Distance Measures

Total Page:16

File Type:pdf, Size:1020Kb

Distance Measures CHAPTER 6 Distance Measures Background • One can calculate distances among either the The first step of most multivariate analyses is to rows of your data matrix or the columns of calculate a matrix of distances or similarities among a your data matrix. With community data this set of items in a multidimensional space. This is means you can calculate distances among your analogous to constructing the triangular "mileage sample units (SUs) in species space or among chart" provided with many road maps. But in our case, your species in sample space. we need to build a matrix of distances in hyperspace, Figure 6.1 shows two species as points in sample rather than the two-dimensional map space. Fortu­ space, corresponding to the tiny data set below (Table nately, it is just as easy to calculate distances in a 6.1). We can also represent sample units as points in multidimensional space as it is in a two-dimensional species space, as on the right side of Figure 6.1. using space. the same data set. This first step is extremely important. If There are many distance measures. A selection of information is ignored in this step, then it cannot be the most commonly used and most effective measures expressed in the results. Likewise, if noise or outliers are described below. It is important to know the are exaggerated by the distance measure, then these domain of acceptable data values for each distance unwanted features of our data will have undue measure (Table 6.2). Many distance measures are not influence on the results, perhaps obscuring meaningful compatible with negative numbers. Other distance patterns. measures assume that the data are proportions ranging between zero and one, inclusive Distance concepts Distance measures are flexible: Table 6.1. Example data set Abundance of two • Resemblance can be measured either as a species in two sample units. distance (dissimilarity) or a similarity. Species • Most distance measures can readily be con­ verted into similarities and vice-versa. Sample unit 1 2 • All of the distance measures described below A 1 4 can be applied to either binary (presence- B 5 2 absence) or quantitative data. 5 j Sam ple space Species space ш q ♦SUA 3 3 <L> Sp 2 SUB ex 2 tU CL C / 3 C / 3 -I------------------1------------------1------------------L_ » 1 2 3 4 5 1 2 3 Sample Unit A Species 1 Figure 6.1. Graphical representation of the data set in Table 6.1. The left-hand graph shows species as points in sample space. The rieht-hand nranh shnwc eamnip unite ·>- Chapter 6 Table 6.2 Reasonable and acceptable domains of input data. л\ and ranges of distance measures, d - fix). Domain Name (synonyms) of X Range of d=fix) Com ments Sorensen x > 0 0 < d < 1 proportion coefficient in city- (Brav & Curtis; (or()<x< 100%) block space, semimetric Czekanovvski) Relative Sorensen x > 0 0 <d< 1 proportion coefficient in city - (Kulczyński; Quantitative (or 0 < x < 100%) block space; same as Sorensen Symmetric) but data points relativized by sample unit totals; semimetric Jaccard x > 0 ()< £ /< 1 proportion coefficient in city- (orO <d< 100%) block space; metric Euclidean (Pythagorean) all non-negative metric Relative Euclidean all 0 < d <42 for quarter Euclidean distance between (Chord distance, points on unit hy persphere: hypersphere; 0 < d < 2 standardized Euclidean) metric for full hvpersphere Correlation distance all 0 < d < I converted from correlation to distance; proportional to arc distance between points on unit hypersphere; cosine of angle from centroid to points; metric Chi-square x > 0 d> 0 Euclidean but doubly weighted by variable and sample unit totals; metric Squared Euclidean all d> 0 metric Mahalanobis all d> 0 distance between groups weighted by within-group dispersion: metric Distance measures can be categorized as metric, Kulczyński distances Seimmctrics are extremely use­ scmimetric. or nonmetric A metric distance measure ful in community ecology but obey a non-Euclidean must satisfy the following rules: geometry Nonmetrics violate one or more of the other rules and are seldom used in ecology 1 The minimum value is zero when two items are identical. Distance measures 2 When two items differ, the distance is positive (negative distances are not allowed). The equations use the following conventions: Our data matrix A has q rows, which are sample units and 3 Symmetry: the distance from objects A to object p columns, which are species. Each element of the B is the same as the distance from B to A. matrix, a, ,, is the abundance of species j in sample unit 4 Triangle inequality axiom: With three objects. i. Most of the following distance measures can also be the distance between two of these objects used on binary data (1 or 0 for presence or absence). cannot be larger than the sum of the two other In each of the following equations, we are calculating distances. the distance between sample units / and h. Distance Λ le asu res Euclidean distance oo EUCLIDEAN E E ) , , и ' У ' ( α , , ~ ай ,]} Щ DISTANCE D ω Cu oo This formula is simply the Pythagorean theorem applied to p dimensions rather than the usual two dimensions (Fig. 6.2). SPE C IE S 1 City-block distance (= Manhattan distance) 00 ω 0 f CITY-BLOCK ω [ d i s t a n c e CBo a¡, " Ok,, α, ъ oo 7=1 ln city-block space you can only move along one dimension of the space at a time (Fig. 6.2). By SPE C IE S 1 analogy, in a city of rectangular blocks, you caimot cut diagonally through a block, but must walk along either of the two dimensions of the block. In the mathema­ tical space, size of the blocks does not affect distances cos α in the space Note also that many equal-length paths 00 exist between two points in city-block space. Euclidean distance and city-block distance are c e n t r o i d special cases (different values of k) of the Minkowski SPE C IE S 1 metric. In two dimensions: Figure 6.2. Geometric representations of basic dis­ Distance \[xk + yk tance measures between two sample units (A and B) in species space In the upper two graphs the axes meet where x and v are distances in each of two dimensions. at the origin; in the lowest graph, at the centroid Generalizing this to p dimensions, and using the form of the equation for ED: cos(18()“). Two sample units lying on the same radius from the centroid have r = 1 = cos(0°). If two sample Distance,h = at] - ahjt units form a right angle from the centroid then r = 0 = cos(90°). The correlation coefficient can be rescaled to a Note that k = 1 gives city-block distance, k = 2 gives distance measure of range 0-1 by Euclidean distance. As k increases, increasing emphasis is given to large differences in individual t distance ( Î ~ ~ dimensions Correlation Proportion coefficients The correlation coefficient (r) is cosine a (third Proportion coefficients are city-block distance panel in Fig. 6.2) where the origin of the coordinate measures expressed as proportions of the maximum system is the mean species composition of a sample distance possible. The Sorensen. Jaccard. and QSK tuut in species space (the "centroid"; see Fig. 6.2). If coefficients described below are all proportion coef­ the data have not been transformed and the origin is at ficients. One can represent proportion coefficients as (0,0). then this is a noncentered correlation coefficient the overlap between the area under curves. This is easiest to visualize with two curves of species abun­ For example, if two sample units lie at 180“ from dance on an environmental gradient (Fig. 6 3). If A is each other relative to the centroid, then r - -1 = the area under one curve, B is the area under the other. ( 'h apt u r 6 and n' is the overlap (intersection) of the two areas, (hen the Sorensen coefficient is 2и>/(. I · lí). The tin - Cthj Jaccard coefficient is will - И-w). Σ IX, Written in set notation: Σ». +Σ a. 2(A Г) B) Sorensen similarity Another way of writing this, where MÍN is the (A'u B) - ( ArsB) smaller of two values is: АглВ Jaccard similarity A s j B Clh,) Proportion coefficients as distance measures are D,h foreign to classical statistics, which are based on +Σ squared Euclidean distances. Ecologists latched onto proportion coefficients for their simple, intuitive appeal despite their falling outside of mainstream statistics. Nevertheless, Roberts (1986) showed how proportion One can convert this dissimilarity' (or any of the coefficients can be derived from the mathematics of following proportion coefficients) to a percentage dissi­ fuzzy sets an increasingly important branch of milarity (rø): mathematics For example, when applied to quantita­ tive data. Sorensen similarity is the intersection between two fuzzy sets. Sorensen distance ■=- BC, PD,i 100 D„ Sorensen similarity(also known as "BC" for Bray-Curtis coefficient) is thus shared abundance Jaccard dissimilarity is the proportion of the combined abundance that is not shared, or u (A ■ B w) (Jaccard 1901): -Σ a ¡j “ ci JD, а ч ^ ο ~ Qhj Environmental Gradient Σ + Σ ^ ”^Σ Figure 6.3. Overlap between two species abundances along an environmental gradient. The abundance shared between species A and B is shown by w. Quantitative symmetric dissimilarity(also known as the Kulczyński or QSK coefficient: see Faith et al. 1987): divided by total abundance. It is frequently known as "2u/(A+B)" for short. This logical-seeming measure was also proposed by Czekanowski (1913).
Recommended publications
  • Glossary Physics (I-Introduction)
    1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay.
    [Show full text]
  • Forces Different Types of Forces
    Forces and motion are a part of your everyday life for example pushing a trolley, a horse pulling a rope, speed and acceleration. Force and motion causes objects to move but also to stay still. Motion is simply a movement but needs a force to move. There are 2 types of forces, contact forces and act at a distance force. Forces Every day you are using forces. Force is basically push and pull. When you push and pull you are applying a force to an object. If you are Appling force to an object you are changing the objects motion. For an example when a ball is coming your way and then you push it away. The motion of the ball is changed because you applied a force. Different Types of Forces There are more forces than push or pull. Scientists group all these forces into two groups. The first group is contact forces, contact forces are forces when 2 objects are physically interacting with each other by touching. The second group is act at a distance force, act at a distance force is when 2 objects that are interacting with each other but not physically touching. Contact Forces There are different types of contact forces like normal Force, spring force, applied force and tension force. Normal force is when nothing is happening like a book lying on a table because gravity is pulling it down. Another contact force is spring force, spring force is created by a compressed or stretched spring that could push or pull. Applied force is when someone is applying a force to an object, for example a horse pulling a rope or a boy throwing a snow ball.
    [Show full text]
  • Geodesic Distance Descriptors
    Geodesic Distance Descriptors Gil Shamai and Ron Kimmel Technion - Israel Institute of Technologies [email protected] [email protected] Abstract efficiency of state of the art shape matching procedures. The Gromov-Hausdorff (GH) distance is traditionally used for measuring distances between metric spaces. It 1. Introduction was adapted for non-rigid shape comparison and match- One line of thought in shape analysis considers an ob- ing of isometric surfaces, and is defined as the minimal ject as a metric space, and object matching, classification, distortion of embedding one surface into the other, while and comparison as the operation of measuring the discrep- the optimal correspondence can be described as the map ancies and similarities between such metric spaces, see, for that minimizes this distortion. Solving such a minimiza- example, [13, 33, 27, 23, 8, 3, 24]. tion is a hard combinatorial problem that requires pre- Although theoretically appealing, the computation of computation and storing of all pairwise geodesic distances distances between metric spaces poses complexity chal- for the matched surfaces. A popular way for compact repre- lenges as far as direct computation and memory require- sentation of functions on surfaces is by projecting them into ments are involved. As a remedy, alternative representa- the leading eigenfunctions of the Laplace-Beltrami Opera- tion spaces were proposed [26, 22, 15, 10, 31, 30, 19, 20]. tor (LBO). When truncated, the basis of the LBO is known The question of which representation to use in order to best to be the optimal for representing functions with bounded represent the metric space that define each form we deal gradient in a min-max sense.
    [Show full text]
  • Robust Topological Inference: Distance to a Measure and Kernel Distance
    Journal of Machine Learning Research 18 (2018) 1-40 Submitted 9/15; Revised 2/17; Published 4/18 Robust Topological Inference: Distance To a Measure and Kernel Distance Fr´ed´ericChazal [email protected] Inria Saclay - Ile-de-France Alan Turing Bldg, Office 2043 1 rue Honor´ed'Estienne d'Orves 91120 Palaiseau, FRANCE Brittany Fasy [email protected] Computer Science Department Montana State University 357 EPS Building Montana State University Bozeman, MT 59717 Fabrizio Lecci [email protected] New York, NY Bertrand Michel [email protected] Ecole Centrale de Nantes Laboratoire de math´ematiquesJean Leray 1 Rue de La Noe 44300 Nantes FRANCE Alessandro Rinaldo [email protected] Department of Statistics Carnegie Mellon University Pittsburgh, PA 15213 Larry Wasserman [email protected] Department of Statistics Carnegie Mellon University Pittsburgh, PA 15213 Editor: Mikhail Belkin Abstract Let P be a distribution with support S. The salient features of S can be quantified with persistent homology, which summarizes topological features of the sublevel sets of the dis- tance function (the distance of any point x to S). Given a sample from P we can infer the persistent homology using an empirical version of the distance function. However, the empirical distance function is highly non-robust to noise and outliers. Even one outlier is deadly. The distance-to-a-measure (DTM), introduced by Chazal et al. (2011), and the ker- nel distance, introduced by Phillips et al. (2014), are smooth functions that provide useful topological information but are robust to noise and outliers. Chazal et al. (2015) derived concentration bounds for DTM.
    [Show full text]
  • Hydraulics Manual Glossary G - 3
    Glossary G - 1 GLOSSARY OF HIGHWAY-RELATED DRAINAGE TERMS (Reprinted from the 1999 edition of the American Association of State Highway and Transportation Officials Model Drainage Manual) G.1 Introduction This Glossary is divided into three parts: · Introduction, · Glossary, and · References. It is not intended that all the terms in this Glossary be rigorously accurate or complete. Realistically, this is impossible. Depending on the circumstance, a particular term may have several meanings; this can never change. The primary purpose of this Glossary is to define the terms found in the Highway Drainage Guidelines and Model Drainage Manual in a manner that makes them easier to interpret and understand. A lesser purpose is to provide a compendium of terms that will be useful for both the novice as well as the more experienced hydraulics engineer. This Glossary may also help those who are unfamiliar with highway drainage design to become more understanding and appreciative of this complex science as well as facilitate communication between the highway hydraulics engineer and others. Where readily available, the source of a definition has been referenced. For clarity or format purposes, cited definitions may have some additional verbiage contained in double brackets [ ]. Conversely, three “dots” (...) are used to indicate where some parts of a cited definition were eliminated. Also, as might be expected, different sources were found to use different hyphenation and terminology practices for the same words. Insignificant changes in this regard were made to some cited references and elsewhere to gain uniformity for the terms contained in this Glossary: as an example, “groundwater” vice “ground-water” or “ground water,” and “cross section area” vice “cross-sectional area.” Cited definitions were taken primarily from two sources: W.B.
    [Show full text]
  • Mathematics of Distances and Applications
    Michel Deza, Michel Petitjean, Krassimir Markov (eds.) Mathematics of Distances and Applications I T H E A SOFIA 2012 Michel Deza, Michel Petitjean, Krassimir Markov (eds.) Mathematics of Distances and Applications ITHEA® ISBN 978-954-16-0063-4 (printed); ISBN 978-954-16-0064-1 (online) ITHEA IBS ISC No.: 25 Sofia, Bulgaria, 2012 First edition Printed in Bulgaria Recommended for publication by The Scientific Council of the Institute of Information Theories and Applications FOI ITHEA The papers accepted and presented at the International Conference “Mathematics of Distances and Applications”, July 02-05, 2012, Varna, Bulgaria, are published in this book. The concept of distance is basic to human experience. In everyday life it usually means some degree of closeness of two physical objects or ideas, i.e., length, time interval, gap, rank difference, coolness or remoteness, while the term metric is often used as a standard for a measurement. Except for the last two topics, the mathematical meaning of those terms, which is an abstraction of measurement, is considered. Distance metrics and distances have now become an essential tool in many areas of Mathematics and its applications including Geometry, Probability, Statistics, Coding/Graph Theory, Clustering, Data Analysis, Pattern Recognition, Networks, Engineering, Computer Graphics/Vision, Astronomy, Cosmology, Molecular Biology, and many other areas of science. Devising the most suitable distance metrics and similarities, to quantify the proximity between objects, has become a standard task for many researchers. Especially intense ongoing search for such distances occurs, for example, in Computational Biology, Image Analysis, Speech Recognition, and Information Retrieval. For experts in the field of mathematics, information technologies as well as for practical users.
    [Show full text]
  • Topology Distance: a Topology-Based Approach for Evaluating Generative Adversarial Networks
    Topology Distance: A Topology-Based Approach For Evaluating Generative Adversarial Networks Danijela Horak 1 Simiao Yu 1 Gholamreza Salimi-Khorshidi 1 Abstract resemble real images Xr, while D discriminates between Xg Automatic evaluation of the goodness of Gener- and Xr. G and D are trained by playing a two-player mini- ative Adversarial Networks (GANs) has been a max game in a competing manner. Such novel adversarial challenge for the field of machine learning. In training process is a key factor in GANs’ success: It im- this work, we propose a distance complementary plicitly defines a learnable objective that is flexibly adaptive to existing measures: Topology Distance (TD), to various complicated image-generation tasks, in which it the main idea behind which is to compare the would be difficult or impossible to explicitly define such an geometric and topological features of the latent objective. manifold of real data with those of generated data. One of the biggest challenges in the field of generative More specifically, we build Vietoris-Rips com- models – including for GANs – is the automatic evaluation plex on image features, and define TD based on of the goodness of such models (e.g., whether or not the the differences in persistent-homology groups of data generated by such models are similar to the data they the two manifolds. We compare TD with the were trained on). Unlike supervised learning, where the most commonly-used and relevant measures in goodness of the models can be assessed by comparing their the field, including Inception Score (IS), Frechet´ predictions with the actual labels, or in some other deep- Inception Distance (FID), Kernel Inception Dis- learning models where the goodness of the model can be tance (KID) and Geometry Score (GS), in a range assessed using the likelihood of the validation data under the of experiments on various datasets.
    [Show full text]
  • Euclidean Space - Wikipedia, the Free Encyclopedia Page 1 of 5
    Euclidean space - Wikipedia, the free encyclopedia Page 1 of 5 Euclidean space From Wikipedia, the free encyclopedia In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, and is named for the Greek mathematician Euclid of Alexandria. Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. In modern mathematics, it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. This approach brings the tools of algebra and calculus to bear on questions of geometry, and Every point in three-dimensional has the advantage that it generalizes easily to Euclidean Euclidean space is determined by three spaces of more than three dimensions. coordinates. From the modern viewpoint, there is essentially only one Euclidean space of each dimension. In dimension one this is the real line; in dimension two it is the Cartesian plane; and in higher dimensions it is the real coordinate space with three or more real number coordinates. Thus a point in Euclidean space is a tuple of real numbers, and distances are defined using the Euclidean distance formula. Mathematicians often denote the n-dimensional Euclidean space by , or sometimes if they wish to emphasize its Euclidean nature. Euclidean spaces have finite dimension. Contents 1 Intuitive overview 2 Real coordinate space 3 Euclidean structure 4 Topology of Euclidean space 5 Generalizations 6 See also 7 References Intuitive overview One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle.
    [Show full text]
  • Lesson 3 – Understanding Distance in Space (Optional)
    Lesson 3 – Understanding Distance in Space (optional) Background The distance between objects in space is vast and very difficult for most children to grasp. The values for these distances are cumbersome for astronomers and scientists to manipulate. Therefore, scientists use a unit of measurement called an astronomical unit. An astronomical unit is the average distance between the Earth and the Sun – 150 million kilometers. Using a scale model to portray the distances of the planets will help the students begin to understand the vastness of space. Astronomers soon found that even the astronomical unit was not large enough to measure the distance of objects outside of our solar system. For these distances, the parsec is used. A parsec is a unit of light-years. A parsec equals approximately 3.3 light-years. (Keep in mind that a light-year is the distance light travels in one year – 9.5 trillion km – and that light travels 300,000 km per second!) Teacher Notes and Hints Prior to lesson • Discuss and review measuring length or distance. How far is one meter? • Discuss the use of prefixes that add or subtract to “basic” units of length. Example: meter is the basic metric unit for length “kilo” means 1000 – kilometer is 1000 meters “centi” means 1/100 – centimeter is 1/100 of a meter Adapt the discussion for grade level. For example, fifth graders can learn the metric prefixes for more “lengths” and may add to decimal work in math class: Kilo – 1000 Hecto – 100 Deka – 10 Deci – 1/10 (0.1) Centi – 1/100 (0.01) Milli – 1/1000 (0.001) • When discussing kilometers, it is helpful to know the number of km’s to local places (for example, kilometers to the Mall).
    [Show full text]
  • Distance Between Points on the Earth's Surface
    Distance between Points on the Earth's Surface Abstract During a casual conversation with one of my students, he asked me how one could go about computing the distance between two points on the surface of the Earth, in terms of their respective latitudes and longitudes. This is an interesting exercise in spherical coordinates, and relates to the so-called haversine. The calculation of the distance be- tween two points on the surface of the Spherical coordinates Earth proceeds in two stages: (1) to z compute the \straight-line" Euclidean x=Rcosθcos φ distance these two points (obtained by y=Rcosθsin φ R burrowing through the Earth), and (2) z=Rsinθ to convert this distance to one mea- θ y sured along the surface of the Earth. φ Figure 1 depicts the spherical coor- dinates we shall use.1 We orient this coordinate system so that x Figure 1: Spherical Coordinates (i) The origin is at the Earth's center; (ii) The x-axis passes through the Prime Meridian (0◦ longitude); (iii) The xy-plane contains the Earth's equator (and so the positive z-axis will pass through the North Pole) Note that the angle θ is the measurement of lattitude, and the angle φ is the measurement of longitude, where 0 ≤ φ < 360◦, and −90◦ ≤ θ ≤ 90◦. Negative values of θ correspond to points in the Southern Hemisphere, and positive values of θ correspond to points in the Northern Hemisphere. When one uses spherical coordinates it is typical for the radial distance R to vary; however, in our discussion we may fix it to be the average radius of the Earth: R ≈ 6; 378 km: 1What is depicted are not the usual spherical coordinates, as the angle θ is usually measure from the \zenith", or z-axis.
    [Show full text]
  • Lecture 24 Angular Momentum
    LECTURE 24 ANGULAR MOMENTUM Instructor: Kazumi Tolich Lecture 24 2 ¨ Reading chapter 11-6 ¤ Angular momentum n Angular momentum about an axis n Newton’s 2nd law for rotational motion Angular momentum of an rotating object 3 ¨ An object with a moment of inertia of � about an axis rotates with an angular speed of � about the same axis has an angular momentum, �, given by � = �� ¤ This is analogous to linear momentum: � = �� Angular momentum in general 4 ¨ Angular momentum of a point particle about an axis is defined by � � = �� sin � = ��� sin � = �-� = ��. � �- ¤ �⃗: position vector for the particle from the axis. ¤ �: linear momentum of the particle: � = �� �⃗ ¤ � is moment arm, or sometimes called “perpendicular . Axis distance.” �. Quiz: 1 5 ¨ A particle is traveling in straight line path as shown in Case A and Case B. In which case(s) does the blue particle have non-zero angular momentum about the axis indicated by the red cross? A. Only Case A Case A B. Only Case B C. Neither D. Both Case B Quiz: 24-1 answer 6 ¨ Only Case A ¨ For a particle to have angular momentum about an axis, it does not have to be Case A moving in a circle. ¨ The particle can be moving in a straight path. Case B ¨ For it to have a non-zero angular momentum, its line of path is displaced from the axis about which the angular momentum is calculated. ¨ An object moving in a straight line that does not go through the axis of rotation has an angular position that changes with time. So, this object has an angular momentum.
    [Show full text]
  • Rotational Motion and Angular Momentum 317
    CHAPTER 10 | ROTATIONAL MOTION AND ANGULAR MOMENTUM 317 10 ROTATIONAL MOTION AND ANGULAR MOMENTUM Figure 10.1 The mention of a tornado conjures up images of raw destructive power. Tornadoes blow houses away as if they were made of paper and have been known to pierce tree trunks with pieces of straw. They descend from clouds in funnel-like shapes that spin violently, particularly at the bottom where they are most narrow, producing winds as high as 500 km/h. (credit: Daphne Zaras, U.S. National Oceanic and Atmospheric Administration) Learning Objectives 10.1. Angular Acceleration • Describe uniform circular motion. • Explain non-uniform circular motion. • Calculate angular acceleration of an object. • Observe the link between linear and angular acceleration. 10.2. Kinematics of Rotational Motion • Observe the kinematics of rotational motion. • Derive rotational kinematic equations. • Evaluate problem solving strategies for rotational kinematics. 10.3. Dynamics of Rotational Motion: Rotational Inertia • Understand the relationship between force, mass and acceleration. • Study the turning effect of force. • Study the analogy between force and torque, mass and moment of inertia, and linear acceleration and angular acceleration. 10.4. Rotational Kinetic Energy: Work and Energy Revisited • Derive the equation for rotational work. • Calculate rotational kinetic energy. • Demonstrate the Law of Conservation of Energy. 10.5. Angular Momentum and Its Conservation • Understand the analogy between angular momentum and linear momentum. • Observe the relationship between torque and angular momentum. • Apply the law of conservation of angular momentum. 10.6. Collisions of Extended Bodies in Two Dimensions • Observe collisions of extended bodies in two dimensions. • Examine collision at the point of percussion.
    [Show full text]