G11-Physics-STB-2023-web.pdf

Physics Student Textbook Grade 11 Take Good Care of This Textbook This textbook is the property of your school. Take good care not to damage or lose it. Here are 10 ideas that help you take care of the book. 1. Cover the book with protective materials. 2. Always keep the book in a clean dry place. 3. Be sure your hands are clean when you use the book. 4. Do not write on the cover or inside pages. 5. Use a piece of paper or cardboard as a bookmark. 6. Never tear or cut out any pictures or pages. 7. Repair any torn pages with paste or tape. 8. Pack the book carefully when you place it in your school bag. 9. Handle the book with care when passing it to another person. 10. When using a new book for the first time, lay it on its back. Open only a few pages at a time. Press lightly along the bound edge as you turn the pages. This will keep the cover in good condition. Physics 11 FDRE, MINISTRY OF EDUCATION HAWASSA UNIVERSITY First Published in 2023 by the Federal Democratic Republic of Ethiopia, Ministry of Education, under the General Education Quality Improvement Program for Equity (GEQIP-E) supported by the World Bank, UK’s Department for International Development/DFID-now merged with the Foreign, Common wealth and Development Office/FCDO, Finland Ministry for Foreign Affairs, the Royal Norwegian Embassy, United Nations Children’s Fund/ UNICEF), the Global Partnership for Education (GPE), and Danish Ministry of Foreign Affairs, through a Multi Donor Trust Fund. ©2023 by the Federal Democratic Republic of Ethiopia, Ministry of Education. All rights reserved. The moral rights of the author have been asserted. No part of this textbook reproduced, copied in a retrieval system, or transmitted in any form or by any means including electronic, mechanical, magnetic, photocopying, recording or otherwise, without the prior written permission of the Ministry of Education or licensing in accordance with the Federal Democratic Republic of Ethiopia as expressed in the Federal Negarit Gazeta, Proclamation No. 410/2004 - Copyright and Neighboring Rights Protection. The Ministry of Education wishes to thank many individuals, groups, and other bodies involved – directly or indirectly – in publishing this Textbook. Special thanks are due to Hawassa University for their huge contribution to the development of this textbook in collaboration with Addis Ababa University, Bahir Dar University, and Jimma University. Copyrighted materials used by permission of their owners. If you are the owner of copyrighted material not cited or improperly cited, please contact the Ministry of Education, Head Office, Arat Kilo, (P.O.Box 1367), Addis Ababa Ethiopia. Printed by: GRAVITY GROUP IND LLC 13th Industrial Area, Sharjah, UNITED ARAB EMIRATES Under Ministry of Education Contract no. MOE/GEQIP-E/LICB/G-01/23 ISBN: 978-99990-0-034-5 CONTENTS Unit 1 1. Physics and Human Society 1 1 U nit 2 2. Vectors 17 17 Unit 3 3. motion in one and two dimensions 47 47 U nit 4 4. Dynamics 95 95 U nit 5 5. Heat Conduction and Calorimetry 175 175 Unit 6 6. Electrostatics and Electric Circuit 217 217 Unit 7 7. Nuclear Physics 287 287 1. PHYSICS AND HUMAN SOCIETY After completing this unit, student will be able to: get acquainted with the impact of physics on society. familiarize themselves with physics communities and their roles. develop basic understanding of the making of physics knowledge. familiarize themselves with basic principles and applications of physics in various disciplines. acquire basic knowledge and understandings of nature and appreciate it. update themselves with the current status of physics. 1.1 Importance of Physics to Society At the end of this section you will be able to: explain the importance of physics to society Brainstorming 1. What is the benefit of physics for society? 2. What are the technologies directly related to physics that benefit society? 1 Physics Student Text Book 2 The Influence of Physics on Society Explain the important of physics in transport, electronics and health. Ä Activity 1.1 1. Explain the influence of physics on society in transport, health, economy, technology, etc 1.2 Physics Communities and Their Roles At the end of this section you will be able to: discuss about physics communities and their roles. Brainstorming 1. What is the purpose of establishment of physics communities? What are the technologies directly related to physics that benefit society? 3 Physics Student Text Book Objectives of EPS ã To promote physics education and research in the country; ã To organize and coordinate various conferences on physics education and exchange of scientific information; ã To popularize physics in order to make students develop interest in physics; ã To promote active participation of Ethiopian physicists and the general public in the design and implementation of the physics curriculum; ã To create a means for disseminating scientific information; etc. equipment, computing facilities, provide scholarship…etc. Group Work Ä Activity 1.2 What are the benefits of physics communities? 4 1.3 Making of Physics Knowledge At the end of this section you will be able to: discuss how scientific knowledge is constructed. discuss the roles that the learning of physics plays to the individual intellectual satisfaction. Brainstorming 1. What is the process of gaining knowledge in physics? Experimental knowledge I. Sensory perception: sensory perception is perhaps the dominant source of experiential knowledge, it immediately raises a critical question. We gather knowledge by seeing, touching hearing, etc. II. Introspection: Introspection is like a sixth sense that looks into the most intimate parts of our minds, which allows us to inspect how we are feeling and how our thoughts are operating. If I go to a doctor complaining of an aching back, she’ll ask me to describe my pain. Through introspection I then might report, “Well, it’s a sharp pain that starts right here and stops right here.” The doctor herself cannot directly experience what I do and must rely on my introspective description. III. Memory: a memory is like a recording device that captures events that one can experience more or less in the order that they occur. IV. Testimony: Testimonies from written sources are usually more reliable than 5 Physics Student Text Book oral sources, but much depends on the integrity of the author, publisher, and the methods of fact-gathering. Non-Experiential Knowledge Scientific method Examples of Scientific Method in Physics Experiment: You decide to heat water at different altitudes and record the boiling temperature. Analysis: Altitude (m) Boiling point of water ( Co) 0 100 150 99.5 305 99 610 98 1524 95 6 Ä Activity 1.3 1. Give some examples in the process of scientific method of knowledge gain. 1.4 The Mission of Physics and Career Awareness At the end of this section you will be able to: explain the job opportunities concerning to physics. Brainstorming 1. Mansion lists of career related to physics. Typical careers in physics Figure 1: Career opportunities 7 Physics Student Text Book Physics careers in space and astrono- my Figure 2: A person observing stars through telescope Physics careers in healthcare Figure 3: observing CT scan result Physics careers in engineering Figure 4: engineers working place 8 Physics careers in energy Figure 5: wind energy production site Physics careers in technology Figure 6: application of machines 9 Physics Student Text Book Geophysics and meteorology careers Figure 7: Metrological report Research scientist careers Figure 8: Data scientist 1.5 Current Status of Physics At the end of this section you will be able to: list at least five recent new developments or discoveries in the fields of physics. Brainstorming Go to library and websites search out what are the newest discoveries in physics 10 Major recent discoveries in Physics Discovery of Exoplanets: Figure 9: Artist’s impressio of exo planet orbiting two stars https://www.nobelprize.org/interactive-visualisations-the-discovery-of-exoplanets/ Black hole: Figure 10: Image of Black Hole https://en.wikipedia.org/wiki/Roger_Penrose, https://en.wikipedia.org/wiki/Reinhard_Genzel https://en.wikipedia.org/wiki/Andrea_M._Ghez 11 Physics Student Text Book Quantum cryptography: High energy physics (Particle Accelerators) Gravitational wave (Large scale structures) Global warming: Figure 11: gravitational wave experimenting areas 12 _ Figure 12: illustrating the effect of global warming. James Webb Space Telescope (JWST) (−223 °C) Figure 13: image of JWST 13 Physics Student Text Book Figure 14: Star formation in Carina Nebula. Future perspectives. Figure 15: Some photographs of the newly inaugurated Ethiopian Science and Art Museum. 14 Figure 16: Interior of the Dome-shaped Science and Art Museum Ä Activity 1.4 1. Google the web or go to the library to find some information to find out what is going on https://en.wikipedia.org/wiki/Ethiopia_Museum_of_Art_and_ Science. Questions 1.1 Explain current status of physics. 1. What do you think the future of physics looks like? Your opinions in one paragraph or two with justification. 2. What is the significances of the two discoveries- gravitational wave, and Higgs boson? 15 Physics Student Text Book UNIT SUMMARY Physics improves our quality of life by providing the basic understanding necessary for developing new instrumentation and techniques for medical applications, such as computer tomography, magnetic resonance imaging, positron emission tomography, ultrasonic imaging, and laser surgery. Transportation vehicles such as automobiles, airplanes and space shuttles could not be constructed without the help of physics experts. Physics is also useful for military purposes, informing the design of weapons. Many of the scientists responsible for inventing the atomic bomb were physicists, and today physicists are involved in the creation of nuclear weapons. Physics communities are an organized group of persons associated together for scientific purposes. This organization is established at the level of national, continent as well as worldwide based on their aims. The scientific method is an ordered series of steps to acquire knowledge based on experimental evidence. The mission of physics is to advance science, engineering, and innovation throughout the world for the benefit of all and serving Society. Physics have a lot of job opportunities in the fields of Astronomy, healthcare, engineering, energy, technology, meteorology, etc New discoveries of Exoplanets, Black hole, Quantum cryptography, High energy physics, Gravitational wave, Global warming, James Webb Space Telescope, etc have been seen in recent years. END OF UNITQUESTIONS 1. What is the importance of learning physics in your opinion? 2. What did you benefit from knowing physics at this level? 3. List at least ten technological benefits that physics has contributed for society. 4. What is the benefit of establishing physics communities? 5. Is there a physics community in your high school? If the answer is yes, what is it helpful for? 6. What process skills should you use investigating a phenomenon? 7. What are the process skills you are using in your in investigating a phenomenon? 8. What are the career opportunities physics have? 9. At the forefront in physics research, there are many countries collaborating. Why is that? 16 2. VECTORS After completing this unit you should be able to: familiarize themselves with basic principles of vector operations interpret physical phenomena using the concept of vector develop skills of using the concept of vector in solving various problems. 2.1 Vectors and Types of Vectors After completing this section, you should be able to: describe the difference between vector and scalar quantities. list down the common vector quantities in our everyday life. discuss geometric representation of vectors. give the definitions of the different types of vectors. Brainstorming 1. Why do we need to know the direction of some quantities in order to have a complete sense of them? If a girl told you that the velocity of a delivery truck was 50 km/h, has she fully expressed the motion of the truck? 17 Physics Student Text Book Ä Activity 2.1 Categorize the quantities listed in Table 2.1 as scalars or vectors and prepare your own list of vector and scalar quantities that are common in our daily practices. Table 2.1 Table of physical quantities Quantity Scalar/vector Quantity Scalar/Vector Area Density Distance Weight Pressure Representation of vectors Figure 2.1 Representing a vector by an arrow drawn to scale. Ruler, Protractor and Graph paper used 18 Illustration of the concept of vectors in practice Figure 2.2 Hydro power dam and force vectors Figure 2.3 Force vectors in action Ä Activity 2.2 Ask your friend to stand at a mark in the middle of the volleyball field. Tell her to walk through a certain distance in any direction she chooses. Use a measuring tape to measure the distance between her starting point and the ending point. Take the starting point as the origin of coordinates with the x axis pointing toward East and the y axis toward North. Show the displacement of the girl graphically on a graph paper. Types of vectors 19 Physics Student Text Book - Av A A A A Figure 2.4 (a) Parallel vectors, (b) Antiparallel vectors, (c) Equal vectors, (d) Negative of a vector Example 2.1 Find the negative of each of the following vectors. a. dosplacement vector= 60km toward North b. Velocity vector = 100km/h towards 200 southeast. 20 Solution The negative of a vector has the same magnitude as the original vector in opposite direction. (a) If S = 60 km, toward North, then - S = 60 km, toward South (b) S = 100 km/h, toward 20o South of East, then - S = 100 km/h, toward 20o North of West 1. How do you represent a vector graphically? 2. Why do we use of scale diagram in vector drawing? 3. How does vector A differ from vector - A ? 2.2 Graphical Method of Addition of Vectors in Two Dimensions (2-D) After completing this section, you should be able to: define the term resultant vector. explain the geometric method for addition and subtraction of vectors in a plane. apply geometric method of addition of vectors to find resultant of vectors in two dimensions. Brainstorming 1. In grade 10 physics you have studied the properties of vectors in one dimension. Is it possible to add two vector quantities such as displacement vectors the way we do with scalar quantities like mass? Share your answer with your classmates. Figure 2.5 A vector diagram with scale of a trip in a park 21 Physics Student Text Book Triangle Law of Vector Addition s1 s2 S S s1 s2 S = S1 + S 2 Figure 2.6 Triangle law of addition of vectors: S1 and S 2 are placed head to tail. 22 Ä Activity 2.3 1. Use graph paper, a protractor, and a ruler and apply triangle law of vector addition to show that A + B = B + A. Use vector A of magnitude 5 units and vector B of 9 units, and the angle between the vectors to be 60o. Parallelogram law of vector addition Figure 2.7. Paralelogram Law of addi- tion of vectors: : S and 1 S2 are placed tail to tail. Ä Activity 2.4 Discussion with your classmates It’s a common error to conclude that if C = A + B , then the magnitude C should equal the magnitude. A plus the magnitude B. What is wrong with this conclusion? R=A +B +C +D Figure 2.8 polygon law of vector addition Exercise 2.1 What is the result of adding a vector to the negative of itself? 23 Physics Student Text Book Ä Activity 2.5 How can you add individual displacement vectors to determine total displacement? Materials ‣ A 20 m tape measure, Field marker (or flag), Graph paper ruler and Protractor Procedure ‣ Mark a point at the center of the field with a flag. ‣ Indicate the North-South direction taking the marked point as the origin ‣ Ask your friend to walk the following distances starting from the flag: 12 m toward North, and 16 m toward East, 10 m toward South-East. ‣ Mark the final point and use the measuring tape to measure magnitude of his/her displacement from the starting point. ‣ Show your vector diagram on a graph paper ‣ Use a scale of 1 cm for 1 m of displacement. ‣ Use a protractor to determine the direction if displacement. What is the magnitude and direction of the displacement? Subtraction of Vectors - B B A - B B A - B C = A - B = A + (-B) Figure 2.9 Subtraction of vectors 24 1. Which one of the vector diagrams (Figure 2.10) satisfies the vector equation C =A +B? Figure 2.10 2. Three displacement vectors A, B, C are specified by their magnitudes A = 10 cm, B = 7 cm, C = 8 cm, respectively, and by their respective direction angles (with the +x direction) of a = 53o, b = 120o, c = 210o. Choose a convenient scale and use a ruler and a protractor to find the magnitude and direction of the vector sum: (a) R = A + B (b) P = A - B (c) Q = A - 2B + C a) the magnitude C equal to the sum of the magnitude A and magnitude B? b) the magnitude C equal to the difference of magnitude A and magnitude B? 3. Suppose an insect lands on a sheet of graph paper at a point located 12 cm to the right of the left edge and 8 cm above the bottom edge and walks slowly to a point located 7 cm from the left edge and 4 cm from the bottom edge. Choose the rectangular coordinate system with the origin at the lower left-side corner of the paper and find the displacement vector of the insect. Use a ruler, a graph paper and make use of graphical representation of vectors 4. What is the minimum number of vectors of equal magnitudes required to produce a zero resultant? 25 Physics Student Text Book 2.3 Algebraic Method of addition of vectors in Two Dimensions (2-D) After completing this section, you should be able to: apply algebraic method to find the resultant of two collinear vectors, and perpendicular vectors. apply trigonometry to determine the direction of a resultant vector in terms of the angle it forms with a reference direction. compare the resultant of two given vectors as determined by way of geometric construction and by analytic method. resolve a vector into its components. find the resultant of two or more vectors applying component method. define unit vector and determine a unit vector in the direction of a given vector. apply the unit representation of vectors to determine the resultant of two or more vectors. Brainstorming Do you recall the different ways in which vectors that represent similar quantities can be added to/ subtracted from one another? Mention two ways of vector addition based on your physics lesson in grade 10. Adding two collinear vectors F1 F2 R 26 Figure 2.11 Vectors in the same direction R Rv = Fv1 + Fv2 Fv1 Fv2 v R Figure 2.12 Vectors in opposite directions Exercise 2.2 The magnitudes of two vectors Av and Bv are 13 units and 9 units, respectively. What are the largest and smallest possible values for the magnitude of the v = Av + Bv ? resultant vector R Example 2.2 A man rows his boat along the direction of flow of a river (downstream). If the boat can sail in still water at 0.50 m/s, and the river flows at 0.30 m/s, what is the resultant velocity of the boat? Let the magnitude of velocity of the river be A and that of the boat be B. A = 0.30 m/s B = 0.50m/s The resultant velocity is Rv = Av + Bv 27 Physics Student Text Book Since both vectors are along the same direction, the magnitude of the resultant velocity is given by: R = A + B = 0.30 m/s + 0.50m/s. R = 0.80 m/s. The direction of the resultant is along the direction of the river flow. Therefore, the resultant velocity is: v = 0.8m/s, downstream R Exercise 2.3 If the man in Example 2.2 rows his boat upstream (in opposite direction to the river flow), what would his resultant velocity be? Example 2.3 Two men are pushing a block along a horizontal surface by exerting oppositely directed forces of magnitudes 200 N and 100 N respectively, as shown in Figure 2.13. What is the resultant force applied on the block by the two men? Figure 2.13 Two men pushing the block in opposite directions Solution: The two men applied two different forces on the same block. Let the magnitude of the force exerted by the man on the right be F1 = 200 N and that of the man on the left be F2 = 100 N. Rv = Fv1 + Fv2 The magnitude of the resultant force is R = F1-F2 = 200N - 100N = 100 N The direction of the resultant is along the direction of the 200 N (larger) force. v = 100N, toward the right. R 28 Adding two perpendicular vectors Recall Figure 2.14 A Right-angled triangle c 2 = a 2 + b 2, c = a 2 + b 2 opposite b sin i = hypotenuse = c adjacent a cos i = hypotenuse = c opposite b tan i = adjacent = a Example 2.4 An ant, starting at a point, walked through a distance of 40 cm due South and then 50 cm due West. What is the resultant displacement of the ant from its starting point? Solution Placing the tail of the vectors at the origin of coordinates, the vector diagram of the motion is shown in Figure 2.15. East Figure 2.15 Vector diagram of motion of the ant 29 Physics Student Text Book using pythagorean theorem, the magnitude of the resultant displacement is 2 S = S12 + S22 = (40cm) 2 + (50cm) = 4100cm2 = 64cm Direction of the reultant displacement is shown by the angle i , where opposite 40cm tan i = adjacent = 30cm = 0.8, i = 38.6 o the resultant is Sv = 64cm, 38.6 o South of West Components of a vector Av Av Av Av x Av y Av Av = Av x + Av y y y 𝐴𝐴⃗ 𝐴𝐴⃗ 𝐴𝐴⃗y 𝐴𝐴⃗y α θ x x o o 𝐴𝐴⃗x 𝐴𝐴⃗x (a) (b) v x and Av y Figure 2.16 Vector \vec{A} in terms of its vector components A Av Ax = A cos i Ay = A sin i 30 Note that: In finding the x- and y- components of a vector, we associate cosine with the x-component and sine with the y-component, as in Figure 2.16a. This as- sociation is due to the fact that we chose to measure the angle a with respect to the positive x-axis. If the angle were measured with respect to the y-axis, as in Figure 2.16b, the components would be given by Ax = A sin a and Ay = A cos a Key terms Component of a vector: the projection of a vector on the coordinate axes. Resolution of a vector: Breaking a vector into its components Example 2.5 A force vector of magnitude of 35 N is pushing on a box placed on the horizontal ground as shown in Figure 2.17a. If the force makes an angle of 40o above the negative x axis, what are the x and y components of the force? Figure 2.17 A pushing force Solution Force F can be A placed on the xy plane as shown in Figure 2.17b. Remember that translation of a vector does not alter the magnitude and direction of the vector. The x component of the force is Fx = F cos 40o = 35 N x (0.766) = 26.8 N, and the y component is Fy = -F sin 40o = -35 N x (0.643) = -22.5 N Finding the resultant of vectors by component method Figure 2.18 Vectors \vec{A} and \vec{B} added by Component Method 31 Physics Student Text Book Rv = Rv x + Rv y, where Rv x = Av x + Bv x, and Rv y = Av y + Bv y R = Rx2 + Ry2 v x Rv y R i = tan -1 c m Ry Ry tan i = R x Rx Attention ã if both Rx and Ry are positive numbers, angle i is above the +x axis ã if Rx is negative number and Ry is positive number, angle i is above the -x axis ã if both Rx and Ry are negative numbers, angle i is below the -x axis ã if Rx is positive number and Ry is negative number, angle i is be- low the +x axis Exercise 2.4 Under what circumstances would a vector have components that are equal in magnitude? Exercise 2.5 A vector has an x-component of - 35 units and a y-component of 40 units. Find the magnitude and direction of the vector considering that the positive x direction indicates East and that the positive y direction indicates North. Example 2.6 Consider two displacement vectors Av and Bv of magnitudes 100 m and 200 m respectively, Figure 2.21. Find the magnitude and direction of the resultant displacement vector. 32 Figure 2.19 Solution We first resolve the vectors into their x and y components. Using trigonometry, we get Ax = A cos 50o = 100(0.643) m = 64.3 m, and Ay = A sin 50o = 100(0.766) m = 76.6 m Bx = -B cos 30o = -200 (0.866) m = -173.2 m, and By = B sin 30o = 200(0.50)m = 100 m The vector components of the resultant are R v x = Av x + Bv x , and Rv y = Av y + Bv y , where Rx = Ax + Bx = 64.3 m + (-173.2 m) = -108.9 m and Ry =Ay + By = 76.6 m + 100 m = 176.6 m. Then we place the components of the resultant on the xy-plane as in Figure 2.19 (b). The magnitude of the resultant is R = Rx2 + Ry2 = (- 108.9m) 2 + (176.6m) 2 = 207.5m and i = tan -1 b 108.9m l = tan -1 (1.62) = 58.3c 176.6m Therefore, Rv = 207.5m, i = 58.3c above the negative x axis Exercise 2.6 Vector Av has a magnitude of 30 units and points in the positive y-direction. When vector Bv is added to Av , the resultant vector Av + Bv points in the negative y-direction with a magnitude of 10 units. Find the magnitude and direction of Bv. 33 Physics Student Text Book Example 2.7 A girl walked across a large field through the following distance in the given order:72 m toward 32o East of North, 57 m toward 370South of West, 18 m toward South. Find the magnitude and direction of the resultant displacement of the girl? Solution The displacement vectors are placed on the xy-plane as shown in Figure 2.20. (a) (b) Figure 2.20 (a) The actual path the girl followed, (b) Vector diagram Solution The components of the displacement vectors are calculated as in Table 2.2 below. Table 2.2 x component y component S1 72 (sin 32 ) m = 38.2 m o 72 (cos 32o) m = 61.1 m S2 -57(cos 37o) m = - 45.6 m -57 (sin37o) m = -34.2 m S3 0m -18 m Rx = S1x + S2x + S3x = -7.4 m Rx = S1y + S2y + S3y = 8.9 m R = Rx2 + R y2 = (- 7.4m) 2 + (8.9m) 2) = 11.6m (8.9m) and i = tan -1 (7.4m) = tan -1 (1.2) = 50.2c above the nagative x axis Therefore, Rv = 11.6m, i = 50.2c N of W Did you know? Analytic methods of vector algebra are used routinely in mechanics, electricity, and mag- netism. They are important mathematical tools of physics. 34 Exercise 2.7 A car travels 20 km due north and then 25 km in a direction 60° West of North. Use both graphical and algebraic methods to find the magnitude and direction of a single vector that gives the net effect of the car’s trip. Unit Vectors ^ it Figure 2.21 Unit vectors on the xy-plane Av x = Ax it Av y = Ay jt Av = Ax it + Ay jt Av = Ax it + Ay jt Bv = Bx it + By jt Rv = Av + Bv Rv = (Ax it + Ay jt) + (Bx it + By jt) = (Ax + Bx) it + (Ay + By j) jt = Rx it + Ry jt 35 Physics Student Text Book Av = Ax it + Ay jt Av Av UvA = A Example 2.8 Given force vector Fv = (12it - 16jt) N , what is a unit vector along Fv ? solution Fv 12it - 16jt 12it - 16it UvF = F = 2 2 = 20 = (0.6it - 0.8jt) N 12 + (- 16) See that the magnitude of UvF is UF = ]0.6g2 + (0.8) 2 = 1 Example 2.9 Vector Av has x and y components of 4 units and 2 units, respectively and vector Bv has the corresponding components of -9 units and 3 units respectively. Find (a) the vector components of their resultant, (b) the magnitude and direction of their resultant. solution Given Av = Ax it + Ay jt = (4it + 2jt) units Bv = Bx it + By jt = (- 9it + 3jt) units The resultant Rv = (Ax + Bx) it + (Ay + By j) jt = (4 - 9) it + (2 + 3) jt = (- 5jt + 5jt) units (a) The x and y components of Rv are = Rx =- 5 units, Rv x = Rx it =- 5it Ry =- 5 units, Rv y = Ry ijt =- 5jt (b) the magniture of Rv is R = Rx2 + Ry2 = (- 5) 2 + 52 = 50 = 5 2 units i = tan -1 c R m = tan -1 b 5 l = 45c Ry 5 x As R v is along the negative x direction and Rv y is along the positive y direction, we see that Rv is in the second quadrant and it makes angle of 45o above the negative x axis. 36 Example 2.10 An unknown vector D is added to vector C = (- 4it + 5jt) units and the resultant R = C + D has x and y components of each -1 and 1 units, respectively Find the magnitude of the unknown vector. Solution Given Cv = (- 4it + 5jt), Dv = (Dx it + Dy jt) and Rv = (- it + jt) For Rv = Cv + Dv , we have Rx = Cx + Dx, - 1 =- 4 + Dx, Dx = 3 units Ry = Cy + Dy, 1 = 5 + Dy, Dy =- 4 units Vector Dv = (3it - 4jt) units Exercise 2.8 The vector sum Pv + Qv is a unit vector along the positive x axis. If Pv = it - jt , find Qv. Exercise 2.9 If vector 0.4 it + b jt is a unit vector, then what is the value of b? Example 2.11 Traditional Maresha A farmer is ploughing the field using traditional Maresha (plow) pulled by two oxen Figure 2.24a. The two animals are pulling the beam (Mofer) with a force Fv2 of 1200 N at angle of 40o from the horizontal and the farmer’s force Fv1 on the handle (Erf) is 150 N at 60o above the horizontal as shown in Figure 2.24b. Find the resultant horizontal pulling force exerted by the farmer and the oxen on the Maresha? At this point assume all the forces are acting at point K. For the whole system to work the resultant downward vertical force must be slightly greater than the upward. Explain why. 37 Physics Student Text Book Figure 2.22 (a) A farmer plowing his (b) Schematic diagram of the Maresha farm Solution The forces that are acting on the Maresha are the force exerted by the farmer on the Erf, Fv1 , and the pulling force on the Mofer, Fv2. Both Fv1 and Fv2 have x components and the sum of these components gives the resultant horizontal pulling force. Both F1x and F2x are along the positive x direction. Fx = F1x + F2x = F1 cos i1 + F2 cos i2 = 150N cos 60c + 1200N cos 40c = 75N + 919N = 994.3 N. If the upward forces on the marseha exceed the downward focrces(which is the sum of weight of the plow and the downward component of the force exerted on the Erf), the maresha will be lifted above the ground and plaughing will not be possible. 1. Two vectors give a resultant of magnitude 8 units when are parallel and 2 units when they are antiparallel. Determine the possible values of each of the vectors. 2. State the condition where the vector sum of two vectors is a null vector. 3. Suppose you are in a large hall and that you mark a point on the floor and start walking the following distances in the given order: 50 m [forward], 10 m [backward], 10 m [forward], 10 m [backward], 20 m [forward], 10 m [backward], 40 m [forward], and 10 m [backward]. A. What is the total distance you traveled? B. What is the magnitude and direction of your displacement from the starting point? 4. What vector must be added to vector Cv of 10km, East in order to give a resultant vector of A. 15 km, East? B. 15 km, West 38 5. A sailor boards a paddle boat and heads the boat Westward directly across a river. The river flows South at 50 cm/s and the woman paddles the boat with a speed of 100 cm/s. A. Determine the resultant velocity of the boat – both magnitude and direction. B. How far down stream relative to the straight-across direction will woman be when she reaches the opposite shore? 6. Given three displacement vectors Bv , Cv, and Dv such that B = (- i + 2j) m, C = 3i - 2j) m D of unknown components, determine v t t v t t v the magnitude and direction of D v for 3Bv - Cv + Dv = 0 7. Given displacement vectors of Sv1 and Sv2 as Sv1 = (3it + 6jt) m , Sv2 = (- 2it - 4vj ) m , find a unit vector along the direction of Sv1 + Sv2. 8. Five displacement vectors Av , Bv , Cv, D v and Ev are placed on the coordinate axes as shown in Figure 2.23. Fill Table 2.3 with the correct values of the components of the vectors. (Hint: A vector on either of the coordinate axis has only one component). y x component y component ⃗ 𝐵𝐵 𝐴𝐴⃗ ⃗ 𝐵𝐵 37o 𝐴𝐴⃗ x 𝐶𝐶⃗ 30o 53o 𝐶𝐶⃗ 𝐷𝐷⃗ ⃗ 𝐸𝐸 ⃗ 𝐷𝐷 𝐸𝐸 ⃗ Figure 2.23 The five vectors on the xy plane 2.4 Product of Vectors After completing this section, you should be able to: define dot product of a vector apply the technique of dot product to solve practical problems 39 Physics Student Text Book Brainstorming Do you know any physical quantity that can be expressed as a product of two vector quantities? Multiplying or dividing a vector by a number or scalar v A A v A v A v A v A v A v Av v A Av A v v A A v v A A v A v v and 2\ A Figure 2.26 (a) A v are parallel (b) A v and -2 A v are antiparallel Example 2.12 Displacement vector given as Sv =35km, towards 60 60c North of east. Find the magnitude direction of a) 1.2 Sv and b) -1.2 Sv ? Solution Given Sv =35km, towards 60 60c North of east a) Vector 1.2 Sv is a displacement with magnitude 1.2 times that of S along the direction of Sv. 1.2S= 1.2 X 35=42km and 1.2 Sv =42km, 600 North of East. b) Vector -1.2 Sv a displacement vector with magnitude 1.2 times that of Sv pointing opptisite to the direction Sv. Therefore -1.2 Sv =42km, 600 South of West. 40 Dot product A v :B v Av Bv Av.Bv Av Bv v :B A v cos i i Figure 2.25 Vectors are placed tail to tail to form angle θ v :B A v Key terms Scalar (dot) product: product of vectors that yields a scalar. Angle between vectors: the angle two vectors form when placed tail to tail B A v and Bv Figure 2.26: Alternative ways of finding the dot product of A 41 Physics Student Text Book Exercise 2.10 The dot product of A and B can be defined alternatively as the magnitude of multiplied by the component B of in the direction of A. Find the angle between each of the following pairs of vectors. a) 3it - jt and it - 2jt b) 3it - 2jt and it - 2jt c) 3it - 2jt and 4it + 6jt Ä Activity 2.6 Group discussion Apply the definition of dot product to show that the dot product of a unit vector with itself is one (unity) while the dot product of a unit vector with another unit vector is zero, and use the result to prove that Pv : Qv = Px Qx + Py Qy, where Pv = Px iv + Py vj and Qv = Qx iv + Qy vj Note that v :B A v cos i v :B A v Exercise 2.11 Given a position vector Pv =- 4it + 6jt , What is the cosine of the angle between Pv and (a) the x-axis? (b) the y-axis? (The cosine of the angle between a vector and the positive coordibate axis is called direction cosine of the vector. Example 2.13 Vector Av = it + jt and vector Bv =- 2it + 3jt. What is (a) the scalar product of Av and Bv ? (b) the nagle between Av and Bv ? Solution (a) A.B = Ax.Bx + Ay.B y = (1) (- 2) + (1) (3) A.B (b) From A.B = AB cosi, cosi = AB A = Ax2 + Av2 = 1 2 + 1 2 = 2 B = Bx2 + Bv2 = (- 2) 2 + (3) 2 = 13 AB 1 = 0.196, i = cos -1 (0.196) = 78.7 a cos AB = 2 12 42 Example.2.14: Force and displacement Consider a block placed on a horizontal surface and that force Fv is applied on the block to move the block through displacement Sv. The work done by the force is defined as the dot product of the force and the displacement. If Fv = (5it + 3jt) N and Sv = (- 2it + 4jt) m, what is the work done by the applied force? solution Work W = Fv.Sv = (5it + 3jt) N. (- 2it + 4jt) m = (5 (- 2) + 3 (4) = 2Nm = 2J Example 2.15: Force and velocity Consider force Fv of a certain machine is applied on a body to move the body with an average velocity Vv. The power developed by the machine is defined as the dot product of the force and the average velocity. If Fv = (50it + 30jt) and Vv = (3it + 4jt) m/s, what is the power developed by the machine? Solution The power P = Fv.Vv = (50it + 30jt) N. (3it + 4jt) m/s = 50 (3) + 30 (4) = 270Nm/s = 270W 1. What is the dot product of vector Av with itself? 2. Show that the dot product of two vectors obeys: (a) commutative property, Av : Bv = Bv : Av ? (b) distributive property, A v. (B v = (A v + C) v. B) v. C) v + (A v 3. Is Av : Bv = Bv : Av ? 4. For what angle between Av and Bv will Av.Bv be equal to Bv.Av 5. The force applied on a body and the displacement it underwent are given respectively as Fv = (25it - 30jt) N and Sv = (- 2it - 3jt) m. What is the work done by the force? 6. For what values of the angle between Av and Bv will the dot product Av.Bv be positive, negative or zero? 7. If Dv = (5it - 3jt) N and Ev = (2it + jt) , what is the angle between Dv + Ev and the x axis? 8. If Pv.Qv = PQ, what is the angle between Pv and Qv ? 43 Physics Student Text Book UNIT 2 SUMMARY When a physical quantity is described by a single number, we call it a scalar quantity. In contrast, a quantity described by both a magnitude (the “how much” or “how big” part) and direction in space is a vector. Depending on their direction, magnitude and orientation we can categorize vectors as collinear, coplanar, parallel, antiparallel, equal, null, unit vector etc. Graphical method of vector addition makes use of translation of vectors. Triangle law – Vectors are connected head to tail to form a triangle. Parallelogram law – vectors are connected tail to tail and a parallelogram is constructed using the vectors as its sides. Polygon law - used to find the resultant of more than two vectors. The vectors are joined head to tail without regard to the order they are taken. The vector drawn from the tail of the first vector to the head of the final vector represents the resultant. Analytic methods of vector addition/subtraction allow us to find resultant of sums or differences of vectors without having to draw them. They are exact, contrary to graphical methods which are approximate. The resultant of two vectors in the same direction has a magnitude equal to the sum of the magnitude of each of the vectors and it is directed along the direction of any one of the vectors. The resultant of two vectors at right angles has a magnitude equal to the square root of the sum of the squares of the magnitude of each of the vectors and the direction of the resultant is determined by the angle it makes with respect to a reference line (commonly the x axis). A unit vector vector along the x axis is denoted by it and that along the y axis is denoted by jt. A vector that is represented by its vector components as A v = Av x + Av y v can also be represented interms of its components as A = Ax it + Ay jt. If Rv is the resultant of Av and Bv then Rv = Rx it + Ry jt , where Rx = Ax + Bx and Ry = Ay + By. The angle that Rv makes with the x axis is i = tan -1 c Ry m v and Bv is defined as Av.Bv = AB cos Rx The scalar product of two vectors A i and also v v A : B =AxBx+AyBy. The scalar product of two vectors can be used to determine the angle between the vectors and it can also be used to determine the component of one of the vectors in the direction of the other. 44 END OF UNIT 2 QUESTION 1. Identify the following quantities as vectors and scalars: Speed, velocity, Displace- ment, Volume, Acceleration, Power. 2. Which one of the following is not true about the three vectors shown in Figure 2.27? v = Bv + Cv 𝐴𝐴⃗ ⃗ 𝐵𝐵 a. - A b. Bv =- Cv - Av c. Cv =- A v - Bv v + Bv - Cv = 0 𝐶𝐶⃗ d. A Figure 2.27 3. If you and your friend have to apply a force to move an object, what would be the best way to apply the forces to maximize your resultant force? 4. What is the least number of unequal vectors required to produce a zero resultant? 5. If you and your friend have to apply a force to move an object, what would be the best way to apply the forces to maximize your resultant force? 6. What is the least number of unequal vectors required to produce a zero resultant? 7. Suppose you are adding two vectors Pv and Q v , how would you add them to get the resultant with the least magnitude? 8. What will happen to the magnitude of the resultant of two vectors as the angle between them increases from 0o to 180o? 9. A vector A v lies in the xy plane. For what orientations of Av will both of its compo- nents be negative? For what orientations will its components have opposite signs? 10.Two vectors have unequal magnitudes. Can their sum be zero? Explain. 11. Given displacement vectors Sv1 = (- 2it + 3jt) m, Sv3 = (3it - 2jt) m , Sv1 = (- 2it + 3jt) m. Find the magnitude and direction of the resultant displace- ment Sv = Sv1 + Sv2 + Sv3 12. Three force vectors A v , Bv and Cv are acting at a point. If Av = 4it - 3jt , Bv = 2iv + vj v + Bv - Cv = 0. Find , and A a) the magnitude and direction of vector Cv and b) a unit vector along Cv. 13.Determine a velocity vector that has a magnitude of 5 m/s directed along the di- rection of vector D v = 1.5it + 2jt. 14. The angle between vector A v and vector Bv is 20o. If the magnitude of vector Av is 6 units and that of B v is 4 units, what is the component of vector along the direc- tion of vector B v 15. The initial position vector of a particle on the xy plane is (5,2) and its final position vector is (−2,−3). If distance is measured in centimeter, a) what is the displacement of the particle? b) what is the magnitude and direction of the displacement? 45 Physics Student Text Book 16. A postal employee drives a delivery truck along the route shown in Figure 2.30. De- termine the magnitude and direction of the resultant displacement by drawing a scale diagram. Figure 2.28 Path of the truck 17.A girl pushes a box placed on the floor with a force of 50 N at an angle of 37o below the horizontal. If the box moves toward the positive x direction, what are the vector components of the force? 18.The Tug-of-War Game Four dogs nicknamed A, B, C, and D play a tug-of-war game with a toy, Figure 2.31. A pulls strongly with a force of magnitude A = 160 N. B pulls even stronger than A with a force of magnitude B = 200 N, and C pulls with a force of magnitude C = 140 N. When D pulls on the toy in such a way that his force balances out the re- sultant of the other three forces, the toy does not move in any direction. With how big a force and in what direction must D pull on the toy for this to happen? (Hint: If the toy does not move, the vector sum of the four forces, FvA + FvB + FvC + FvD = 0. Figure 2.29 Dogs enjoying the tug-of-war game 19. Given two vectors X v and Yv such that Xv = 4it + jt and Yv = 6it + 3jt , (a) what is the angle between the two vectors? (b) Find the component of X v along the direction of Yv ? 20. What is the vector product A v.Bv of the vectors shown in Figure 2.30? Figure 2.32 Vectors on the xy plane – x axis as a reference line for measuring angles 46 3. MOTION IN ONE AND TWO DIMENSIONS Learning outcomes: Students will be able to: gain an understanding of the fundamental principles of kinematics in one and two dimensions. develop skills in applying equations of motions to solve practical problems. recognize the effect of air resistance and force of gravity on motion of a body. describe technological advances related to motion; and identify the effects of societal influences on transportation and safety issues. 47 Physics Student Text Book 3.1 Uniformly Accelerated Motion in 1D By the end of this section, you will be able to: explain a uniformly accelerated motion in 1D; explain the difference between average velocity and instantaneous velocity; and solve problems involving average velocity, instantaneous velocity and acceleration. Brainstorming After revising what you have learnt in grades 9 and 10 about motion in a straight line and uniformly accelerated motion discuss about the following questions in group and present your group’s answers to the class. 1. Define the terms position, distance, displacement, speed, and velocity in motion. 2. Explain the relationship between ȑ position and displacement, ȑ distance and displacement, ȑ distance and speed, ȑ displacement and velocity, ȑ average velocity and instantaneous velocity. 3. When do we say an object is accelerating? 4. How will it be possible for a car to move but not accelerating? Acceleration 48 Ä Activity 3.1: Discuss in group and present your group’s understanding of the following questions to the class. 1. Does negative acceleration necessarily mean slowing down? 2. Does positive acceleration necessarily mean speeding up? Average acceleration avav 9 v = v f - vi 9tv = tv f - tvi v f - vt aav = Dt v f v i m/s2 = ms -2 Key term Acceleration is the rate of change of velocity. It is a vector quantity, with a unit of m/s2. Example: 3.1 If the speed of a certain car is increased from rest to 20m/s in 6 seconds, what is the average acceleration of the car? Solution We have v 0 = 0 , at and at t0 = 0 , and v f = 0 at tf = 6s ∆v v f − v0 20m / s − 0 = a = = = 3.67 m / s 2 ∆t t f − t0 6s − 0 Instantaneous acceleration  a ins Δt Δt Dv av = lim Dt " 0 Dt 49 Physics Student Text Book Example 3.2. The velocity of a particle that moves along a straight line varies as a function of time with a velocity equation as: v(t = ) t 2 − 2t. Find the acceleration of the particle at t = 2 s. Solution: To find the acceleration at t =2s, we first find the acceleration at any time t, that is ∆v  v(t + ∆ t) − v(t)  = = lim  a (t ) lim  ∆t → 0 ∆t ∆t ∆t → 0   Sin ce, v(t = ) t − 2t 2 ⇒ v(t + ∆t )= (t + ∆t ) 2 − 2(t + ∆t )  v(t + ∆ t) − v(t)  a (t ) = lim 

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