Chapter 8: Rotational Motion
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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. -
Motion Projectile Motion,And Straight Linemotion, Differences Between the Similaritiesand 2 CHAPTER 2 Acceleration Make Thefollowing As Youreadthe ●
026_039_Ch02_RE_896315.qxd 3/23/10 5:08 PM Page 36 User-040 113:GO00492:GPS_Reading_Essentials_SE%0:XXXXXXXXXXXXX_SE:Application_File chapter 2 Motion section ●3 Acceleration What You’ll Learn Before You Read ■ how acceleration, time, and velocity are related Describe what happens to the speed of a bicycle as it goes ■ the different ways an uphill and downhill. object can accelerate ■ how to calculate acceleration ■ the similarities and differences between straight line motion, projectile motion, and circular motion Read to Learn Study Coach Outlining As you read the Velocity and Acceleration section, make an outline of the important information in A car sitting at a stoplight is not moving. When the light each paragraph. turns green, the driver presses the gas pedal and the car starts moving. The car moves faster and faster. Speed is the rate of change of position. Acceleration is the rate of change of velocity. When the velocity of an object changes, the object is accelerating. Remember that velocity is a measure that includes both speed and direction. Because of this, a change in velocity can be either a change in how fast something is moving or a change in the direction it is moving. Acceleration means that an object changes it speed, its direction, or both. How are speeding up and slowing down described? ●D Construct a Venn When you think of something accelerating, you probably Diagram Make the following trifold Foldable to compare and think of it as speeding up. But an object that is slowing down contrast the characteristics of is also accelerating. Remember that acceleration is a change in acceleration, speed, and velocity. -
Frames of Reference
Galilean Relativity 1 m/s 3 m/s Q. What is the women velocity? A. With respect to whom? Frames of Reference: A frame of reference is a set of coordinates (for example x, y & z axes) with respect to whom any physical quantity can be determined. Inertial Frames of Reference: - The inertia of a body is the resistance of changing its state of motion. - Uniformly moving reference frames (e.g. those considered at 'rest' or moving with constant velocity in a straight line) are called inertial reference frames. - Special relativity deals only with physics viewed from inertial reference frames. - If we can neglect the effect of the earth’s rotations, a frame of reference fixed in the earth is an inertial reference frame. Galilean Coordinate Transformations: For simplicity: - Let coordinates in both references equal at (t = 0 ). - Use Cartesian coordinate systems. t1 = t2 = 0 t1 = t2 At ( t1 = t2 ) Galilean Coordinate Transformations are: x2= x 1 − vt 1 x1= x 2+ vt 2 or y2= y 1 y1= y 2 z2= z 1 z1= z 2 Recall v is constant, differentiation of above equations gives Galilean velocity Transformations: dx dx dx dx 2 =1 − v 1 =2 − v dt 2 dt 1 dt 1 dt 2 dy dy dy dy 2 = 1 1 = 2 dt dt dt dt 2 1 1 2 dz dz dz dz 2 = 1 1 = 2 and dt2 dt 1 dt 1 dt 2 or v x1= v x 2 + v v x2 =v x1 − v and Similarly, Galilean acceleration Transformations: a2= a 1 Physics before Relativity Classical physics was developed between about 1650 and 1900 based on: * Idealized mechanical models that can be subjected to mathematical analysis and tested against observation. -
Velocity-Corrected Area Calculation SCIEX PA 800 Plus Empower
Velocity-corrected area calculation: SCIEX PA 800 Plus Empower Driver version 1.3 vs. 32 Karat™ Software Firdous Farooqui1, Peter Holper1, Steve Questa1, John D. Walsh2, Handy Yowanto1 1SCIEX, Brea, CA 2Waters Corporation, Milford, MA Since the introduction of commercial capillary electrophoresis (CE) systems over 30 years ago, it has been important to not always use conventional “chromatography thinking” when using CE. This is especially true when processing data, as there are some key differences between electrophoretic and chromatographic data. For instance, in most capillary electrophoresis separations, peak area is not only a function of sample load, but also of an analyte’s velocity past the detection window. In this case, early migrating peaks move past the detection window faster than later migrating peaks. This creates a peak area bias, as any relative difference in an analyte’s migration velocity will lead to an error in peak area determination and relative peak area percent. To help minimize Figure 1: The PA 800 Plus Pharmaceutical Analysis System. this bias, peak areas are normalized by migration velocity. The resulting parameter is commonly referred to as corrected peak The capillary temperature was maintained at 25°C in all area or velocity corrected area. separations. The voltage was applied using reverse polarity. This technical note provides a comparison of velocity corrected The following methods were used with the SCIEX PA 800 Plus area calculations using 32 Karat™ and Empower software. For Empower™ Driver v1.3: both, standard processing methods without manual integration were used to process each result. For 32 Karat™ software, IgG_HR_Conditioning: conditions the capillary Caesar integration1 was turned off. -
Create an Absence/OT Request (Timekeeper)
Create an Absence/OT Request (Timekeeper) 1. From the Employee Self Service home page, select the drop-down at the top of the screen, and choose Time Administration. a. Follow these instructions to add the Time Administration page/tile to your homepage. 2. Select the Time Administration tile. a. It might take a moment for the Time Administration page to load. Create Absence/OT Request (Timekeeper) | 1 3. Select the Report Employee Time tab. 4. Click on the Employee Selection arrow to make the Employee Selection Criteria section appear. Enter full or partial search items to refine the list of employees, and select the Get Employees button. a. If you do not enter search criteria and simply click Get Employees, all employees will appear in the Search Results section. Create Absence/OT Request (Timekeeper) | 2 5. A list of employees will appear. Select the employee for whom you would like to create an absence or overtime request. 6. The employee’s timesheet will appear. Navigate to the date of the leave request using the Date field or Previous Period/Next Period hyperlinks, and select the refresh [ ] icon. Create Absence/OT Request (Timekeeper) | 3 7. Select the Absence/OT tab on the timesheet. 8. Select the Add Absence Event button. Create Absence/OT Request (Timekeeper) | 4 9. Enter the Start and End Dates for the absence/overtime event. 10. Select the Absence Name drop-down to choose the appropriate option. Create Absence/OT Request (Timekeeper) | 5 11. Select the Details hyperlink. 12. The Absence Event Details screen will pop up. Create Absence/OT Request (Timekeeper) | 6 13. -
Chapter 8 Relationship Between Rotational Speed and Magnetic
Chapter 8 Relationship Between Rotational Speed and Magnetic Fields 8.1 Introduction It is well-known that sunspots and other solar magnetic features rotate faster than the surface plasma (Howard & Harvey, 1970). The sidereal rotation speeds of weak magnetic features and plages (Howard, Gilman, & Gilman, 1984; Komm, Howard, & Harvey, 1993), individual sunspots (Howard, Gilman, & Gilman, 1984; Sivaraman, Gupta, & Howard, 1993) and sunspot groups (Howard, 1992) have been studied for many years by different research groups by use of different observatory data (see reviews of Howard 1996 and Beck 2000). Such studies are deemed as diagnostics of subsurface conditions based on the assumption that the faster rotation of magnetic features was caused by the faster-rotating plasma in the interior where the magnetic features were anchored (Gilman & Foukal, 1979). By matching the sunspot surface speed with the solar interior rotational speed robustly derived by helioseismology (e.g., Thompson et al., 1996; Kosovichev et al., 1997; Howe et al., 2000a), several studies estimated the depth of sunspot roots for sunspots of different sizes and life spans (e.g., Hiremath, 2002; Sivaraman et al., 2003). Such studies are valuable for understanding the solar dynamo and origin of solar active regions. However, D’Silva & Howard (1994) proposed a different explanation, namely that effects of buoyancy 111 112 CHAPTER 8. ROTATIONAL SPEED AND MAGNETIC FIELDS and drag coupled with the Coriolis force during the emergence of active regions might lead to the faster rotational speed. Although many studies were done to infer the rotational speed of various solar magnetic features, the relationship between the rotational speed of the magnetic fea- tures and their magnetic strength has not yet been studied. -
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. -
The Origins of Velocity Functions
The Origins of Velocity Functions Thomas M. Humphrey ike any practical, policy-oriented discipline, monetary economics em- ploys useful concepts long after their prototypes and originators are L forgotten. A case in point is the notion of a velocity function relating money’s rate of turnover to its independent determining variables. Most economists recognize Milton Friedman’s influential 1956 version of the function. Written v = Y/M = v(rb, re,1/PdP/dt, w, Y/P, u), it expresses in- come velocity as a function of bond interest rates, equity yields, expected inflation, wealth, real income, and a catch-all taste-and-technology variable that captures the impact of a myriad of influences on velocity, including degree of monetization, spread of banking, proliferation of money substitutes, devel- opment of cash management practices, confidence in the future stability of the economy and the like. Many also are aware of Irving Fisher’s 1911 transactions velocity func- tion, although few realize that it incorporates most of the same variables as Friedman’s.1 On velocity’s interest rate determinant, Fisher writes: “Each per- son regulates his turnover” to avoid “waste of interest” (1963, p. 152). When rates rise, cashholders “will avoid carrying too much” money thus prompting a rise in velocity. On expected inflation, he says: “When...depreciation is anticipated, there is a tendency among owners of money to spend it speedily . the result being to raise prices by increasing the velocity of circulation” (p. 263). And on real income: “The rich have a higher rate of turnover than the poor. They spend money faster, not only absolutely but relatively to the money they keep on hand. -
PHYSICS of ARTIFICIAL GRAVITY Angie Bukley1, William Paloski,2 and Gilles Clément1,3
Chapter 2 PHYSICS OF ARTIFICIAL GRAVITY Angie Bukley1, William Paloski,2 and Gilles Clément1,3 1 Ohio University, Athens, Ohio, USA 2 NASA Johnson Space Center, Houston, Texas, USA 3 Centre National de la Recherche Scientifique, Toulouse, France This chapter discusses potential technologies for achieving artificial gravity in a space vehicle. We begin with a series of definitions and a general description of the rotational dynamics behind the forces ultimately exerted on the human body during centrifugation, such as gravity level, gravity gradient, and Coriolis force. Human factors considerations and comfort limits associated with a rotating environment are then discussed. Finally, engineering options for designing space vehicles with artificial gravity are presented. Figure 2-01. Artist's concept of one of NASA early (1962) concepts for a manned space station with artificial gravity: a self- inflating 22-m-diameter rotating hexagon. Photo courtesy of NASA. 1 ARTIFICIAL GRAVITY: WHAT IS IT? 1.1 Definition Artificial gravity is defined in this book as the simulation of gravitational forces aboard a space vehicle in free fall (in orbit) or in transit to another planet. Throughout this book, the term artificial gravity is reserved for a spinning spacecraft or a centrifuge within the spacecraft such that a gravity-like force results. One should understand that artificial gravity is not gravity at all. Rather, it is an inertial force that is indistinguishable from normal gravity experience on Earth in terms of its action on any mass. A centrifugal force proportional to the mass that is being accelerated centripetally in a rotating device is experienced rather than a gravitational pull. -
Basic Galactic Dynamics
5 Basic galactic dynamics 5.1 Basic laws govern galaxy structure The basic force on galaxy scale is gravitational • Why? The other long-range force is electro-magnetic force due to electric • charges: positive and negative charges are difficult to separate on galaxy scales. 5.1.1 Gravitational forces Gravitational force on a point mass M1 located at x1 from a point mass M2 located at x2 is GM M (x x ) F = 1 2 1 − 2 . 1 − x x 3 | 1 − 2| Note that force F1 and positions, x1 and x2 are vectors. Note also F2 = F1 consistent with Newton’s Third Law. Consider a stellar− system of N stars. The gravitational force on ith star is the sum of the force due to all other stars N GM M (x x ) F i j i j i = −3 . − xi xj Xj=6 i | − | The equation of motion of the ith star under mutual gravitational force is given by Newton’s second law, dv F = m a = m i i i i i dt and so N dvi GMj(xi xj) = − 3 . dt xi xj Xj=6 i | − | 5.1.2 Energy conservation Assuming M2 is fixed at the origin, and M1 on the x-axis at a distance x1 from the origin, we have m1dv1 GM1M2 = 2 dt − x1 1 2 Using 2 dv1 dx1 dv1 dv1 1 dv1 = = v1 = dt dt dx1 dx1 2 dx1 we have 2 1 dv1 GM1M2 m1 = 2 , 2 dx1 − x1 which can be integrated to give 1 2 GM1M2 m1v1 = E , 2 − x1 The first term on the lhs is the kinetic energy • The second term on the lhs is the potential energy • E is the total energy and is conserved • 5.1.3 Gravitational potential The potential energy of M1 in the potential of M2 is M Φ , − 1 × 2 where GM2 Φ2 = − x1 is the potential of M2 at a distance x1 from it. -
Rotating Objects Tend to Keep Rotating While Non- Rotating Objects Tend to Remain Non-Rotating
12 Rotational Motion Rotating objects tend to keep rotating while non- rotating objects tend to remain non-rotating. 12 Rotational Motion In the absence of an external force, the momentum of an object remains unchanged— conservation of momentum. In this chapter we extend the law of momentum conservation to rotation. 12 Rotational Motion 12.1 Rotational Inertia The greater the rotational inertia, the more difficult it is to change the rotational speed of an object. 1 12 Rotational Motion 12.1 Rotational Inertia Newton’s first law, the law of inertia, applies to rotating objects. • An object rotating about an internal axis tends to keep rotating about that axis. • Rotating objects tend to keep rotating, while non- rotating objects tend to remain non-rotating. • The resistance of an object to changes in its rotational motion is called rotational inertia (sometimes moment of inertia). 12 Rotational Motion 12.1 Rotational Inertia Just as it takes a force to change the linear state of motion of an object, a torque is required to change the rotational state of motion of an object. In the absence of a net torque, a rotating object keeps rotating, while a non-rotating object stays non-rotating. 12 Rotational Motion 12.1 Rotational Inertia Rotational Inertia and Mass Like inertia in the linear sense, rotational inertia depends on mass, but unlike inertia, rotational inertia depends on the distribution of the mass. The greater the distance between an object’s mass concentration and the axis of rotation, the greater the rotational inertia. 2 12 Rotational Motion 12.1 Rotational Inertia Rotational inertia depends on the distance of mass from the axis of rotation.