Rolling Motion: Experiments and Simulations Focusing on Sliding Friction Forces
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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 ................................................................................................................................................ -
Explain Inertial and Noninertial Frame of Reference
Explain Inertial And Noninertial Frame Of Reference Nathanial crows unsmilingly. Grooved Sibyl harlequin, his meadow-brown add-on deletes mutely. Nacred or deputy, Sterne never soot any degeneration! In inertial frames of the air, hastening their fundamental forces on two forces must be frame and share information section i am throwing the car, there is not a severe bottleneck in What city the greatest value in flesh-seconds for this deviation. The definition of key facet having a small, polished surface have a force gem about a pretend or aspect of something. Fictitious Forces and Non-inertial Frames The Coriolis Force. Indeed, for death two particles moving anyhow, a coordinate system may be found among which saturated their trajectories are rectilinear. Inertial reference frame of inertial frames of angular momentum and explain why? This is illustrated below. Use tow of reference in as sentence Sentences YourDictionary. What working the difference between inertial frame and non inertial fr. Frames of Reference Isaac Physics. In forward though some time and explain inertial and noninertial of frame to prove your measurement problem you. This circumstance undermines a defining characteristic of inertial frames: that with respect to shame given inertial frame, the other inertial frame home in uniform rectilinear motion. The redirect does not rub at any valid page. That according to whether the thaw is inertial or non-inertial In the. This follows from what Einstein formulated as his equivalence principlewhich, in an, is inspired by the consequences of fire fall. Frame of reference synonyms Best 16 synonyms for was of. How read you govern a bleed of reference? Name we will balance in noninertial frame at its axis from another hamiltonian with each printed as explained at all. -
Sliding and Rolling: the Physics of a Rolling Ball J Hierrezuelo Secondary School I B Reyes Catdicos (Vdez- Mdaga),Spain and C Carnero University of Malaga, Spain
Sliding and rolling: the physics of a rolling ball J Hierrezuelo Secondary School I B Reyes Catdicos (Vdez- Mdaga),Spain and C Carnero University of Malaga, Spain We present an approach that provides a simple and there is an extra difficulty: most students think that it adequate procedure for introducing the concept of is not possible for a body to roll without slipping rolling friction. In addition, we discuss some unless there is a frictional force involved. In fact, aspects related to rolling motion that are the when we ask students, 'why do rolling bodies come to source of students' misconceptions. Several rest?, in most cases the answer is, 'because the didactic suggestions are given. frictional force acting on the body provides a negative acceleration decreasing the speed of the Rolling motion plays an important role in many body'. In order to gain a good understanding of familiar situations and in a number of technical rolling motion, which is bound to be useful in further applications, so this kind of motion is the subject of advanced courses. these aspects should be properly considerable attention in most introductory darified. mechanics courses in science and engineering. The outline of this article is as follows. Firstly, we However, we often find that students make errors describe the motion of a rigid sphere on a rigid when they try to interpret certain situations related horizontal plane. In this section, we compare two to this motion. situations: (1) rolling and slipping, and (2) rolling It must be recognized that a correct analysis of rolling without slipping. -
Numerical Methods Applied Id Metallurgical Processing
NUMERICAL METHODS APPLIED ID METALLURGICAL PROCESSING by JUAN HECTOR BIANCHI A thesis submitted for the degree of Doctor of Philosophy of the University of London and for the Diploma of Membership of the Imperial College John Percy Research Grou Department of Metallurgy Royal School of Mines, Imper i a 1 Co 11ege London JULY 1983 In bulk forming operations the plastic deformation is very large compared with the elastic one. This fact allows the use of constitutive laws based upon both current stress-strain rate measures. A Finite Element formulation of the mechanics involved in quasi-steady state conditions was implemented, introducing the incompressibi1ity via a penalty on the volumetric strain rate. The perfectly plastic behaviour was considered first, and the extrusion process was thoroughly analyzed. Solutions for direct and and indirect modes of operation up to very high extrusion ratios are presented. Both plane strain and axisymmetric geometries were considered. Frictional conditions at the billet-container interface were introduced in two ways. Comparison between results obtained using these lines of approach and numerical problems associated to them are examined. Pressure results are compared with previously reported solutions resulting from Slip Lines, Upper Bounds and Finite Elements. The behaviour of the internal mechanics as a function of changes of geometrical and frictional conditions are examined. Next, a non-linear flow stress resulting from hot torsion experi- ments was considered. The thermal analysis involves the solution of another equation, which is coupled with the mechanical problem due to convection, heat generation and flow stress dependence on temperature. After an appropriate transformation, that equation is formulated in a "weak" form and implemented to be solved jointly with the mechanical part. -
Work, Energy, Rolling, SJ 7Th Ed.: Chap 10.8 to 9, Chap 11.1
A cylinder whose radius is 0.50 m and whose rotational inertia is 10.0 kgm^2 is rotating around a fixed horizontal axis through its center. A constant force of 10.0 N is applied at the rim and is always tangent to it. The work done by on the disk as it accelerates and rotates turns through four complete revolutions is ____ J. Physics 106 Week 4 Work, Energy, Rolling, SJ 7th Ed.: Chap 10.8 to 9, Chap 11.1 • Work and rotational kinetic energy • Rolling • Kinetic energy of rolling Today • Examples of Second Law applied to rolling 2 1 Goal for today Understanding rolling motion For motion with translation and rotation about center of mass ω Example: Rolling vcm KKKtotal=+ rot cm 1 2 1 2 KI= ω KMvcm= cm rot2 cm 2 UM= gh Emech=+KU tot gravity cm 2 A wheel rolling without slipping on a table • The green line above is the path of the mass center of a wheel. ω • The red curve shows the path (called a cycloid) swept out by a point on the rim of the wheel . • When there is no slipping, there are simple vcm relationships between the translational (mass center) and rotational motion. s = Rθ vcm = ωR acm = αR First point of view about rolling motion Rolling = pure rotation around CM + pure translation of CM a) Pure rotation b) Pure translation c) Rolling motion 3 Second point of view about rolling motion Rolling = pure rotation about contact point P • Complementary views – a snapshot in time • Contact point “P” is constantly changing vtang = 2ωPR vA = ωP 2Rcos(ϕ) A v = ω R φ cm P R ωP φ P vtang = 0 vRRcm==ω Pω cm ∴ωP = ωcm ∴α pcm= α Angular velocity and acceleration are the same about contact point “P” or about CM. -
Mechanik Solution 9
Mechanik HS 2018 Solution 9. Prof. Gino Isidori Exercise 1. Rolling cones In the following exercise we consider a circular cone of height h, density ρ, mass m and opening angle α. (a) Determine the inertia tensor in the representation of the axes of Fig.1. (b) Compute the kinetic energy of the cone rolling on a level plane. (c) Repeat the exercise, only this time the cone's tip is fixed to a point on the z axis such that its longitudinal axis is parallel with the plane. Figure 1: The cone with the reference frame for Exercise 1(a). Solution. (a) We start with the inertia around the z-axis. To this end, we have to compute: Z ZZZ 2 2 3 Iz = ρ (x + y )dV = ρ r dz dr dφ Z 2π Z R Z h π = ρ dφ r3dr dz = ρhR4 (S.1) 0 0 hr=R 10 3 = mh2 tan2 α : 10 where we used R = h tan α and m = πhR2ρ/3 in the last step. By the symmetry of the problem, the other two moments of inerta are identical to each other. So it suffices to compute: Z ZZZ 2 2 2 2 2 I1 = ρ (y + z )dV = ρ (r sin φ + z )r dz dr dφ Z 2π Z R Z h = ρ dφ rdr (r2 sin2 φ + z2)dz 0 0 hr=R (S.2) π = ρ hR2(R2 + 4h2) 20 3 = mh2 tan2 α + 4 : 20 1 Figure 2: The rolling cone in part a) of the problem. -
Dynamics of a Rolling and Sliding Disk in a Plane. Asymptotic Solutions, Stability and Numerical Simulations
ISSN 1560-3547, Regular and Chaotic Dynamics, 2016, Vol. 21, No. 2, pp. 204–231. c Pleiades Publishing, Ltd., 2016. Dynamics of a Rolling and Sliding Disk in a Plane. Asymptotic Solutions, Stability and Numerical Simulations Maria Przybylska1* and Stefan Rauch-Wojciechowski2** 1Institute of Physics, University of Zielona G´ora, Licealna 9, PL-65–417 Zielona G´ora, Poland 2Department of Mathematics, Link¨oping University, 581 83 Link¨oping, Sweden Received October 27, 2015; accepted February 15, 2016 Abstract—We present a qualitative analysis of the dynamics of a rolling and sliding disk in a horizontal plane. It is based on using three classes of asymptotic solutions: straight-line rolling, spinning about a vertical diameter and tumbling solutions. Their linear stability analysis is given and it is complemented with computer simulations of solutions starting in the vicinity of the asymptotic solutions. The results on asymptotic solutions and their linear stability apply also to an annulus and to a hoop. MSC2010 numbers: 37J60, 37J25, 70G45 DOI: 10.1134/S1560354716020052 Keywords: rigid body, nonholonomic mechanics, rolling disk, sliding disk 1. INTRODUCTION The problem of a disk rolling (without sliding) in a plane is integrable [1, 4, 9, 15]. It is described in the Euler angles by four dynamical equations and this system admits three additional integrals of motion. It can be in principle reduced to a single quadrature. The difficulty is, however, that only the energy integral is given explicitly, while two other integrals, of angular momentum type, are defined implicitly as constants of integrations for the Legendre equation for the dependence of ω3 on cos θ [22] and [9, §8]. -
Experiment 4: Newton's 2Nd
Experiment 4: Newton's 2nd Law - Incline Plane and Pulley In this lab we will further investigate Newton's 2nd law of motion by using an incline-pulley system. The incline-pulley system, shown in Figure 1, can be classified as a simple machine, that is, one of the classic elementary devices that more complicated and advanced machines are built around. As shown in Figure 1, the acceleration of the mass along the inclined plane (M1) can be controlled by using a hanging counterweight (M2) over the pulley and/or varying the angle of the incline. The free body diagrams for the two masses are shown in Figure 2. We will use the airtrack to create a frictionless plane and also assume that the pulley is frictionless with uniform tension in the string. With these assumptions, the acceleration P of the two masses are the same (a1;x = a2;y). Applying Newton's second law, F = ma, to the freebody diagram, we can write a system of equations describing the motion of the two masses (T is the tension in the string): m1a = T − m1gsinθ (1) m2a = m2g − T (2) Solving these equations for the acceleration: (M − M sin(θ))g a = 2 1 (3) M1 + M2 M1 M2 θ Figure 1: A mass M1 slides along a frictionless incline of angle θ with a counterweight M2 passing over a pulley. 1 a1,x a2,y T N T y x y θ x W1 = m1g W2 = m2g Figure 2: Freebody diagram for the two masses. Experimental Objectives The objective of the lab is to experimentally test the theoretical acceleration (Eq. -
1 Graded Problems
PHY 5246: Theoretical Dynamics, Fall 2015 Assignment # 9, Solutions 1 Graded Problems Problem 1 (1.a) In order to find the equation of motion of the triangle, we need to write the Lagrangian, with generalized coor- dinate θ . The potential energy is going to be based on y { } the location of the center of mass, and we find that via l/2 l/2 ρ l/2 0 − x yCM (θ =0) = 2 dx dyy m 0 √3(l/2 x) Z Z− − 2 m l/2 1 l 2 = dx 3 x CM 1 √3 −m 0 2 2 − ! 2 l 2 l Z 2 8 3 1 l l/2 θ = x √3l2 2 3 2 − ! 0 3 4 l l √3l = = , − 2 −√3l2 8 −2√3 where the density is ρ. This is as expected by symmetry considerations and by the fact that the CM has to be at the geometrical center of the triangle, i.e. at one third of it’s height, 1 √3 l l = . −3 2 −2√3 Note the negative sign is because the triangle lies below the x-axis. So, for generalized θ, ρ yCM = cos θ, −2√3 and the potential energy is l U = mgyCM = mg cos θ. − 2√3 To calculate the kinetic energy, 1 T = T (about 0) = I θ˙2, rot 2 3 we need first to find the principal moment of inertia about the axis of rotation, which is an axis perpendicular to the plane of the triangle, through 0. This is the only moment of inertia we need, since ˙ ~ω = θˆe3 and therefore, 1 1 1 T = ~ωT ˆI ~ω = θ˙2I = θ˙2I . -
Rolling Rolling Condition for Rolling Without Slipping
Rolling Rolling simulation We can view rolling motion as a superposition of pure rotation and pure translation. Pure rotation Pure translation Rolling +rω +rω v Rolling ω ω v -rω +=v -rω v For rolling without slipping, the net instantaneous velocity at the bottom of the wheel is zero. To achieve this condition, 0 = vnet = translational velocity + tangential velocity due to rotation. In other wards, v – rω = 0. When v = rω (i.e., rolling without slipping applies), the tangential velocity at the top of the wheel is twice the 1 translational velocity of the wheel (= v + rω = 2v). 2 Condition for Rolling Without Big yo-yo Slipping A large yo-yo stands on a table. A rope wrapped around the yo-yo's axle, which has a radius that’s half that of When a disc is rolling without slipping, the bottom the yo-yo, is pulled horizontally to the right, with the of the wheel is always at rest instantaneously. rope coming off the yo-yo above the axle. In which This leads to ω = v/r and α = a/r direction does the yo-yo move? There is friction between the table and the yo-yo. Suppose the yo-yo where v is the translational velocity and a is is pulled slowly enough that the yo-yo does not slip acceleration of the center of mass of the disc. on the table as it rolls. rω 1. to the right ω 2. to the left v 3. it won't move -rω X 3 4 Vnet at this point = v – rω Big yo-yo Big yo-yo, again Since the yo-yo rolls without slipping, the center of mass The situation is repeated but with the rope coming off the velocity of the yo-yo must satisfy, vcm = rω. -
Physics 20 Homework 3 SIMS 2016
Physics 20 Homework 3 SIMS 2016 Due: Thursday, August 25th Special thanks to Sebastian Fischetti for problems 1, 5, and 6. Edits in red made by Keith Fratus. 1. The ballistic pendulum is a device used to measure the speed of a bullet. A bullet of mass m is fired into a block of mass M hanging on a pendulum of length L, into which it embeds itself; this causes the pendulum to swing up to some maximum angle θ, as shown. Calculate the initial speed of the bullet in terms of m, M, θ, L, and g. Hint: What conservation laws might you be able to apply in this problem? When can you apply each one, and when can you not? Figure 1: The operation of a ballistic pendulum. Special thanks to Sebastian Fischetti for the figure. 1 2. In lecture, we discussed the definition of potential energy for a single particle which is subjected to an external force. However, we know that in reality, forces arise from the interactions between multiple particles. It is possible to generalize the definition of potential energy to describe two particles interacting with each other as a closed system. As an example, the potential energy of a pair of neutral atoms can be modelled, to a very good approximation, by the Lennard-Jones potential, given by r 12 r 6 U(r) = U 0 − 2 0 0 r r Here U0 and r0 are constants, and r is the distance between the two atoms. Despite the fact that the atoms are electrically neutral overall, the charge due to the protons and electrons is spread out over a non-zero region, so there is still a small, residual electrical interaction between the two. -
Implementation of Rolling Wheel Deflectometer (RWD) in PMS and Pavement Preservation
TECHNICAL REPORT STANDARD PAGE 1. Report No. 2. Government Accession No. 3. Recipient's Catalog No. FHWA/11.492 4. Title and Subtitle 5. Report Date Implementation of Rolling Wheel Deflectometer August 2012 (RWD) in PMS and Pavement Preservation 6. Performing Organization Code LTRC Project Number: 09-2P SIO Number: 30000143 7. Author(s) 8. Performing Organization Report No. Mostafa Elseifi, Ahmed M. Abdel-Khalek , and Karthik Dasari 9. Performing Organization Name and Address 10. Work Unit No. Department of Civil and Environmental Engineering Louisiana State University 11. Contract or Grant No. Baton Rouge, LA 70803 12. Sponsoring Agency Name and Address 13. Type of Report and Period Covered Louisiana Department of Transportation and Final Report Development 2009-2011 P.O. Box 94245 14. Sponsoring Agency Code Baton Rouge, LA 70804-9245 LTRC 15. Supplementary Notes Conducted in Cooperation with the U.S. Department of Transportation, Federal Highway Administration 16. Abstract The rolling wheel deflectometer (RWD) offers the benefit to measure pavement deflection without causing any traffic interruption or compromising safety along tested road segments. This study describes a detailed field evaluation of the RWD system in Louisiana in which 16 different test sites representing a wide array of pavement conditions were tested. Measurements were used to assess the repeatability of RWD measurements, the effect of truck speeds, and to study the relationship between RWD and falling weight deflectometer (FWD) deflection measurements and pavement conditions. Based on the results of the experimental program, it was determined that the repeatability of RWD measurements was acceptable with an average coefficient of variation at all test speeds of 15 percent.