Grade 11 Physics Textbook PDF

Physics Student Textbook Grade 11 Authors: Jim Newall Susan Gardner Graham Bone Advisers: Tilahun Tesfaye Deressu (PhD) Endeshaw Bekele Buli Evaluators: Yosef Mihiret Gebremeskel Gebreegziabher Yusuf Mohame...

Physics Student Textbook Grade 11 Authors: Jim Newall Susan Gardner Graham Bone Advisers: Tilahun Tesfaye Deressu (PhD) Endeshaw Bekele Buli Evaluators: Yosef Mihiret Gebremeskel Gebreegziabher Yusuf Mohamed Federal Democratic Republic of Ethiopia Ministry of Education Acknowledgments The development, printing and distribution of this student textbook has been funded through the General Education Quality Improvement Project (GEQIP), which aims to improve the quality of education for Grades 1–12 students in government schools throughout Ethiopia. The Federal Democratic Republic of Ethiopia received funding for GEQIP through credit/financing from the International Development Associations (IDA), the Fast Track Initiative Catalytic Fund (FTI CF) and other development partners – Finland, Italian Development Cooperation, the Netherlands and UK aid from the Department for International Development (DFID). 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We would be pleased to insert the appropriate acknowledgement in any subsequent edition of this publication. © Federal Democratic Republic of Ethiopia, Ministry of Education First edition, 2002 (E.C.) ISBN: 978-99944-2-020-9 Developed, Printed and distributed for the Federal Democratic Republic of Ethiopia, Ministry of Education by: Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England In collaboration with Shama Books P.O. Box 15 Addis Ababa Ethiopia All rights reserved; no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without the prior written permission of the copyright owner or a licence permitting restricted copying in Ethiopia by the Federal Democratic Republic of Ethiopia, Federal Negarit Gazeta, Proclamation No. 410/2004 Copyright and Neighboring Rights Protection Proclamation, 10th year, No. 55, Addis Ababa, 19 July 2004. Disclaimer Every effort has been made to trace the copyright owners of material used in this document. We apologise in advance for any unintentional omissions. We would be pleased to insert the appropriate acknowledgement in any future edition Printed in Malaysia Contents Unit 1 Measurement and practical work 1 1.1 Science of measurement 2 1.2 Errors in measurement 9 1.3 Precision, accuracy and significance 12 1.4 Report writing 17 Unit 2 Vector quantities 22 2.1 Types of vector 22 2.2 Resolution of vectors 26 2.3 Vector addition and subtraction 28 2.4 Multiplication of vectors 33 Unit 3 Kinematics 38 3.1 Motion in a straight line 39 3.2 Motion in a plane 51 Unit 4 Dynamics 66 4.1 The force concept 67 4.2 Basic laws of dynamics 69 4.3 Law of conservation of linear momentum and its applications 75 4.4 Elastic and inelastic collisions in one and two dimensions 82 4.5 Centre of mass 85 4.6 Momentum conservation in a variable mass system 90 4.7 Dynamics of uniform circular motion 92 Unit 5 Work, energy and power 98 5.1 Work as a scalar product 99 5.2 Work done by a constant and variable force 101 5.3 Kinetic energy and the work-energy theorem 103 5.4 Potential energy 107 Grade 11 iii  5.5 Conservation of energy 111 5.6 Conservative and dissipative forces 114 5.7 Power 116 Unit 6 Rotational motion 118 6.1 Rotation about a fixed axis 119 6.2 Torque and angular acceleration 120 6.3 Rotational kinetic energy and rotational inertia 124 6.4 Rotational dynamics of a rigid body 129 6.5 Parallel axis theorem 133 6.6 Angular momentum and angular impulse 135 6.7 Conservation of angular momentum 138 6.8 Centre of mass of a rigid body (circular ring, disc, rod and sphere) 140 Unit 7 Equilibrium 143 7.1 Equilibrium of a particle 144 7.2 Moment or torque of a force 146 7.3 Conditions of equilibrium 150 7.4 Couples 155 Unit 8 Properties of bulk matter 158 8.1 Elastic behaviour 159 8.2 Fluid statics 166 8.3 Fluid dynamics 175 8.4 Heat, temerature and thermal expansion 184 Index 200 iv Grade 11 Measurement and Unit 1 practical work Contents Section Learning competencies 1.1 Science of Explain the importance of measurement. measurement Identify and use appropriate units for data that will be collected. (page 2) Describe what is meant by the term significant figures and how it is related to precision. Identify rules concerning the number of significant figures that a numeral has. Define the term scientific method. State the steps of scientific methods. State the uncertainty in a single measurement of a quantity. Identify the orders of magnitude that will be appropriate and the uncertainty that may be present in the measurement of data. Distinguish between random error and systematic error. 1.2 Errors in Describe sources of errors. measurement (page 9) Identify types of errors. Distinguish between random uncertainties and systematic errors. 1.3 Precision, Distinguish between precision and accuracy. State what is meant by the degree of precision of a measuring instrument. accuracy and Use scientific calculators efficiently. significance (page 12) 1.4 Report writing Describe the procedures of report writing. (page 17) Use terminology and reporting styles appropriately and successfully to communicate information and understanding. Present information in tabular, graphical, written and diagrammatic form. Report concisely on experimental procedures and results. One of the most important developments in the history of science was the scientific method, the procedure scientists use to acquire knowledge in any field of science. Science is all about observing and experimenting. We need a framework to add relevance to observations and experiments. Experiments may vary in size and expense, including using huge particle accelerators, making observations using space telescopes, testing simple circuits, or even just bouncing a ball! In order to be considered scientific, they must follow a key set of principles and be presented in a suitable manner. This section looks at how experiments should be represented and how the data should be analysed before drawing conclusions. Grade 11 1 UNIT 1: Measurement and practical work 1.1 Science of measurement By the end of this section you should be able to: Explain the importance of measurement. Identify and use appropriate units for data that will be collected. Describe what is meant by the term significant figures and how it is related to precision. Identify rules concerning the number of significant figures that a numeral has. Define the term scientific method. Figure 1.1 The Hubble space telescope State the steps of scientific methods. State the uncertainty in a single measurement of a quantity. Identify the orders of magnitude that will be appropriate and the uncertainty that may be present in the measurement of data. The scientific method The scientific method is exceptionally important to the process of science. It ensures a rigorous, evidence-based structure where only ideas that have been carefully tested are accepted as scientific theory. Ask a question (maybe based on an observation) Use existing knowledge or do background research Form a hypothesis Make predictions from your hypothesis Design an experiment to test your predictions Analyse your experimental data Draw conclusions (Was your hypothesis correct? If not, construct a new hypothesis and repeat.) Figure 1.2 The scientific method Your hypothesis will have to be tested by others before it becomes Figure 1.3 This scientist is an accepted scientific theory. This process of peer review is very researching ways of making important and prevents scientists making up data. energy-efficient lighting. 2 Grade 11  UNIT 1: Measurement and practical work The process of science begins with a question. For example: Why is the sky blue? Why does the Sun shine? Scientists are curious about the world around them; it is this curiosity that is the spark of the process. When you have a question, scientists may have already looked into it and devised an explanation. So the first step is to complete some preliminary research into the existing theories. These theories may provide answers to your question. It is quite probable most of the questions you encounter in your physics course already have answers. However, there are still some big unanswered questions in physics. They are waiting for someone like you to answer them! Are there any questions that science cannot answer? Using your existing knowledge or information collected via research, the next step is to form a hypothesis. Figure 1.4 Scientists A hypothesis is just an idea that might provide an answer to your investigating renewable energy question. A scientific hypothesis is based on scientific knowledge, not just made up! Activity 1.1: Using the For example: Why does the Sun shine? scientific method You might form two hypotheses: Look around your classroom Nuclear fusion reactions in the Sun release heat and light. or outside. Make three There is a large lamp in the centre of the Sun powered by observations, and using electricity. your existing scientific knowledge, form a The first is clearly a more scientific hypothesis using some hypothesis for each of them. thoroughly tested existing ideas. That is not to say you shouldn’t Discuss these with a partner. be creative in making a hypothesis but you should include some scientific reasoning behind your ideas. It is important to note your hypothesis might be incorrect, and this is what makes science special. An investigation must be carried out to test the ideas before you either rule out an idea or accept it. Why does the Sun shine? Once you have a hypothesis, you should use it to form a series of predictions that can be tested through experiment. These may range from ‘easy to test’ to ‘hugely complex’ predictions. You then design an experiment to test your predictions. Figure 1.5 We all know that the Discussion activity Sun shines. But why? Some simple questions lead to massively complex experiments costing billions of dollars. One of these is the ATLAS project in KEY WORDS Switzerland. This costs billions of dollars; do you think it is right hypothesis a proposed to spend such a large amount of money on scientific experiments? explanation for an observation experiment a test under known conditions to investigate the truth of a hypothesis Grade 11 3 UNIT 1: Measurement and practical work KEY WORDS analyse examine in detail to discover the meaning of a set of results conclusions the overall result or outcome of an experiment. The hypothesis being tested may be supported by the results or may be proven incorrect significant figures the number of digits used in a measurement, regardless of the location of the decimal point Figure 1.6 The ATLAS detector in Switzerland It is important that your experiment is clearly planned. This will enable others to test your experiment and check your ideas (more detail on this can be found in Section 1.4). Once you have carefully conducted your experiment, you will need to analyse your results and draw conclusions. At this stage you need to decide if your results support your prediction. If they do, then perhaps your hypothesis was correct. This will need to be confirmed by several other scientists before it becomes accepted as scientific fact. If your results do not support your prediction, then perhaps your hypothesis was wrong. There is nothing wrong with that, you just go back and form a different hypothesis. This process continues and it Activity 1.2: Choosing the may take years to come up with a correct hypothesis! right unit In a group, discuss what Making measurements units you would use when As part of your experiment you will have to make measurements measuring each of the and collect data. This process is very important and needs to be following: conducted carefully. length of a football pitch, width of this book, Choosing units diameter of a small seed, When you are planning your experiment, you need to choose width of a finger nail. units that are appropriate to the size of the quantity you are area of a page in this measuring. For example, you would measure the length of a finger book, area of the floor in in centimetres. your classroom, area of a Explain why you have chosen the units. football pitch. volume of this book, Significant figures volume of your classroom, All digits in a number that are not zero are called significant volume of a bottle, volume figures. For example, the number 523 has three significant figures of a soccer ball. and the number 0.008 has one significant figure. The zeroes in 0.008 are not significant figures but they are important as they tell you how big or small the number is. 4 Grade 11  UNIT 1: Measurement and practical work When a number is given in standard form, the number of digits Table 1.1 Prefixes and their tells you how many significant figures there are. For example, 0.008 symbols in standard form is 8 × 103 – it’s now much more obvious that the number has one significant figure. Prefix Power Symbol of 10 For any measurement you take, the number of significant figures giga- 109 G (s.f.) must be consistent with the instrument precision. For example, if you are measuring length with a 1 m ruler that has mm on it, then mega- 106 M all readings should be expressed to the nearest mm. For example: kilo- 103 k 0.6 m ✗ hecto- 102 h 0.64 m ✗ deca- 10 da 0.643 m ✓ deci- 10–1 d If your reading is exactly on an increment this still applies! centi- 10–2 c 0.5 m ✗ milli- 10–3 m 0.50 m ✗ micro- 10–6 μ 0.500 m ✓ nano- 10–9 n You must be consistent with your use of significant figures in your results tables. If your data is to two significant figures, so should be your average. For example: Reading one: 62 Reading two: 61 Average: ?? The average here should be 62, not 61.5 as this is going from two to three significant figures. If you then go on to calculate something using your data, you must express your answer to the lowest number of significant figures in your data. For example, if you are calculating average speed you might have the following: Average speed = distance travelled time taken Distance travelled = 4.345 m Time taken = 1.2 s The distance travelled is to four significant figures but the time taken is only to two. This means your answer should only be to two Discussion activity significant figures: Is 100 V to one, two or three Average speed = distance travelled significant figures? The time taken answer is it could be any of Average speed = 4.345 m those! If your reading is to 1.2 s a whole number you may Average speed = 3.62083333… need to specify its number of significant figures. Average speed = 3.6 m/s (to 2 significant figures) Zeroes between the significant figures and the decimal point are also significant as well as demonstrating the magnitude of the quantity. ‘Trailing zeroes’ (e.g. the zeroes in 3.20000) are also considered significant. Grade 11 5 UNIT 1: Measurement and practical work KEY WORDS Uncertainties uncertainty the amount of Every measurement you take will have an uncertainty associated doubt in a measurement with it. It does not mean it is wrong, it is just a measure of your confidence in your measurement. If you were measuring the height of a friend, you might write: Activity 1.3: Measuring 1.80 m bounce height of a ball Does this mean exactly 1.8 m? Does it mean Collect a ball and ruler. In 1.80000000000000000000000000000 m? When you write 1.80 m small groups drop the ball you mean your friend’s height is between 1.805 m and 1.795 m. You from various heights (around could write this as: ten different heights) and 1.80 m ± 0.005 m record how high it bounces. Notice how difficult it is to The 0.005 m is the uncertainty in your reading. You have measured determine the bounce height the height to the nearest 5 mm. You should try to keep this to anything less than the uncertainty as small as possible (more on this in Section 1.3). nearest cm. The uncertainty in every measurement will be related to the nature If you were using the same of the task and the precision of the instrument you are using. ruler to measure the width of For example, measuring the height of a ball bouncing is very a piece of paper, you might difficult. You might measure the height using a ruler with mm on it also get 16.0 cm; however, as 16.0 cm, but what is the uncertainty? your uncertainty would be much less than 1.0 cm! This is 16.0 cm ± 0.1 cm ➞ Even though the ruler may measure to mm it because the piece of paper is is very hard to measure the height to the nearest mm as this is too not moving, so measuring is small. easier than with the bouncing 16.0 cm ± 0.5 cm ➞ This is possible, if you ensure to measure the ball. Perhaps you would write bounce height carefully; by getting down to eye level and doing a 16.0 cm ± 0.1 cm. test drop it might be fair to say you can measure the height to the nearest ½ cm. 16.0 cm ± 1.0 cm ➞ This is a realistic uncertainty for this experiment. 16.0 cm ± 2.0 cm ➞ This might be a little unrealistic, but is also acceptable, since determining the bounce height by eye is quite difficult. 16.0 cm ± 5.0 cm ➞ Hopefully you have designed the experiment to enable you to measure to more than the nearest 5 cm! Figure 1.7 Measuring bounce Which uncertainty you use is a judgement you will have to make height of a ball depending on the results. Discussion activity Can you think of some other examples of measurements you might take using a ruler and what the uncertainty might be in each case? 6 Grade 11  UNIT 1: Measurement and practical work Percentage uncertainties Activity 1.4: Percentage You may need to calculate the percentage uncertainty in one of your uncertainty for the readings. This is just the uncertainty of the reading expressed as a bouncing ball percentage. If you measured the current through a bulb, you might Using the data collected on express your measurement as: the bounce height of the 4.32 A ± 0.05 A ball, calculate the percentage So the percentage uncertainty would be: uncertainty in each case. 0.05 × 100 = 1.157407…% What do you notice about the 4.32 smaller readings? The larger So you would write 4.32 A ± 1.2%. your reading, the smaller the percentage uncertainty. So As a rule of thumb, percentage uncertainties should be to two small distances have a greater significant figures. percentage uncertainty than You should always aim to keep your percentage error under 10%, large distances, if measured although this may not always be possible. with the same instrument. Whenever possible you should measure multiple values instead of just one. For example, the time for 20 swings of a pendulum rather than just one, or several thicknesses of card rather than just one. This KEY WORDS has the effect of reducing the percentage uncertainty as shown below: multiple values several One piece of card Ten pieces of card readings of the same measurement 1.03 ± 0.05 mm ➞ 4.9% 10.41 ± 0.05 mm ➞ 0.48% Notice the uncertainty is still the same but the percentages are very different. Calculations You need to determine the uncertainty in a quantity you have calculated. For example, when calculating resistance from values of p.d. (potential difference) and current: p.d. = 4.32 V ± 1.157407…% Current = 2.3 A ± 4.534641…% p.d. Resistance = current 4.32 V Resistance = 2.3 A Resistance = 1.9 Ω (2 s.f., as the value of the current is to 2 s.f.). To find the percentage uncertainty in the resistance you just add the percentage uncertainties of the p.d. and current. This gives 5.692047…%, and so you would write: 1.9 Ω ± 5.7%. Do not round up until the end! You could express the resistance (1.9 Ω ± 5.7%) as 1.9 ± 0.1 Ω. This is because 5.7% of 1.9 is 0.108...Ω and also as the resistance reading is to 1/10 of an ohm, you would write 0.1 Ω not 0.11 Ω. This is same if you are multiplying quantities together. For example, calculating distance travelled using distance travelled = average speed × time taken. Grade 11 7 UNIT 1: Measurement and practical work Average speed: ± 4.1….% Time taken: ± 3.4…% Activity 1.5: Uncertainty in the dimensions of a Therefore: distance travelled: ± 7.5% book Be careful if there is a square in the equation. For example, area of a Using a ruler, measure the circle = πr2. If r has a percentage uncertainty of 2.312…%, then the dimensions of a book in area will have a percentage uncertainty of 4.6% as the equation is centimetres. Write down the effectively: area = π × r × r. So the error in r must be counted twice dimensions, the uncertainty (2.312…% + 2.312…% = 4.624…% so 4.6% to 2 s.f.). for each, and express this as a percentage. Summary Use your readings to calculate the volume of the book. In this section you have learnt that: Calculate the percentage The scientific method includes: observing, researching, uncertainty in the volume hypothesising, predicting, experimenting, analysing and and express this in cm3. concluding. Measurements must always be recorded to an appropriate number of significant figures (this depends on the equipment you are using). All measurements have an uncertainty associated with them. This is effectively a quantification of the amount of doubt in a measurement. To determine the uncertainty of a calculated value, you add the percentage uncertainties of the quantities used to perform the calculation. Review questions 1. Describe each part of the scientific method. Explain why it is important to follow this structure when conducting a scientific investigation. 2. How many significant figures do the following numbers have: a) 258 b) 0.2 c) 12 000 d) 0.084 3. How can you reduce the percentage uncertainty in measurements that you make? 4. Nishan and Melesse have measured the voltage across a resistor to be 5.26 V and the current flowing through it to be 0.41 A. They work out the resistance. Nishan says that the resistance is 12.8 Ω. Melesse disagrees and says that the resistance is 13 Ω. Who is correct? Explain your answer. 5. A bulb is connected as part of a circuit. The following data is collected: Electric current: 3.2A ± 0.1 A Potential difference: 12.3 V ± 0.1 V 8 Grade 11  UNIT 1: Measurement and practical work Use this data and the equation potential difference Resistance = electric current to determine the resistance. Express the uncertainty in your answer. 1.2 Errors in measurement By the end of this section you should be able to: Distinguish between random error and systematic error. Describe sources of errors. Identify types of errors. Distinguish between random uncertainties and systematic errors. What are errors? KEY WORD An experimental error (or just referred to as an error) is not the accepted/true value the same as a mistake. An example of a mistake would be to measure actual value of the property the height of a desk when asked to measure the height of a chair. It being measured, made is just plain wrong! without any experimental errors The measurements you take as part of your investigations random errors unpredictable will contain experimental errors, but hopefully no mistakes. errors that have no pattern or Errors occur in every scientific investigation; they affect your bias and which may be above measurements, making them different from the accepted value or below the true value (sometimes called true value) of the item being measured. There are several different types of experimental error. Accepted or true value This is the actual value of the physical property you are measuring. It is the value you would get if it were possible to make the measurement with no experimental errors. Random error Random errors are errors with no pattern or bias. They cause measurements to vary in an unpredictable manner. Importantly, they cause your measurements to be sometimes above the accepted value, sometimes below the accepted value. For example, if you were measuring the acceleration due to gravity, random errors will cause your readings to vary both above and below the accepted value. Accepted value for acceleration due to gravity = 9.80665 m/s2 (to 6 s.f.) Recorded values (m/s2) 9.81 9.78 9.65 9.87 9.80 9.86 9.83 Grade 11 9 UNIT 1: Measurement and practical work Activity 1.6: Testing random error with a ruler Make yourself a ruler by cutting a strip of card 15.0 cm long. Use a ruler to carefully mark on the centimetre divisions. Use your ruler to make several different length measurements of items in your classroom. You must resist the temptation to record your readings to the nearest mm. A suitable uncertainty will be to the nearest 0.5 cm. Repeat the experiment using a real ruler. You will find about half of your readings were too high, the other half too low. This kind of random error happens with all measurements. Even those taken with the real ruler will either be 0.5 mm too high or 0.5 mm too low. Figure 1.8 Home-made ruler Another example of a random error could be encountered when completing investigations into heat. The surrounding temperature will vary depending on the time of day and general weather conditions. If you are conducting an experiment over a number of days, this will produce random errors in your measurements. To reduce the effect of random errors, wherever possible you should take several reading and average them. The more repeats you take, the lower the impact of random errors. Parallax errors with scales The use of a ruler for length is not without its problems at times. If you wanted to measure the diameter of a table-tennis ball, how might you do it? When the object and the scale lie at different distances from you, Figure 1.9 Viewing from directly it is essential to view them from directly above if you are to avoid above avoids a parallax error. what we call parallax errors (Figure 1.9). A clock in a public place has to be read from many different angles. A neat way of avoiding parallax errors in that case is shown in Figure 1.10. With an instrument designed to be read by a single experimenter, Figure 1.10 How does this you must take care to position your head correctly. Two ways of arrangement avoid parallax achieving the same thing with a current meter are shown in errors? Figure 1.11. 10 Grade 11  UNIT 1: Measurement and practical work pointer Did you know? Parallax is the name we mirror give to an effect that If you can see either you are familiar with in scale side of this pointer, everyday life. As you travel you are not reflection in mirror along a road, objects in directly above it. the distance seem to shift position relative to one another. Because of your Figure 1.11 Two ways of preventing parallax errors. (a) You know movement, a distant house you are looking straight down on the pointer when it is hiding its own may disappear behind a reflection in the mirror. (b) A flat pointer is twisted so it is upright at nearer clump of trees, but the tip. as you travel further along Returning to the question of the diameter of the table-tennis ball, it comes back into view the two rectangular wooden blocks would help (Figure 1.12). other side of them. That is parallax. Discussion activity A little thought is still needed for the best possible result. What if the blocks are not quite parallel? The doubt can be removed by measuring both ends of the gap as shown in the drawing; if the Figure 1.12 Measuring using two lengths differ slightly, their average should be taken. wooden blocks Figure 1.13 shows some calipers, which can do the same job as the ruler and the two blocks. They may be made of steel, and the part drawn shaded will slide along the main part. It must fit snugly, so that the shaded prong A is always at right angles to the arm B. The arrow engraved on the sliding part indicates the diameter of the ball, on the millimetre scale. If the ball is removed and the jaws are Figure 1.13 Measuring using the closed, that arrow should then lie on the zero of the scale. Vernier calipers Systematic errors A systematic error is a type of error that shows a bias or a trend. KEY WORD It makes your readings too high every time, or too low every time. Taking repeated readings will not help account for this type of error. systematic errors errors caused by a bias in A simple example might be an ammeter that always reads 0.4 A measurement and which too low. So if your reading was 6.8 A, the true value for the current show a bias or trend would be 7.2 A. More complex examples include ignoring the effect of friction in Newton’s second law experiments, or not measuring to the centre of mass of a simple pendulum. The problem with systematic errors is that they can be quite hard to spot! When you have found the source you then either redesign the experiment or account for the error mathematically. Grade 11 11 UNIT 1: Measurement and practical work This is quite easy to do. Take for example a voltmeter where each reading is 0.2 V too large. To find the corrected value you need to subtract 0.2 V from each of your readings. Recorded value (v) Corrected value (v) 2.8 2.6 6.4 6.2 10.8 10.6 15.4 15.2 20.7 20.5 Zero errors Zero errors are special examples of a systematic error. They are caused by an instrument giving a non-zero reading for a true zero KEY WORDS value. For example, the ammeter mentioned above is a type of zero zero errors errors caused by error. When the current is 0 A it would read –0.4 A. equipment that has not been correctly zeroed Summary In this section you have learnt that: Experimental errors cause readings to be different from their true value. Random errors cause readings to be above and below the true value. Systematic errors cause a bias in your readings (they are all either too high or too low). Parallax errors can cause your readings to be less accurate because of the position of your eye. A zero error is a type of systematic error caused by equipment not being zeroed properly. Review questions 1. Explain the meaning of the term error. 2. Describe different types of errors, give examples, and explain how the effect of these errors might be reduced. 1.3 Precision, accuracy and significance By the end of this section you should be able to: Distinguish between precision and accuracy. State what is meant by the degree of precision of a measuring instrument. Use scientific calculators efficiently. 12 Grade 11  UNIT 1: Measurement and practical work What does ‘accurate’ mean? KEY WORDS Accuracy means how close a reading is to the true value. The more accuracy the closeness of a accurate a reading, the closer it is to the true value. measurement to its true value precision the quality of being Again using the acceleration due to gravity as an example: exact and the degree to which Accepted value for acceleration due to gravity = 9.80665 m/s2 repeated measurements under (to 6 s.f.) the same conditions give the same value If you took three readings you might get: significance the number of 9.76 9.87 9.82 significant figures used in a reading, which should be The most accurate reading is the third one; it is closest to the true appropriate to the precision value. of the measuring instrument In order to obtain more accurate measurements you must ensure you have minimised random errors, taken into account systematic errors and conducted the experiment as carefully as you can. Precision and significance The precision of your reading is a measure of the degree of ‘exactness’ of your value; this is sometimes related to the number of significant figures in the reading. The more precise a reading is, the smaller the uncertainty. A series of precise measurements will have very little variation; they will all be very similar. For example, dealing with lengths: Increasing precision 1.0 m 1.00 m 1.000 m ± 0.1 m ± 0.01 m ± 0.001 m The significance of your reading is indicated by the number of significant figures you can express your data to. This was discussed in Section 1.1 It is very important not to overstate the significance of your readings. If your ruler measures to mm, then your readings should be to mm; it would not be right to give a length of 1.2756 m. This is particularly true when calculating values. Take, for example, calculating the resistance of a light bulb: potential difference Resistance = electric current Potential difference = 10.0 V ± 0.1 V (so a 1.0% uncertainty) Current = 3.0 A ± 0.1 A (so a 3.3% uncertainty) Grade 11 13 UNIT 1: Measurement and practical work This shows a series of imprecise Put this into your calculator and you would get: measurements, they are all quite spread out. In addition, the readings potential difference Resistance are inaccurate as they are not = electric current clustered around the true value. 10.0 V This is what you want to try to avoid! Resistance = 3.0 A Resistance = 3.333333333 Ω As previously discussed you would answer 3.3 Ω to two significant figures, but why did the calculator give 3.3333333? The answer is to do with how calculators treat values. When you enter 10.0 V, you mean 10.0 V ± 0.1 V but the calculator takes the value to be exactly 10. That is 10.0000000000000000…. The same sort of thing is true for your current reading. To express the resistance as 3.333333333 Ω would be wrong. It implies the reading is more precise than it actually is. Figure 1.14 Imprecise and inaccurate Accurate and precise Accuracy and precision are often confused. A common analogy to help overcome this involves using a target. The centre of the target represents the true value and each shot represents a measurement. This shows a series of precise Here the measurements are imprecise This is what we are aiming for! High measurements; there is very little as there is quite a large spread of precision, little spread from readings variation in the readings. However, readings, but at the same time they and all close to the true value. they are also inaccurate as they are are accurate (they are all gathered quite far from the true value (centre). around the true value). A large random A systematic error may give these error may cause this. kinds of results. Figure 1.15 Precise but Figure 1.16 Accurate but Figure 1.17 Accurate and precise inaccurate imprecise 14 Grade 11  UNIT 1: Measurement and practical work Instrument precision Activity 1.7: Determining The precision of an instrument is given by the smallest scale instrument precision for division on the instrument. A normal ruler may have a precision of different instruments 1 mm but a screw gauge micrometer has a precision of 0.01 mm. Look at a range of different When you are taking single readings, the precision of the piece of pieces of measuring equipment you are using usually determines the uncertainty. For equipment. Determine the example, if you are using an ammeter with a precision of 0.01 A, instrument precision in each then your readings might be: case. 0.32 ± 0.01 A or 2.61 ± 0.01 A An exception to this rule would be if the nature of the task meant that there are other random errors that produce an uncertainty greater than the precision of the equipment. For example, the bouncing ball experiment described earlier, or measuring the time of a pendulum swinging. A stopwatch may have a precision of ± 0.001 s but your reaction time is much greater. As a result it might be better to express the uncertainty as ± 0.1 s. Activity 1.8: Uncertainty However, when you are taking multiple readings, for example in the swing of a recording p.d. with a voltmeter which has a precision of ± 0.01 V, pendulum you may obtain the following: Using a piece of string and 4.32 V 4.36 V 4.27 V some plasticine, make a The average would be 4.32 V (to 3 s.f., as the other readings). To simple pendulum. Working express this average as 4.32 ± 0.01 V would not be right as you can in pairs, time how long it tell by looking at your results the variation is more than ± 0.01 V. takes to complete one swing for various different lengths. In this case you would use half the range as your uncertainty. Repeat this three times for In the example above, the range from 4.27 V to 4.36 V is 0.09 V, each length. so therefore half this range is 0.05 V. The average reading would be written as 4.32 ± 0.05 V. If you have no variation in your Calculate the average time repeats, then you would use the precision of the instrument as the of one swing for each uncertainty. length, and determine the uncertainty in this reading. Solving physics problems When you are solving physics problems you need to make sure that Discussion activity you use the correct units. For example, one length may be given in feet and another in metres. You need to convert the length in feet How could you reduce the to metres using a conversion factor. When you do the calculation, percentage error in the timing check that the units are correct using dimensional analysis. of the pendulum experiment? If there are any intermediate stages in the calculation, keep all of the figures on your calculator screen for later stages in the calculation. You should only round your calculation to the appropriate number KEY WORDS of significant figures at the end of the calculation. conversion factor a Always check your calculations, because it is easy to make mistakes. numerical factor used to Check that you are using the conversion factor correctly by making multiply or divide a quantity sure that when the units cancel, you are left with the units that you when converting from one think you should have. system of units to another Grade 11 15 UNIT 1: Measurement and practical work Remember that when you are talking about an order of magnitude for an answer, this is much less precise than saying that it is approximately equal to something. For example, saying that N is ~ (is about) 1025 implies that N is in the range 1024 to 1026, but saying that N ≈ 1025 implies that N is in the range, say 9 × 1024 to 1.1 × 1025. The latter answer is much more precise. Worked example 1.1 Berihun walks 3000 feet in 10 minutes. What speed is he walking at? Give your answer in metres per second. First you need to convert the distance to metres and the time into seconds. Conversion factor for feet to metres = 1 metre/3.28 feet Distance = 3000 feet × 1 metre/3.28 feet = 914.634 14 metres Remember to keep all the digits from the conversion for the next stage in your calculation. Time = 10 minutes = 10 × 60 seconds = 600 seconds Speed = distance ÷ time = 914.634 14 metres ÷ 600 seconds = 1.524 390 2 m/s The greatest number of significant figures is three in the conversion factor, so your answer should be given to three significant figures. Speed = 1.52 m/s Summary In this section you have learnt that: Accuracy is a measure of how close a measurement is to the true value of the quality being measured. Precision is a measure of the degree of ‘exactness’ of your value. A series of precise measurements will have very little variation. The significance of a measurement is indicated by the number of significant figures in your value. The precision of an instrument is usually given by the smallest scale division on the instrument. More precise instruments have smaller scale divisions. 16 Grade 11  UNIT 1: Measurement and practical work Review questions 1. Explain the terms accuracy and precision. Describe how they differ using examples of experiments that you might conduct. 2. Research the precision of a range of instruments in your classroom. 3. Dahnay is 167 cm tall. Abeba is 66 inches tall. Who is taller and by how much. The conversion factor for inches to centimetres is 2.54 cm/1 inch. 4. a) What difference would it make to the answer in the worked example if you rounded the answer to the intermediate step to 3 significant figures? b) What effect do you think rounding the answer to each step might have in a calculation with several intermediate steps? 1.4 Report writing By the end of this section you should be able to: Describe the procedures of report writing. Use terminology and reporting styles appropriately and successfully to communicate information and understanding. Present information in tabular, graphical, written and diagrammatic form. Report concisely on experimental procedures and results. Presenting information Science is a collaborative process. As discussed in the first section, all ideas must be independently tested and verified. It is therefore important to ensure that when you write up reports or write up experiments, you do so carefully. Your results should be recorded in a clear and organised manner. This will usually be in a tabular format. Your tables should include all your raw data, including repeated readings and, where appropriate, columns for processed data (averages, calculations of resistance, etc). It is up to you if you wish to include clearly incorrect readings in your table, or simply repeat the reading. Grade 11 17 UNIT 1: Measurement and practical work Column headings must be labelled with a quantity and unit. You should use the standard convention for this: Quantity (unit). For example: Time (s) or Mass (kg). A sample table can be seen below: Current (A) Length (m) 1st Set 2nd Set 3rd Set Average KEY WORDS framework an outline A framework for writing up reports structure that can be used as There are lots of different ways to write up scientific investigations. the basis for a report Your teacher may have some suggestions. A sample 10-point framework can be seen here. You might not always include every section for every experiment you write up, but instead just focus on three or four of the sections. 18 Grade 11  UNIT 1: Measurement and practical work 1 Title (and date) 2 Aim What theory are you going to test? Or what are you going to investigate, and why? Or what are you going to measure? 3 Theory Explain the theory behind your experiment, with all the equations set out and explained clearly. If you are going to plot a graph to find a quantity, explain how the graph will enable you to do this. You should then be able to refer back to/quote from this section in your method and in your analysis of results. 4 Diagram(s) of experimental arrangement(s) These should be BIG (don’t be afraid to take up a full page), detailed and fully labelled, and showing how the experiment works. 5 Method Don’t repeat information that is already in the diagram! Give a clear, detailed and step-by-step procedure (bullet-point list), including measurements to be taken, any repeats, AND: Accuracy How did you ensure the accuracy of your measurements? (Any zero errors, was the experiment horizontal, etc.) How did you choose appropriate instruments to give readings to an appropriate precision? Mention the precision and range of key instruments used. For example, 1 m rule with a 1 mm scale, or a 0–10 A ammeter reading to 0.01 A. Did you do repeats? Were you looking from the correct angle when making measurements, etc? 6 Results Neatly set out ALL data/measurements recorded in a neat table, and averages (if applicable). Don’t forget headings/explanations of each table, and don’t forget the units either. Grade 11 19 UNIT 1: Measurement and practical work 7 Analysis What does your data show? Draw large graphs (with suitable scales, so that points take up a least half of the paper) on graph paper, with titles, labelled axes (units), and best-fit (not necessarily straight) smooth lines drawn through the points. Describe what your graphs show. When you use information from your graph(s) explain what you are using – e.g. gradient, area, etc. 8 Error/uncertainty analysis Identify all possible sources of error in your measurements. Distinguish between random and systematic uncertainties. Quantify the uncertainty of these (e.g. using ½-range or instrument precision (see Section 1.3). Express the uncertainty as a percentage for important readings. Use these to estimate the uncertainty in your final results. 9 Conclusions This should refer back to the aim – i.e. can you answer the question implied by the aim? If measuring something, quote final value with experimental uncertainty. If there is an accepted value, comment on the difference between your value and the accepted value. Does your experimental data fit the theory within experimental uncertainty? 10 Evaluation How could you improve the experiment – how could you improve the reliability? Be specific and realistic – just saying ‘be more careful’ or ‘use better equipment’ is not enough. 20 Grade 11  UNIT 1: Measurement and practical work Summary In this section you have learnt that: Your experimental results should be recorded in a clear and organised manner. Experimental results are usually recorded in tabular form. You can use a 10-point framework to write up scientific investigations. End of unit questions 1. Explain the importance of the scientific method. 2. Construct a glossary of all the key terms used in this unit. 3. Use the writing frame on pages 19–20 and complete all sections. Carry out a detailed investigation into one of the following: a)  How the height a ball is dropped from affects the height it bounces up to. b)  How the length of a piece of wire affects the electric current passing through it. c)  How the angle of slope affects the time taken for a ball to roll down the slope. 4. For the two activities in question 3 that you did not carry out, identify: a) Possible sources of error. b) The sizes of the uncertainties in the measurements you would take. 5. Makeda says that you should write down all the numbers on the calculator display when recording the final result of a calculation. Is Makeda correct? Explain your answer. Grade 11 21 Vector quantities Unit 2 Contents Section Learning competencies 2.1 Types of vector Demonstrate an understanding of the difference between scalars and (page 22) vectors and give common examples. Explain what a position vector is. Use vector notation and arrow representation of a vector. Specify the unit vector in the direction of a given vector. Determine the magnitude and direction of the resolution of two or 2.2 Resolution of vectors more vectors using Pythagoras’s theorem and trigonometry. (page 26) 2.3 Vector addition Add vectors by graphical representation to determine a resultant. and subtraction Add/subtract two or more vectors by the vector addition rule. (page 27) 2.4 Multiplication Use the geometric definition of the scalar product to calculate the of vectors scalar product of two given vectors. (page 33) Use the scalar product to determine projection of a vector onto another vector. Test two given vectors for orthogonality. Use the vector product to test for collinear vectors. Explain the use of knowledge of vectors in understanding natural phenomena. You have studies vectors in grade 9. An understanding of vectors is essential for an understanding of physics. They help physicists and engineers to build amazing structures and to design spacecraft, and they also help you find your way home! 2.1 Types of vector By the end of this section you should be able to: Demonstrate an understanding of the difference between scalars and vectors and give common examples. Explain what a position vector is. Use vector notation and arrow representation of a vector. Specify the unit vector in the direction of a given vector. 22 Grade 11  UNIT 2: Vector quantities Introduction and recap of basic vectors KEY WORDS All physical quantities are either scalar or vector quantities: magnitude the size of a value A vector quantity has both magnitude (size) and direction. scalar a quantity specified only by its magnitude A scalar quantity has magnitude only. All vector quantities have a direction associated with them. For Discussion activity example, a displacement of 6 km to the West, or an acceleration of 9.81 m/s2 down. Scalars are just a magnitude; for example, a mass of Which of the following do you think are scalars and which 70 kg or an energy of 600 J. are vectors? Electric current, Table 2.1 Some examples of vector and scalar quantities moment, time, potential difference, resistance, volume, Vector quantities Scalar quantities air resistance and charge. Forces (including weight) Distance Displacement Speed Velocity Mass 70 N Acceleration Energy 40° Momentum Temperature Figure 2.1 An arrow representing a force of 70 N at about 40° to the horizontal. Is this a vector or a Representing vectors scalar? All vector quantities must include a direction. For example, a displacement of 8 km would not be enough information. We must write 8 km South. Did you know? Vectors can be represented by arrows, the magnitude (size) of the In 1881 vectors appeared in vector is shown by the length of the arrow. The direction of the a publication called Vector arrow represents the direction of the vector. Analysis by the American J. W. Gibbs. They have been essential to maths and physics ever since! magnitude final point Activity 2.1: Drawing initial vector diagrams point Draw four vector arrows for the following (you will need direction to use different scales): Figure 2.2 A vector has size (magnitude) and direction. 140 km North 2.2 N left Representing vectors as arrows 9.81 m/s2 down Vectors are sometimes written in lowercase boldface, for example, a or a. If the vector represents a displacement from point A to point 8 7 m/s at an angle of B, it can also be denoted as: 75° to the horizontal. AB Grade 11 23 UNIT 2: Vector quantities KEY WORDS Types of vector collinear vectors vectors that There are several different types of vector to consider. These are are parallel to each other and outlined below. which act along the same line coplanar vectors vectors Position vector that act in the same A position vector represents the position of an object in relation to two-dimensional plane another point. position vector a vector that B y B B represents the position of an object in relation to another 45o 14.1 km point 20 km 20 km unit vector a vector with a 45o x A A A 14.1 km length of one unit distance and polar form component form bearing Figure 2.3 Position vectors B is 20 km North East of A. Alternatively this could be written as B is 20 km from A on a bearing of 45o. Remember that bearings are specified as an angle going clockwise from north. The vector can be given in polar form. The angle is given from the positive x-axis, going anticlockwise. The angle can be in degrees or radians. The vector B from A is: B = (20, 45o) The vector can also be given in component form, where it is given in 1N 1 m/s2 terms of the components in the x, y and z directions. The vector B 1m is: B = (14.1 km, 14.1 km, 0 km) Figure 2.4 Unit vectors Unit vector A unit vector is a vector with a length equal to one unit. For example, Figure 2.4 contains three examples of unit vectors, one j each for displacement, force and acceleration. Unit vectors can also have direction. There are three unit vectors which are used to specify direction, as shown in Figure 2.5: i unit vector i is 1 unit in the x-direction unit vector j is 1 unit in the y-direction k unit vector k is 1 unit in the z-direction. Figure 2.5 Unit vectors i, j and k Collinear vector Collinear vectors are vectors limited to only one dimension. Two vectors are said to be collinear if they are parallel to each other and act along the same line. They can be in the same direction or opposite directions. Figure 2.6 These three vectors are collinear 24 Grade 11  UNIT 2: Vector quantities Coplanar vector This refers to vectors in the same two-dimensional plane. This may include vectors at different angles to each other. For example, Figure 2.7 shows two displacement vectors when viewed from above. A xz plane 12 km East y 5 km North xy plane B Figure 2.7 Coplanar displacement vectors A more complex example might involve three forces acting on a x cube. A and B are both in the same plane (the xy plane) so they might z C yz plane be described as coplanar. C is in a different plane and so is not coplanar. Figure 2.8 Three vector forces acting on a cube B and C are in the same plane (the xz plane) so they might be described as coplanar. A is in a different plane and so is not coplanar. A and C are in the same plane (the yz plane) so they might be described as coplanar. B is in a different plane and so is not coplanar. A, B and C cannot be considered to be coplanar with each other as they are in different planes. Summary In this section you have learnt that: A vector quantity has both magnitude (size) and direction. A scalar quantity has magnitude only. Vectors are often represented by arrows. Different types of vectors include position vectors, unit vectors, collinear vectors (along the same line) and coplanar vectors (in the same two-dimensional plane). Review questions 1. Define the terms vector and scalar. Give five examples of each. 2. Explain the differences and similarities between position vectors, unit vectors, collinear vectors and coplanar vectors. Give examples for each. Grade 11 25 UNIT 2: Vector quantities KEY WORDS 2.2 Resolution of vectors component vectors two or more vectors that, when By the end of this section you should be able to: combined, can be expressed as Determine the magnitude and direction of the resolution a single resultant vector of two or more vectors using Pythagoras’s theorem and resolving splitting one vector trigonometry. into two parts that, when combined, have the same effect as the original vector What is resolution? Resolving means splitting one vector into two component vectors. This may be a component in the x direction (horizontal) and 25.0 m/s another in the y direction (vertical). The two components have the same effect as the original vector when combined. An example can be seen in Figure 2.9, the velocity of 25.0 m/s can 65° be resolved into two component vectors that, when combined, have the same effect. The component vectors can be made to form the sides of a right- angled triangle. They make up the opposite and adjacent sides of the triangle. As we know the size of the hypotenuse (in this case 25.0 m/s) and the angle (in this case 65°), we can then use trigonometry to find their relative sizes. Figure 2.9 Component vectors of Using trigonometry to resolve vectors the main vector are shown in blue You will probably remember the following rules from your maths class: sin θ = opposite hypotenuse cos θ = adjacent hypotenuse opposite tan θ = adjacent In the case of Figure 2.9: hypotenuse × sin θ = opposite opposite 25.0 m/s × sin 65 = 22.7 m/s in the y direction opposite hypotenuse × cos θ = adjacent 25.0 m/s × cos 65 = 10.6 m/s in the adjacent x direction adjacent You can check your working by θ hypotenuse Figure 2.10 Component vectors using Pythagoras’s theorem to as a right-angled triangle recombine the vectors. Figure 2.11 The basic rules of trigonometry 26 Grade 11  UNIT 2: Vector quantities Pythagoras’s theorem b a For a right-angled triangle, Pythagoras’s theorem states: a2 = b2 + c2 We can use this to work out the magnitude of two coplanar vectors. We can use trigonometry to work out the direction of the two vectors. c Figure 2.12 A right-angled Worked example 2.1 triangle demonstrates Pythagoras’s theorem What is the a) magnitude and b) direction of the two coplanar vectors in Figure 2.13? a) D2 = 32 + 42 = 9 + 16 = 25 D = √25 = 5 m b) tan θ = opposite/adjacent = 4/3 = 1.333... θ = tan –1 1.333... = 53º 4m D 3m θ Figure 2.13 Two perpendicular coplanar vectors form a right-angled triangle. Summary In this section you have learnt that: Resolving means splitting a vector into two perpendicular components. The components have the same effect as the original vector. Trigonometry can be used to determine the magnitude of the components. Vectors can be added mathematically using Pythagoras’s theorem and trigonometry. Review questions 1. Explain what it means to resolve a vector. 2. Draw simple vector diagrams and resolve them into two components. a) 60 N at an angle of 30° from the horizontal. b) 45 m/s at an angle of 80° from the horizontal. c) 1900 km at an angle of 40° from the vertical. Grade 11 27 UNIT 2: Vector quantities Did you know? 2.3 Vector addition and subtraction Vectors are used in computer games to By the end of this section you should be able to: determine the movement of a character. Software will Calculate vectors by graphical and mathematical methods. convert the commands from Appreciate the parallelogram rule and the triangle rule. the games controller into a Solve more complex examples of vectors. three-dimensional vector to describe how the character should move. Adding vectors It is often necessary to add up vectors to find the resultant vector acting on a body. This may be the resultant velocity of an object, the resultant force acting on an object, or even the resultant Activity 2.2: Drawing displacement after several legs of a journey. scale diagrams Draw a scale diagram to find Vector diagrams the resultant displacement from the following: The first technique for vector addition involves carefully drawing diagrams. This can only be applied to collinear or coplanar vectors 1 2 km at an angle of 0° to (this is because your diagrams will be two-dimensional only!). the vertical 2 4 km at an angle of 90° There are three slightly different techniques that could be used. to the vertical Scale diagrams 6 km at an angle of 120° to the vertical Whenever you are drawing vector diagrams you should draw them to a scale of your own devising. Scale diagrams are very simple: 3 0 km at an angle of 210° to the vertical Select a scale for your vectors. Draw them to scale, one after the other (in any order), lining them up head to tail ensuring the directions are correct. The resultant will then be the arrow drawn from the start of the first vector to the tip of the last. For example: Start End A E Figure 2.14 Scale diagram B showing a resultant vector (the D red arrow) for a series of coplanar C vectors It does not matter in which order you draw your vectors. Check them for yourself! Start Figure 2.15 Adding three Finish Start collinear vectors A Finish E B C B D D E Figure 2.16 The resultant vector C A remains the same If you end up where you started, then all the vectors cancel out and there is no resultant vector. 28 Grade 11  UNIT 2: Vector quantities The fact that you can add vectors in any order and get the same resultant vector is called the commutative law. Start/End Parallelogram rule If you have two coplanar vectors, you could use the parallelogram rule. This involves drawing the two vectors with the same starting point. The two vectors must be drawn to a scale and are made to be the sides of the parallelogram. The resultant will be the diagonal of the parallelogram. Figure 2.17 Scale diagram showing no resultant force KEY WORDS commutative law a process obeys the commutative law when it does not matter which order the quantities Figure 2.18 Two perpendicular coplanar vectors are in. For example, the addition of numbers obeys the If the vectors are perpendicular, the parallelogram will always be a commutative law. rectangle. Activity 2.3: Finding an unknown force There are three forces acting on an object, A, B and C. This object is at equilibrium (there is no resultant force acting on it). Draw a scale diagram Figure 2.19 Two non-perpendicular coplanar vectors to find the magnitude and direction of the unknown If the vectors are still coplanar but not perpendicular, the force. parallelogram will not be a rectangle. F orce A, 45 N at an angle Triangle rule of 0° to the horizontal This is a very similar technique, it involves drawing the two F orce B, 30 N at an angle coplanar vectors but this time drawing them head to tail. The two of 300° to the vertical vectors must again be drawn to a scale. The resultant will be the Force C, unknown missing side from the triangle. If the vectors are perpendicular, the triangle will be a right-angled triangle. Figure 2.20 Two perpendicular coplanar vectors Grade 11 29 UNIT 2: Vector quantities Discussion activity If the triangle is a right- angled triangle, we could use trigonometry to determine the sides and angles mathematically. What if the triangle is not a right-angled triangle? Figure 2.21 Two non-perpendicular coplanar vectors If the vectors are still coplanar but not perpendicular the triangle will not be a right-angled triangle. Activity 2.4: Determining resultant velocity Use the parallelogram rule to determine the resultant velocity of the following two velocities: Velocity A, 30 m/s at an angle of 45° to the horizontal. Velocity B, 40 m/s at an angle of 80° to the horizontal. Repeat, this time use the triangle rule. Activity 2.5: Adding vectors Vector A is 6 m/s along the horizontal and vector B is 9 m/s at 90º to the horizontal. Add vector B to vector A. Add vector A to vector B. Does the order you add the vectors make any difference to the resultant? Adding coplanar vectors mathematically To add coplanar vectors we use more complex mathematics. Since two perpendicular coplanar vectors form a right angled- triangle, they can be added using Pythagoras’s theorem and trigonometry. Pythagoras’s theorem determines the magnitude, and trigonometry can be used to determine the direction. 30 Grade 11  UNIT 2: Vector quantities Component method We can also express vectors as components in the x, y and z directions. The resultant vector shown in Figure 2.13 can be expressed in component form as (3, 4), where the first number is the magnitude in the x-direction and the second number is the magnitude in [] 3 the y-direction. The vector can also be given in the column form 4. We can also add and subtract vectors in this form. Activity 2.6: Adding forces You are going to pull on a block of wood with two forces. You will find the resultant of the two forces, and then check your findings by adding the vectors. C F ind a suitable block of wood and three forcemeters (newtonmeters or spring B balances). Place the block on a sheet of 90º plain paper. A  ttach two of the forcemeters (A and A B) to one end of the block as shown in Figure 2.22. Attach the third (C) to the opposite end. O  ne person pulls on each forcemeter. A and B should be at an angle of 90º Figure 2.22 Testing vector addition of forces to each other. C is in the opposite direction. Pull the forcemeters so that their effects balance. B  ecause force C balances forces A and B, it must be equal and opposite to the O  n the paper, record the magnitudes and resultant of A and B. Did you find this? directions of the three forces. R  epeat the experiment with different N  ow add forces A and B together using forces at a different angle. vector addition. Worked example 2.2 Nishan and Melesse are trying to drag a box. Add the components of the forces together Nishan is using a force of (15, 8, 6) N and using: Melesse is using a force of (12, 10, –6) N. FA + FB = (Ax + Bx, Ay + By, Ax + Bz) What is the resultant force on the box? FA = (15, 8, 6) N First draw a diagram to show the forces. FB = (12, 10, –6) N y So FA + FB = (15 + 12, 8 + 10, 6 + – 6) N −6 10 6 = (27, 18, 0) N x 8 z 12 15 Figure 2.23 Forces acting on the box Grade 11 31 UNIT 2: Vector quantities Worked example 2.3 Summary We can also use the cosine rule In this section you have learnt that: and the sine rule to work out the magnitude and direction Vectors can be added graphically by drawing scale of the sum of two vectors. diagrams. Consider two vectors a (5 m Vectors can be expressed as components. along horizontal) and b (6 m at an angle of 60° above the horizontal). Review questions First draw a diagram showing 1. Vector p is 6 m in the x-direction. Vector q is 10 m in the the vector addition. A y-direction. c b a) Use the parallelogram method to work out p + q. C B 60° b) Use Pythagoras’s theorem and trigonometry to work out a