitment to my education since my arrival in the undergraduate program at Marquette University. They, in addition to the other professors and professionals, working within the Opus College of Engineering have each provided innumerable opportunities to grow and develop. My gratitude goes out to the members of the shock physics lab, especially Jeff Lajeunesse, Merit Schumaker, Peter Sable, Chris Johnson, Logan Beaver, Ashley Hatzenbihler, and Emilie Teitz. Finally, I thank the Defense Threat Reduction Agency and by extension the United States of America for providing the means and freedoms to pursue my education. From Proverbs 25:2, “It is the glory of God to conceal a matter; to search out a matter is the glory of kings.” My opportunity to pursue the glory of kings has been provided in hours of expertise and patience on the part of many people. I continue to pray that I someday might be able to provide to others the same gifts others have shared with myself. -Nathaniel ϟteven Helminiak ii TABLE OF CONTENTS 1. INTRODUCTION......................................................................................................1 1.1 Motivation ........................................................................................................... 1 1.2 History and Review ............................................................................................ 2 1.2.1 Ballistics in History: A Path to Gas Gun Technology and Shock Physics ... 2 1.2.2 A Brief History of Shock Physics and Numerical Tools ............................. 15 1.2.3 The Gas Gun ............................................................................................... 18 1.2.4 Marquette’s History of Shock and Gas Guns ............................................. 32 2. CONSTRUCTION ...................................................................................................54 2.1 Static Gun Components ................................................................................... 55 2.1.1 Breach ......................................................................................................... 55 2.1.2 Barrel .......................................................................................................... 70 2.1.3 Sabot ........................................................................................................... 73 2.1.4 Target Tank ................................................................................................. 77 2.1.5 Catch Tank .................................................................................................. 87 2.2 Active Gun Components .................................................................................. 96 2.2.1 Control System ............................................................................................ 96 2.2.2 Pressurization System ............................................................................... 108 2.2.3 Distribution System ................................................................................... 113 2.2.4 Dual Diaphragm Launch System .............................................................. 117 2.2.5 Light Gate Assembly ................................................................................. 120 2.2.6 Photon Doppler Velocimetry and Dynasen Pin System ............................ 124 2.2.7 Vacuum System ......................................................................................... 136 2.2.8 Some Notes on Machining ........................................................................ 141 3. EXPERIMENTAL SETUP ...................................................................................147 3.1 Firing Procedure and Safety Protocol .......................................................... 148 4. THEORETICAL VS. EXPERIMENTAL GUN BEHAVIOR ..........................157 4.1 Breach Charging Time................................................................................... 157 4.2 Burst Disk Failure .......................................................................................... 160 iii 4.3 Taylor Testing ................................................................................................. 175 4.4 Catch Tank Pull Down ................................................................................... 178 4.5 Muzzle Velocity .............................................................................................. 186 4.6 Post Shot Pressure .......................................................................................... 212 4.7 Gun Recoil ....................................................................................................... 215 5. CONCLUSIONS ....................................................................................................216 5.1 Future Work ................................................................................................... 216 5.1.1 Gas Gun Improvements............................................................................. 216 5.1.2 Reinforcement of the Impact Plate and Target Tank ................................ 216 5.1.3 Target Alignment ...................................................................................... 217 5.1.4 Examination of Surface Finishes for use with PDV.................................. 222 5.1.5 Use of CTH and Shock Wave Theory for Experimental Design ............... 223 5.1.6 Experimental Testing ................................................................................ 224 5.2 Closing ............................................................................................................. 227 6. WORKS CITED.....................................................................................................229 Appendix A: Catch Tank Dimensions .........................................................................244 Appendix B: MATLAB Program for Anticipated Experimental Characteristics ..245 Appendix C: MATLAB Program for Non-dimensional Pressure Velocity Profiles 248 Appendix D: MATLAB Program for Agilent 602s Vacuum Curves ........................255 Appendix E: MATLAB Program for Light Gate Interpretation Single .csv file .....261 Appendix F: MATLAB Program for PDV Interpretation ........................................267 Appendix G: MATLAB Program for Burst Disk Selection .......................................271 Appendix H: CTH Flyer Plate Simulation ..................................................................275 iv LIST OF TABLES Table 2.1 Mechanical Properties of AISI 4135 Steel [142] .....................................
Load more

Recommended publications
  • Ballistic Pendulum
    PH 1113: Ballistic Pendulum Ballistic Pendulum Objective The purpose of this experiment1 is to determine the muzzle velocity of a projectile launcher, using the conservation of momentum and the conservation of energy (where applicable). Materials 1. 1-meter stick 4. Projectile (metal ball) 2. 30-cm ruler 5. Triple-beam balance 3. Ballistic pendulum machine Introduction The ballistic pendulum is a classic method for determining the velocity of a projectile; origi- nally, it was used to determine the muzzle velocity of firearms. It is also a good demonstration of many basic principles of physics. In ballistic pendula experiments a projectile (in our case a ball) is fired into a stationary, freely hanging pendulum, which swings up by a measured amount after being impacted by the projectile. From that measured angle and from the length of the pendulum, you can determine the height reached by the center of mass of the pendulum; then you can calculate its gravitational potential energy. We know that mechanical energy (ME) is conserved, so we know that the gravitational potential energy at the maximum height must be equal to the kinetic energy of the pendulum at the bottom of the swing, just after the ball collides with the pendulum. In symbolic terms, MEi = MEf ; (1) where ME = PE + KE. So, 0 0 ¨¨* ¨¨* ¨PEi + KEi = PEf + ¨KE¨f (2) 1Some of the text of this manual was taken from the Pasco Ballistic Pendulum/Projectile Launcher Instruction Manual. 1 Mississippi State University Department of Physics and Astronomy You cannot, however, equate the kinetic energy of the pendulum after the collision with the kinetic energy of the ball before the swing, because the collision between ball and pendulum is inelastic, and kinetic energy is not conserved in inelastic collisions.
    [Show full text]
  • Ballistic Pendulum March 2021 Lancaster/Basnet/Brown
    Ballistic Pendulum March 2021 Lancaster/Basnet/Brown 1 Introduction This lab simulation explores the concept of conservation of energy by way of a ballistic pendulum and a projectile. 1.1 Tools This lab requires the use of the Google Chrome browser. Make certain it is installed on your computer. Navigate to http://physics.bu.edu/~duffy/HTML5/ballistic_pendulum.html to use the simulation. Intall the following Google Chrome extensions: Measure-it and Web Paint. They can be found at: https://chrome.google.com/webstore/detail/measure-it/jocbgkoackihphodedlefohapackjmna? hl=en https://chrome.google.com/webstore/detail/web-paint/emeokgokialpjadjaoeiplmnkjoaegng? hl=en. 1.2 Relevant Equations Kinetic energy: 1 K = mv2 (1) 2 Gravitational potential energy: U = mgh (2) Total energy: 1 E = K + U = mv2 + mgh (3) T otal 2 The velocity of the bullet-pendulum system immediately after the collision is: mb v = vb( ) (4) mb + M Where mb and vb are the mass and initial velocity of the bullet, and M is the mass of the ball hanging at the end of the pendulum. The kinetic energy of the bullet-pendulum system immediately after the collision is: 2 2 1 mb 2 2 1 mb vb K = (mb + M)( ) vb = (5) 2 mb + M 2 (mb + M) 2 Procedure Note: This simulation moves fast. For this reason, we'll take data using successive presses of the "step" button. Another note: For the purposes of this lab, assume that the initial gravitational potential of the pendulum is 0. 1 Ballistic Pendulum March 2021 Lancaster/Basnet/Brown 2.1 Part I: Initial Measurements 1.
    [Show full text]
  • The Dawn of Fluid Dynamics a Discipline Between Science and Technology
    Titelei Eckert 11.04.2007 14:04 Uhr Seite 3 Michael Eckert The Dawn of Fluid Dynamics A Discipline between Science and Technology WILEY-VCH Verlag GmbH & Co. KGaA Titelei Eckert 11.04.2007 14:04 Uhr Seite 1 Michael Eckert The Dawn of Fluid Dynamics A Discipline between Science and Technology Titelei Eckert 11.04.2007 14:04 Uhr Seite 2 Related Titles R. Ansorge Mathematical Models of Fluiddynamics Modelling, Theory, Basic Numerical Facts - An Introduction 187 pages with 30 figures 2003 Hardcover ISBN 3-527-40397-3 J. Renn (ed.) Albert Einstein - Chief Engineer of the Universe 100 Authors for Einstein. Essays approx. 480 pages 2005 Hardcover ISBN 3-527-40574-7 D. Brian Einstein - A Life 526 pages 1996 Softcover ISBN 0-471-19362-3 Titelei Eckert 11.04.2007 14:04 Uhr Seite 3 Michael Eckert The Dawn of Fluid Dynamics A Discipline between Science and Technology WILEY-VCH Verlag GmbH & Co. KGaA Titelei Eckert 11.04.2007 14:04 Uhr Seite 4 The author of this book All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and Dr. Michael Eckert publisher do not warrant the information Deutsches Museum München contained in these books, including this book, to email: [email protected] be free of errors. Readers are advised to keep in mind that statements, data, illustrations, proce- Cover illustration dural details or other items may inadvertently be “Wake downstream of a thin plate soaked in a inaccurate. water flow” by Henri Werlé, with kind permission from ONERA, http://www.onera.fr Library of Congress Card No.: applied for British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library.
    [Show full text]
  • Ballistic Pendulum Is a Wonderful Way of Demonstrating Both the Constant Horizontal Velocity in Projectile Motion and the Conservation of Momentum
    Conservation of Momentum Objective: To study the conservation of energy and momentum using projectile motion. Theory: The ballistic pendulum is a wonderful way of demonstrating both the constant horizontal velocity in projectile motion and the conservation of momentum. Because there is no acceleration in the horizontal direction, the horizontal component vx of the projectile’s velocity remains unchanged from its initial value throughout the motion. In a closed isolated system if no net external force acts on a system of particles, the total linear momentum of the system cannot change. There are two simple types of collisions, elastic and inelastic. If the total kinetic energy of the two systems is conserved then the collision is known as elastic. If the kinetic energy is not conserved then the collision is inelastic. Apparatus: PASCO Ballistic pendulum, meter stick, level, carbon paper, sheet of white paper, plumb bob, clamp. Procedure: Warning: Do not under any circumstance look into the barrel of the launcher. The people carrying out the experiment should wear safety googles at all times. Part 1: Projectile Motion – The goal is to determine the initial velocity vx of the ball 1. Make sure the pendulum is secured by the clamp and does not slide on the table. 2. Lift the pendulum arm until it is fully horizontal so that the ball can freely fire from the spring gun. 3. Set the apparatus near the edge of a table and level the apparatus. If the plump bob is fully vertical and overlapping with the 0 degree line it means it is properly leveled.
    [Show full text]
  • Conservation of Linear Momentum: the Ballistic Pendulum
    Conservation of Linear Momentum: the Ballistic Pendulum I. Discussion a. Determination of Velocity from Collision In the typical use made of a ballistic pendulum, a projectile, having a small mass, m, and a horizontal velocity, v, strikes and imbeds itself in a pendulum bob, having a large mass, M, and an initial horizontal velocity of zero. Immediately after the collision occurs the combined mass, M + m, of the bob and projectile has a small horizontal velocity, V. During this collision the conservation of linear momentum holds. If rotational effects are ignored, and if it is assumed that the time required for this event to take place is too short to allow the projectile to be substantially influenced by the earth's gravitational field, the motion remains horizontal, and mv = ( M + m ) V (1) After the collision has occurred, the combined mass, (M + m), rises along a circular arc to a vertical height h in the earth's gravitational field, as shown in Fig. l (c). During this portion of the motion, but not during the collision itself, the conservation of energy holds, if it is assumed that the effects of friction are negligible. Then, for motion along the arc, 1 2 ( M + m ) V = ( M + m) gh , or (2) 2 V = 2 gh (3) When V is eliminated using equations 1 and 3, the velocity of the projectile is v = M + m 2 gh (4) m b. Determination of Velocity from Range When a projectile, having an initial horizontal velocity, v, is fired from a height H above a plane, its horizontal range, R, is given by R = v t x o (5) and the height H is given by 1 2 H = gt (6) 2 where t = time during which projectile is in flight.
    [Show full text]
  • Backyard Ballistics by Gurstelle.Pdf
    Build potato cannons, paper match rockets, Cincinnati fire kites, tennis ball mortars, and more dynamite devices "One is tempted to dub it 'the official manual for real boys'!" —DAVA SOBEL, author of Longitude and Galileo's Daughter William Gurstelle POPULAR SCIENCE/HOBBY What happens when you duct-tape a couple of potato chip tubes together, then add an energy source, a tennis ball, and a match? Well, not much—unless you know the secret to building the fabled tennis ball mortar. This step-by-step guide enables ordinary folks to construct 13 awesome ballistic devices using inexpensive household or hardware store materials. Clear instructions, diagrams, and photographs show how to build projects ranging from the simple—a match-powered rocket—to the more complex—a tabletop catapult—to the classic—the infamous potato cannon—to the offbeat—a Cincinnati fire kite. With a strong emphasis on safety, Backyard Ballistics also provides troubleshooting tips, explains the physics behind each project, and profiles such scientists and extra- ordinary experimenters as Alfred Nobel, Robert Goddard, and Isaac Newton, among others. This book will be indispensable for the legions of backyard toy-rocket launchers and fireworks fanatics who wish every day were the Fourth of July. WILLIAM GURSTELLE is a professional engineer who has designed, constructed, and collected ballistics experiments for over 20 years. $16.95 (CAN $25.95) ISBN 1-55652-375-0 Distributed by Independent Publishers Group www.ipgbook.com 9 781556 523755 BACKYARD BALLISTICS Build potato Cannons, paper match rockets, Cincinnati fire kites, tennis ball mortars, and more dynamite devices William Gurstelle Library of Congress Cataloging-in-Publication Data Gurstelle, William.
    [Show full text]
  • Chapter 4 Energy and Momentum
    Chapter 4 Energy and Momentum - Ballistic Pendulum 4.1 Purpose . In this experiment, energy conservation and momentum conservation will be investigated with the ballistic pendulum. 4.2 Introduction One of the basic underlying principles in all of physics is the concept that the total energy of a system is always conserved. The energy can change forms (i.e. kinetic energy, potential energy, heat, etc.), but the sum of all of these forms of energy must stay constant unless energy is added or removed from the system. This property has enormous consequences, from describing simple projectile motion to deciding the ultimate fate of the universe. Another important conserved quantity is momentum. Momentum is especially important when one considers the collision between to objects. We generally divide collisions into two types: elastic and inelastic. In an inelastic collision, the colliding objects stick together and move as one object after the collision, whereas in an elastic collision the two objects move independently after the collision. Both types of collisions display conservation of momentum. In this experiment we will be using both conservation of momentum and conservation of energy to find the velocity of an object (ball) being fired from a spring loaded gun. The apparatus we will be using consists of a spring loaded gun which fires a small metal ball into a hanging pendulum. See Figure 4.1. The momentum and energy of the ball then causes the pendulum to swing up a ramp, which will catch the pendulum so that we can measure how high it swung. When the metal ball is initially fired from the gun, it will have a kinetic energy, KEi: 1 2 KE = mv0 (4.1) i 2 where m is the mass of the metal ball and v0 is the initial velocity of the ball before it strikes the hanging pendulum.
    [Show full text]
  • 611-1720 (40-130) Ballistic Pendulum
    ©2012 - v 4/15 ______________________________________________________________________________________________________________________________________________________________________ 611-1720 (40-130) Ballistic Pendulum Introduction: The ballistic pendulum is a classic in the Physics lab with an impressive military history. Its premise had long been known by the makers of munitions. Invented in 1742 by Benjamin Robins to measure the speed of bullets, he soon realized that, by knowing the weight of the pendulum with its target, the weight of the shot fired, and the distance the pendulum moved when struck, he could calculate the velocity of the shot. By performing experiments at various distances, he was also able to determine the effects of air density and gravity as the range increased. Your ballistic pendulum is modified from the ones used in those early days. An ideal instrument would have a bob to catch the projectile, and a massless, perfectly rigid rod connecting it to a scale for readings. That way, the only mass the shot has to move is the bob, giving more accurate readings. Unfortunately, such a unit is impossible to manufacture. To improve accuracy, we have used a carbon fiber rod to attach the bob to the scale. This material is exceptionally stiff and very light, making it close to our ideal. In addition, leveling feet have been added to mitigate the effects of gravity. Operation: To use your ballistic pendulum, you will first need to prepare it. Level the unit. You will see a small omnidirectional bubble level and three thumbscrews attached to the feet. Slowly turn these screws until the bubble is in the exact center of the level.
    [Show full text]
  • CM2 Ballistic Pendulum and Projectile
    EP06 Ballistic Pendulum and Projectile Ballistic Pendulum – Theory Overview The ballistic pendulum is a classic method of determining the velocity of a projectile. It is also a good demonstration of some of the basic principles of physics. The ball is fired into the pendulum, which then swings up a measured amount. From the height reached by the pendulum, we can calculate its potential energy. This potential energy is equal to the kinetic energy of the pendulum at the swing, just after the collision with the ball. We can’t equate the kinetic energy of the pendulum after the collision with the kinetic energy of the ball before the swing, since the collision between the ball and pendulum is inelastic and kinetic energy is not conserved in inelastic collisions. Momentum in conserved is all forms of collision, though; so we know that the momentum of the ball before the collision is equal to the momentum of the pendulum after the collision. Once we know the momentum of the ball and its mass, we can determine the initial velocity. There are two ways of calculating the velocity of the ball. The first method (approximate method) assumes that the pendulum and ball together act as a point mass located at their combined center of mass. This method does not take rotational inertia into account. It is somewhat quicker and easier than the second method, but not as accurate. The second method (exact method) uses the actual rotational inertia of the pendulum in the calculations. The equations are slightly more complicated, and it is necessary to take more data in order to find the moment of inertia of the pendulum; but the results obtained are generally better.
    [Show full text]
  • Lab 10. Ballistic Pendulum
    Lab 10. Ballistic Pendulum Goals • To determine the launch speed of a steel ball for the short, medium, and long range settings on the projectile launcher apparatus using the equations for projectile motion. • To use the concepts of gravitational potential energy and conservation of mechanical energy to determine the speed of the ball plus pendulum as it first begins to swing away from the vertical position after the “collision.” • To explore the relationships between the momentum and kinetic energy of the ball as launched and the momentum and kinetic energy of the ball plus pendulum immediately after the ball is caught by the pendulum apparatus. Introduction The “ballistic pendulum” carries this name because it provides a simple method of determining the speed of a bullet shot from a gun. To determine the speed of the bullet, a relatively large block of wood is suspended as a pendulum. The bullet is shot into the wooden block so that it does not penetrate clear through it. This is a type of “sticky” collision, where the two masses (bullet and block) stick to one another and move together after the collision. By noting the angle to which the block and bullet swing after the collision, the initial speed can be determined by using conservation of momentum. This observation incorporates some predictions that we can check. In this experiment, the ballistic pendulum apparatus will be used to compare the momentum of the steel ball before the “collision” to the momentum of the ball and pendulum apparatus, equivalent to the wooden block plus the bullet, after the collision.
    [Show full text]
  • Energy Module 2R2019
    Name (printed) _______________________________ First Day Stamp INTRODUCTION TO MOMENTUM AND IMPULSE 1. Momentum depends on mass as well as velocity. Find the momentum of a 50-kg person walking at 2.0 m/s and the momentum of a 10-g bullet moving at 1,000 m/s. Person Bullet BIG IDEA #1: 2. Now find the kinetic energy of the person and the bullet. Person Bullet BIG IDEA #2: 3. To change the momentum of an object, you must apply … 4. A 0.25 kg soccer ball is rolling at 8.0 m/s toward a player. The player kicks the ball back in the opposite direction, giving it a speed of 16 m/s. What is the average force, during the kick, between the player’s foot and the ball if the kick lasts -2 3.0 x 10 s? 5. 6. Raw eggs are fragile. They break easily. Can you think of a way to throw one as hard as possible without it breaking? Be specific. 7. Egg tosses are fun, especially when the distance between the participants is large. In one egg toss, a 0.60 kg egg is moving at 15 m/s. The egg can only take 24 N of force before breaking. How much time must the catch last in order for the egg not to break? BIG IDEA #3: 8. The photograph to the right shows a Pelton water wheel attached to an electric generator. Notice that the water wheel blades are more like little bowls rather than being traditionally flat. This causes the water to be splashed upward after striking the blades.
    [Show full text]
  • Conservation Laws and the Ballistic Pendulum
    Conservation Laws and The Ballistic Pendulum Introduction: Many times it is impossible to measure a quantity directly. Perhaps the equipment has insufficient resolution, or perhaps the measuring device hasn't been invented. Sometimes the equipment isn't at hand, or it is too expensive. In any case, the scientist has to improvise. In this lab we shall measure the speed of a projectile indirectly using conservation laws. The Theory: Conservation laws apply here to energy and momentum. Conservation means a quantity (energy, momentum) is the same before and after an event. Mechanical energy can be divided into two categories: potential energy and kinetic energy. Potential energy is the energy of position. It may be a position in a gravitational or electromagnetic field or compressed position against a spring. It is important to realize that an object does not have an absolute quantity of this energy, it only has an amount measured against some reference point. Hence the potential energy of an object varies according to what you chose as a reference or zero point. The formula for potential energy (U) in a gravitational field is: eq. 1 where m is mass, g is the acceleration by gravity and h is the change in height measured from the reference point. Consider an object suspended above a table. It has a certain U referenced to the table but a different U referenced to the floor. What is inviolate is a change in potential energy. If the object falls through a distance y < h (so it doesn't hit the table or floor), the change is U is mgy regardless of its reference point.
    [Show full text]