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Submitted By akuzaty

Words 2739

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Words 2739

Pages 11

1

Vector and tensor analysis

Vectors and scalars Vector methods have become standard tools for the physicists. In this chapter we discuss the properties of the vectors and vector ®elds that occur in classical physics. We will do so in a way, and in a notation, that leads to the formation of abstract linear vector spaces in Chapter 5. A physical quantity that is completely speci®ed, in appropriate units, by a single number (called its magnitude) such as volume, mass, and temperature is called a scalar. Scalar quantities are treated as ordinary real numbers. They obey all the regular rules of algebraic addition, subtraction, multiplication, division, and so on. There are also physical quantities which require a magnitude and a direction for their complete speci®cation. These are called vectors if their combination with each other is commutative (that is the order of addition may be changed without aecting the result). Thus not all quantities possessing magnitude and direction are vectors. Angular displacement, for example, may be characterised by magnitude and direction but is not a vector, for the addition of two or more angular displacements is not, in general, commutative (Fig. 1.1). In print, we shall denote vectors by boldface letters (such as A) and use ordinary italic letters (such as A) for their magnitudes; in writing, vectors are usually ~ ~ represented by a letter with an arrow above it such as A. A given vector A (or A) can be written as

^ A AA;

1:1

^ where A is the magnitude of vector A and so it has unit and dimension, and A is a dimensionless unit vector with a unity magnitude having the direction of A. Thus ^ A A=A.

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...leading watchmaker of his day, and he built up a clientele that included many leading public figures and members of the European nobility. 2. Games that related in phisics. Attack Vector: Tactical Attack Vector: Tactical (AV:T) is a space combat war game published by Ad Astra Games. The game is consciously designed to model comparatively realistic space combat and eschew common conventions of space warfare. Attack Vector: Tactical is set in the "Ten Worlds," a region of space expanded ten parsecs from Sol. Some factors, unusual to the genre of space combat games, that AV:T includes: Beyond its physics model, is notable for being one of the few war games that attempts true three dimensional movement. While prior attempts (Battle Fleet Mars) have been made, most have failed in playability. AV:T handles three dimensionality with several play aids, such as the Attitude/Vector Information Display (AVID), which is a color coded polar projection of a sphere fixed relative to the hex map handling sighting angles and ship orientation for thrust, the Range/Angle Lookup Table (RALT), which is a table of the Pythagorean Theorem that's been color coded to reflect bearing angles used by the AVID, and tilt blocks, box miniatures and stacking tiles for on-map display of ship orientation. Attack Vector: Tactical won the Origins Award in 2004 for "Best New Miniatures Game". The Ten Worlds LESS....

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...coordinates. 4. CONVENTIONS OF REPRESENTATION For a given particle, position, velocity, and acceleration can be specified in three primary forms: vector form, rectangular coordinate form, and unit vector form. • The rectangular coordinate form: Quantity F at (x, y, z) • The vector form: F=Fxi+Fyj+Fzk • Unit vector form: [pic] 5. LINEAR PARTICLE MOTION A linear or rectilinear system is one in which particle move only in straight lines. The relationship among position, velocity, and acceleration for a linear system are given by [pic] [pic] [pic] The average velocity and acceleration over a period from [pic] to [pic] are [pic] [pic] 6. DISTANCE AND SPEED displacement (or linear displacement) is the net change in a particle’s position, as determined from the position function. Distance is a scalar quantity, equal to the magnitude of displacement. When specifying distance, the direction is not considered. Displacement = [pic] Similarly velocity and speed have different meanings: velocity is a vector quantity; Speed is a scalar quantity and is equal to the magnitude of velocity. 7. UNIFORM MOTION The term uniform motion means uniform velocity. The velocity......

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...1500 words, giving credible references on the use of physics in your daily activities. You need to mention 5 or more activities where physics is used. Remember to follow the APA style and give references. Physics is used in so many ways that most people do not even realize that they are using it. Even a stay at home mom uses physics more than one would think. Daily activities that many people do include physics without thinking about it, such as driving a car, using a headrest in a car, walking and running, flushing the toilet, and washing and drying clothes. Driving a car has many different aspects of physics involved, but today only acceleration, speed, and velocity will be discussed. People talk in terms of physics everyday without even knowing that is what they are discussing. For example, “speed” limit, how quickly a car can “accelerate,” and when they add a direction, they are actually talking about the velocity of a vehicle because velocity has a magnitude and direction, not just magnitude. According to Barry Parker in Issac Newton School of Driving, “you are accelerating and decelerating most of the time when you take a trip through the busy streets of a city, either by stepping on the gas, braking, or turning the steering wheel.” Basically, if someone gets in the driver seat of a car and drives, that person is changing the acceleration, speed, and velocity of the car that is being driven. A speed is just the scalar, magnitude with no direction, of velocity. The......

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... #2 The Study of Concurrent Forces with the Force Table Apparatus: Force table with 4 pulleys, centering ring and string, 50 g weight hangers, slotted weights, protractors, and rulers. Discussion: The force table is designed to help you study the properties of forces at known angles. Only when forces are along the same line do they add by ordinary algebra. If two or more forces on the same body form angles with each other, it is necessary to use geometry to find the amount and direction of their combined effect. Prior to Lab: Complete the calculations in the following. The component method of adding vectors is given here for three sample forces as follows: A = 2.45 N @ 40o B = 3.92 N @ 165o C = 3.43 N @ 330o Overview of the component method of vector addition. First make a neat drawing, not necessarily to exact scale, but reasonably accurate as to sizes and angles, placing the three forces on a diagram with a pair of x and y axes. Find the angle of each force with the x-axis. This angle is called the reference angle, and is the one used to calculate sines and cosines. Next compute the x- and y-components of the three forces, placing like components in columns. Place plus or minus signs on the various quantities according to whether an x-component is to the right or left of the origin, or whether a y-component is up or down relative to the origin. Add the columns with regard to sign (subtracting the minus quantities), and place the correct sign on each sum. The resulting...

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... 9 a. Computing Eigenvalues 9 b. Computing Eigen Vectors 10 5. Applications 10 a. Geology and Glaciology 10-11 b. Vibration Analysis 11-12 c. Tensor of Moment of Inertia 12 d. Stress Tensor 12 e. Basic Reproduction Number. 12 6. Conclusion 13 7. References 13 3 Abstract In abstract linear algebra, these concepts are naturally extended to more general situations, where the set of real scalar factors is complex numbers); the set of Cartesian the continuous functions, the multiplication is replaced by any the derivative from calculus). In such cases, the "vector" in "eigenvector" may be replaced by a more specific term, such as " This paper is about the various calculations various civil engineering problems like vibrational analysis,......

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...VECTOR FUNCTIONS VECTOR FUNCTIONS Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal vectors. MOTION IN SPACE: VELOCITY AND ACCELERATION Here, we show how the ideas of tangent and normal vectors and curvature can be used in physics to study: The motion of an object, including its velocity and acceleration, along a space curve. VELOCITY AND ACCELERATION In particular, we follow in the footsteps of Newton by using these methods to derive Kepler’s First Law of planetary motion. VELOCITY Suppose a particle moves through space so that its position vector at time t is r(t). VELOCITY Vector 1 Notice from the figure that, for small values of h, the vector r(t h) r(t ) h approximates the direction of the particle moving along the curve r(t). VELOCITY Its magnitude measures the size of the displacement vector per unit time. VELOCITY The vector 1 gives the average velocity over a time interval of length h. VELOCITY VECTOR Equation 2 Its limit is the velocity vector v(t) at time t : r(t h) r(t ) v(t ) lim h 0 h r '(t ) VELOCITY VECTOR Thus, the velocity vector is also the tangent vector and points in the direction of the tangent line. SPEED The speed of the particle at time t is the magnitude of the velocity vector, that is, |v(t)|. SPEED This is appropriate because, from Equation......

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...Tutorial 1 – Vector Calculus 1. Find the magnitude of the vector PQ with P (−1,2) and Q (5,5) . 2. Find the length of the vector v = 2,3,−7 . 3. Given the points in 3-dimensional space, P ( 2,1,5), Q (3,5,7), R (1,−3,−2) and S ( 2,1,0) . Does r PQ = RS ? ˆ ˆ 4. Find a vector of magnitude 5 in the direction of v = 3i + 5 ˆ − 2k . j r r ˆ ˆ ˆ j ˆ 5. Given u = 3i − ˆ − 6k and v = −i + 12k , find (a) u • v , r r (b) the angle between vectors u and v , r (c) the vector proju v , r r r r (d) the scalar component of v in the direction of u . 6. Given P (1,−1,3), Q ( 2,0,1) and R (0,2,−1) , find (a) the area of the triangle determined by the points P, Q and R. (b) the unit vector perpendicular to the plane PQR. 7. Find the volume of the parallelepiped determined by the vectors u = 4,1,0 , v = 2,−2,3 and r r r r r w = 0,2,5 . 8. Find the area of the parallelogram whose vertices are given by the points A (0, 0, 0), B (3, 2, 4), C (5, 1, 4) and D (2, -1, 0). ˆ j 9. Find the equation of the line through (2, 1, 0) and perpendicular to both i + ˆ and ˆ + k . j ˆ 10. Find the parametric equation of the line through the point (1, 0, 6) and perpendicular to the plane x+3y+z=5. 11. Determine whether the given lines are skew, parallel or intersecting. If the lines are intersecting, what is the angle between them? L1: x −1 y −3 z−2 = = 2 2 −1 x−2 y−6 z+3 L2 : = = 1 −1 3 12. Find the point in which the line x = 1 –t, y = 3t, z = 1 + t meets...

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...Physics PHYSICS FORMULAE AND PRINCIPLES Fundamental quantities and SI units The standard units were set up by the international system of units, SI The fundamental quantities, there symbols and there SI units a listed below Quantity symbol SI unit & unit symbol Length l metre (m) Mass m kilogram (k g) Time t second (s) Temperature T Kelvin (k) Current I Ampere (A) Amt. substance mol mole 6.02 x 10 23 (molecules) Luminous intensity - candela (c d) MULTIPLES AND SUBMULTIPLES Multiples are larger than the base units and submultiples are smaller than the base units Multiples Symbol Prefix Meaning Example Deca d 10 1 0 Decameter Hecto h 102 100 Hectometer ...

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...Introductory Physics I Elementary Mechanics by Robert G. Brown Duke University Physics Department Durham, NC 27708-0305 rgb@phy.duke.edu Copyright Notice Copyright Robert G. Brown 1993, 2007, 2013 Notice This physics textbook is designed to support my personal teaching activities at Duke University, in particular teaching its Physics 141/142, 151/152, or 161/162 series (Introductory Physics for life science majors, engineers, or potential physics majors, respectively). It is freely available in its entirety in a downloadable PDF form or to be read online at: http://www.phy.duke.edu/∼rgb/Class/intro physics 1.php It is also available in an inexpensive (really!) print version via Lulu press here: http://www.lulu.com/shop/product-21186588.html where readers/users can voluntarily help support or reward the author by purchasing either this paper copy or one of the even more inexpensive electronic copies. By making the book available in these various media at a cost ranging from free to cheap, I enable the text can be used by students all over the world where each student can pay (or not) according to their means. Nevertheless, I am hoping that students who truly ﬁnd this work useful will purchase a copy through Lulu or a bookseller (when the latter option becomes available), if only to help subsidize me while I continue to write inexpensive textbooks in physics or other subjects. This textbook is organized for ease of presentation and ease of learning. In particular, they......

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...LABORATORY MANUAL FHSP1014 PHYSICS I FOUNDATION IN SCIENCE (S) TRIMESTER 1 UNIVERSITI TUNKU ABDUL RAHMAN CENTRE FOR FOUNDATION STUDIES UNIVERSITI TUNKU ABDUL RAHMAN CENTRE FOR FOUNDATION STUDIES LABORATORY SAFETY RULES The following rules must be obeyed by all students in the science laboratory of the faculty. Willful or repeated inadvertent noncompliance may result in dismissal or suspension from the laboratories. I. No entry without permission: i) Outsiders are not allowed to enter the laboratory without permission. ii) Visitor must request for a lab coat from the laboratory officer before enteringinto the laboratory. iii) No student is allowed to enter the laboratory unless permission has been given by the laboratory officer or lecturer. II. At work in the laboratory: i) No experiment may be attempted without the knowledge and permission of the lecturer or lab officer. ii) Laboratory coat must be worn at all times in the laboratory. iii) Students must wear covered shoes in the laboratory. Students wearing open- toed shoes such as slippers or sandals are not allowed to work in the laboratory. iv) Safety glasses must be worn when necessary. v) Mobile phones are to be switched off at all times in the laboratory. vi) Do not smoke, drink, eat, bite nails or pencils, or apply cosmetics in the laboratory. vii) Do not pipette chemicals using the mouth. viii) Do not taste any chemicals, including diluted solutions....

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...Find parametric equations for the line L which contains A(1, 2, 3) and B(4, 6, 5). Solution: To get the parametric equations of L you need a point through which the line passes and a vector parallel to the line. −→ Take the point to be A and the vector to be the AB. The vector equation of L is −→ −→ r(t) = OA + t AB = 1, 2, 3 + t 3, 4, 2 = 1 + 3t, 2 + 4t, 3 + 2t , where O is the origin. The parametric equations are: x = 1 + 3t y = 2 + 4t, z = 3 + 2t t ∈ R. Problem 1(b) - Fall 2008 Find parametric equations for the line L of intersection of the planes x − 2y + z = 10 and 2x + y − z = 0. Solution: The vector part v of the line L of intersection is orthogonal to the normal vectors 1, −2, 1 and 2, 1, −1 . Hence v can be taken to be: i j k v = 1, −2, 1 × 2, 1, −1 = 1 −2 1 = 1i + 3j + 5k. 2 1 −1 Choose P ∈ L so the z-coordinate of P is zero. Setting z = 0, we obtain: x − 2y = 10 2x + y = 0. Solving, we ﬁnd that x = 2 and y = −4. Hence, P = 2, −4, 0 lies on the line L. The parametric equations are: x =2+t y = −4 + 3t z = 0 + 5t = 5t. Problem 2(a) - Fall 2008 Find an equation of the plane which contains the points P(−1, 0, 1), Q(1, −2, 1) and R(2, 0, −1). Solution: Method 1 −→ −→ Consider the vectors PQ = 2, −2, 0 and PR = 3, 0, −2 which lie parallel to the plane. Then consider the normal vector: i j k −→ −→ n = PQ × PR = 2 −2 0 3 0 −2 = 4i + 4j + 6k. So the equation of the plane is given by: 4, 4, 6 · x + 1, y , z − 1 = 4(x + 1) + 4y + 6(z − 1) = 0. Problem 2(a) - Fall......

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...*Physics Prelims (1-7).qxd 12/11/08 1:00 PM Page 1 SCIENCE VISUAL RESOURCES PHYSICS An Illustrated Guide to Science The Diagram Group *Physics Prelims (1-7).qxd 12/11/08 1:00 PM Page 2 Physics: An Illustrated Guide to Science Copyright © 2006 The Diagram Group Author: Derek McMonagle BSc PhD CSci CChem FRSC Editors: Catherine Gaunt, Jamie Stokes Design: Anthony Atherton, Richard Hummerstone, Lee Lawrence, Tim Noel-Johnson, Phil Richardson Illustration: Peter Wilkinson Picture research: Neil McKenna Indexer: Martin Hargreaves All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval systems, without permission in writing from the publisher. For information contact: Chelsea House An imprint of Infobase Publishing 132 West 31st Street New York NY 10001 For Library of Congress Cataloging-in-Publication Data, please contact the Publisher ISBN 0-8160-6167-X Chelsea House books are available at special discounts when purchased in bulk quantities for businesses, associations, institutions, or sales promotions. Please call our Special Sales Department in New York at 212/967-8800 or 800/322-8755. You can find Chelsea House on the World Wide Web at http://www.chelseahouse.com Printed in China CP Diagram 10 9 8 7 6 5 4 3 2 This book is printed on acid-free paper. *Physics Prelims (1-7).qxd 12/11/08 1:00......

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...No. Information on Every Subject 1. Unit Name: Physics I 2. Code: FHSP1014 3. Classification: Major 4. Credit Value: 4 5. Trimester/Year Offered: 1/1 6. Pre-requisite (if any): No 7. Mode of Delivery: Lecture, Tutorial, Practical 8. Assessment System and Breakdown of Marks: Continuous assessment: 50% - Theoretical Assessment (Tests/Quizzes/Case Studies) (30%) - Practical Assessment (Lab reports/Lab tests) (20%) Final Examination 9. 10. 50% Academic Staff Teaching Unit: Objective of Unit: The aims of this course are to enable students to: • appreciate the important role of physics in biology. • elucidate the basic principles in introductory physics enveloping mechanics, motion, properties of matter and heat. • resolve and interpret quantitative and qualitative problems in an analytical manner. • acquire an overall perspective of the inter-relationship between the various topics covered and their applications to the real world. • acquire laboratory skills including the proper handling and use of laboratory apparatus and materials. 11. Learning Outcome of Unit: At the end of the course, students will be able to: 1. Identify and practice the use of units and dimensional analysis, uncertainty significant figures and vectors analysis. 2. Apply and solve problems related to translational and rotational kinematics and dynamics in one and two dimensions. 3. Apply and solve problems related to......

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...PhysRozz Midterm 2012 1. Which is not a vector quantity? 1) electric charge 2) displacement 5. As the angle between two concurrent forces decreases, the magnitude of the force required to produce equilibrium 1) decreases 3) velocity [via06-07] 2) increases 3) remains the same 4) magnetic field strength 2. An astronaut standing on a platform on the Moon drops a hammer. If the hammer falls 6.0 meters vertically in 2.7 seconds, what is its acceleration? 6. A child walks 5.0 meters north, then 4.0 meters east, and finally 2.0 meters south. What is the magnitude of the resultant displacement of the child after the entire walk? 1) 4.4 m/s 2 2) 1.6 m/s 2 1) 1.0 m 2) 5.0 m 3) 2.2 m/s 2 4) 9.8 m/s 2 3) 3.0 m 4) 11.0 m 3. A 2.00-kilogram object weighs 19.6 newtons on Earth. If the acceleration due to gravity on Mars is 3.71 meters per second 2 , what is the object’s mass on Mars? 1) 2.64 kg 2) 2.00 kg 3) 19.6 N 7. The diagram above represents a spring hanging vertically that stretches 0.075 meter when a 5.0newton block is attached. The spring-block system is at rest in the position shown. 4) 7.42 N 4. A car moves with a constant speed in a clockwise direction around a circular path of radius r, as represented in the diagram above. The value of the spring constant is 1) 38 N/m When the car is in the position shown, its acceleration is directed toward the 1) south 2) east 3) west 2) 650...

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...COURSE PHYSICS 1 (CORE MODULES) Coordinators Dr. Oum Prakash Sharma Sh. R.S. Dass NATIONAL INSTITUTE OF OPEN SCHOOLING A-25, INSTITUTIONAL AREA, SECTOR-62, NOIDA-201301 (UP) COURSE DESIGN COMMITTEE CHAIRMAN Prof. S.C. Garg Former Pro-Vice Chancellor IGNOU, Maidan Garhi, Delhi MEMBERS Prof. A.R. Verma Former Director, National Physical Laboratory, Delhi, 160, Deepali Enclave Pitampura, Delhi-34 Dr. Naresh Kumar Reader (Rtd.) Deptt. of Physics Hindu College, D.U. Dr. Oum Prakash Sharma Asstt. Director (Academic) NIOS, Delhi Prof. L.S. Kothari Prof. of Physics (Retd.) Delhi University 71, Vaishali, Delhi-11008 Dr. Vajayshree Prof. of Physics IGNOU, Maidan Garhi Delhi Sh. R.S. Dass Vice Principal (Rtd.) BRMVB, Sr. Sec. School Lajpat Nagar, New Delhi-110024 Dr. G.S. Singh Prof. of Physics IIT Roorkee Sh. K.S. Upadhyaya Principal Jawahar Navodaya Vidyalaya Rohilla Mohammadabad (U.P.) Dr. V.B. Bhatia Prof. of Physics (Retd.) Delhi University 215, Sector-21, Faridabad COURSE DEVELOPMENT TEAM CHAIRMAN Prof. S.C. Garg Former Pro-Vice Chancellor IGNOU, Delhi MEMBERS Prof. V.B. Bhatia 215, Sector-21, Faridabad Prof. B.B. Tripathi Prof. of Physics (Retd.), IIT Delhi 9-A, Awadhpuri, Sarvodaya Nagar Lucknow-226016 Sh. K.S. Upadhyaya Principal Navodaya Vidyalaya Rohilla Mohammadabad, (U.P.) Dr. V.P. Shrivastava Reader (Physics) D.E.S.M., NCERT, Delhi EDITORS TEAM CHAIRMAN Prof. S.C. Garg Former Pro-Vice Chancellor IGNOU, Delhi MEMBERS Prof. B.B. Tripathi Prof. of Physics......

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