CGP A-Level Physics Year 1 & AS PDF Edexcel

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CGP CGP CGP P G C A-Level Year 1 & AS Physics Edexcel The new A-Levels A-Level Year 1 & AS are seriously tough... But don’t worry — CGP have come to the rescue with this fantastic all-in-one book! Short, sharp revision notes for every topic... Physics Exam Board: Edexcel No pointless rambling, just the relevant info Exam-style questions to test your skills... Of course — all with detailed answers included 100% matched to the new A-Level courses... Perfect for Edexcel AS Physics and Year 1 of the full A-Level If you can find a better revision guide from anyone else, we’ll eat our hats ☻ Complete Revision & Practice Complete Revision & Practice P G C www.cgpbooks.co.uk C G P CGP C G P Even more brilliant CGP books Our bestselling books that might just save your life... are now available on... To find out more (or place an order with fantastic next-day delivery), just visit our website or give our friendly team a call: www.cgpbooks.co.uk 0800 1712 712...perfect for revision on your computer or tablet! And of course, our books are available from all good booksellers, including: See our full Kindle range at cgpbooks.co.uk/kindle or browse our books at 0816 - 14855  A-Level Year 1 & AS Physics Exam Board: Edexcel Revising for Physics exams is stressful, that’s for sure — even just getting your notes sorted out can leave you needing a lie down. But help is at hand... This brilliant CGP book explains everything you’ll need to learn (and nothing you won’t), all in a straightforward style that’s easy to get your head around. We’ve also included exam questions to test how ready you are for the real thing. P G C A-Level revision? It has to be CGP!  Published by CGP Editors: Emily Garrett, David Maliphant, Rachael Marshall, Sam Pilgrim, Charlotte Whiteley and Sarah Williams. Contributors: Tony Alldridge, Jane Cartwright, Peter Clarke, Barbara Mascetti, John Myers and Andy Williams. PER53DK ISBN: 978 1 78294 293 1 With thanks to Mark A. Edwards for the proofreading. With thanks to Jan Greenway for the copyright research. www.cgpbooks.co.uk Clipart from Corel® Based on the classic CGP style created by Richard Parsons. Text, design, layout and original illustrations © Coordination Group Publications Ltd. (CGP) 2015 All rights reserved. 0800 1712 712 www.cgpbooks.co.uk  Contents Topic 1 — Working as a Physicist Topic 4 — Materials The Scientific Process 2 Hooke’s Law 56 Quantities and Units 4 Stress, Strain and Elastic Strain Energy 58 Planning and Implementing 6 The Young Modulus 60 Analysing Results 8 Stress-Strain and Force-Extension Graphs 62 Measurements and Uncertainties 10 Density, Upthrust and Viscosity 64 Evaluating and Concluding 12 Topic 5 — Waves and Topic 2 — Mechanics Particle Nature of Light Scalars and Vectors 14 The Nature of Waves 66 Motion with Uniform Acceleration 16 Types of Wave 68 Free Fall 18 Polarisation of Waves 70 Projectile Motion 20 Ultrasound Imaging 72 Displacement-Time Graphs 22 Superposition and Coherence 74 Velocity-Time and Acceleration-Time Graphs 24 Stationary (Standing) Waves 76 Forces 26 Refractive Index 78 Newton’s Laws of Motion 28 Lenses 80 Drag and Terminal Velocity 30 Diffraction 82 Momentum 32 Diffraction Gratings 84 Work and Power 34 Light — Wave or Photon? 86 Conservation of Energy and Efficiency 36 The Photoelectric Effect 88 Mass, Weight and Centre of Gravity 38 Wave-Particle Duality 90 Moments 40 Answers 92 Index 100 Topic 3 — Electric Circuits Charge, Current and Potential Difference 42 Resistance and Resistivity 44 I-V Characteristics 46 Electrical Energy and Power 48 E.m.f and Internal Resistance 50 Conservation of Charge & Energy in Circuits 52 The Potential Divider 54 2 Topic 1 — Working as a Physicist The Scientific Process You’ll need the skills covered in Topic 1 throughout your course, so make sure you’re clear on everything that comes up on the next 12 pages, and can apply it to the physics you meet in the rest of your AS or A-level. Scientists Come Up with Theories — Then Test Them... Science tries to explain how and why things happen — it answers questions. It’s all about seeking and gaining knowledge about the world around us. Scientists do this by asking questions, suggesting answers and then testing their suggestions to see if they’re correct — this is the scientific process. 1) Ask a question about why something happens or how something works. E.g. what is the nature of light? 2) Suggest an answer, or part of an answer, by forming a theory (a possible explanation of the observations) — e.g. light is a wave. (Scientists also sometimes form a model too — a simplified picture of what’s physically going on.) 3) Make a prediction or hypothesis — a specific testable statement, based on The evidence supported the theory, about what will happen in a test situation. For example, if light is a Quentin’s Theory of wave, it will interfere and diffract when it travels through a small enough gap. Flammable Burps. 4) Carry out a test — to provide evidence that will support the prediction | || | | | | | | | | | | | | | | A theory is only | | | | | | | || (or help to disprove it). E.g. shining light through a diffraction grating to show | | | | | | | || diffraction and interference (p.84). scientific if it can be tested. | | | | | | | | | | | | | | | | |...Then They Tell Everyone About Their Results... The results are published — scientists need to let others know about their work. Scientists publish their results as reports (similar to the lab write-ups you do in school) written up in scientific journals. Scientific journals are just like normal magazines, only they contain scientific reports (called papers) instead of the latest celebrity gossip. 1) It’s important that the integrity (trustworthiness) of the reports published in scientific journals is checked. Scientists (like anyone else) might be dishonest or biased, or an investigation might have made invalid conclusions (see page 13). 2) The report is sent out to peers — other scientists that are experts in the same area. They examine the data and results, and if they think that the conclusion is reasonable it’s published. This makes sure that work published in scientific journals is of a good standard. This process is known as peer review. 3) But peer review can’t guarantee the science is correct — other scientists still need to reproduce it. 4) Sometimes mistakes are made and bad work is published. Peer review isn’t perfect but it’s probably the best way for scientists to self-regulate their work and to publish quality reports....Then Other Scientists Will Test the Theory Too Other scientists read the published theories and results, and try to test the theory themselves. This involves: Repeating the exact same experiments. Using the theory to make new predictions and then testing them with new experiments. If the Evidence Supports a Theory, It’s Accepted — for Now 1) If all the experiments in all the world provide good evidence to back it up, the theory is thought of as scientific ‘fact’ (for now). 2) But it will never become totally indisputable fact. Scientific breakthroughs or advances could provide new ways to question and test the theory, which could lead to new evidence that conflicts with the current evidence. Then the testing starts all over again... And this, my friend, is the tentative nature of scientific knowledge — it’s always changing and evolving. Topic 1 — Working as a Physicist  3 The Scientific Process So scientists need evidence to back up their theories. They get it by carrying out experiments, and when that’s not possible they carry out studies. Evidence Comes From Controlled Lab Experiments... || | | | | | | | | | | | | | | | || | | | | | | | | Pages 6-13 are all about | | | | | || 1) Results from controlled experiments in laboratories are great. | | | | | | | how to design and carry 2) A lab is the easiest place to control variables so that they’re all kept constant out a lab experiment. || | | (except for the one you’re investigating). | | | | | | | | | | | | | | | | | | | | | | | | | |...That You can Draw Meaningful Conclusions From 1) You always need to make your experiments as controlled as possible so you can be confident that any effects you see are linked to the variable you’re changing. 2) If you do find a relationship, you need to be careful what you conclude. You need to decide whether the effect you’re seeing is caused by changing a variable (this is known as a causal relationship), or whether the two are just correlated. There’s more about drawing conclusions on page 13. Society Makes Decisions Based on Scientific Evidence 1) Lots of scientific work eventually leads to important discoveries or breakthroughs that could benefit humankind. 2) These results are used by society (that’s you, me and everyone else) to make decisions — about the way we live, what we eat, what we drive, etc. 3) All sections of society use scientific evidence to make decisions, e.g. politicians use it to devise policies and individuals use science to make decisions about their own lives. Other factors can influence decisions about science or the way science is used: 1) Economic factors: Society has to consider the cost of implementing changes based on scientific conclusions — e.g. the cost of reducing the UK’s carbon emissions to limit the human contribution to climate change. Scientific research is often expensive. E.g. in areas such as astronomy, the Government has to justify spending money on a new telescope rather than pumping money into, say, the NHS or schools. 2) Social factors: Decisions affect people’s lives — e.g. when looking for a site to build a nuclear power station, you need to consider how it would affect the lives of the people in the surrounding area. 3) Environmental factors: Many scientists suggest that building wind farms would be a cheap and environmentally friendly way to generate electricity in the future. But some people think that because wind turbines can harm wildlife such as birds and bats, other methods of generating electricity should be used. Practice Questions Q1 What is peer review? Q2 Explain why a theory supported by a paper published in a scientific journal is not necessarily scientific ‘fact’? Q3 Explain why considering scientific evidence alone is not enough when making decisions in society. Exam Question Q1* A scientist has developed a proposal for a power station which generates electricity renewably from the energy of the tides, without producing greenhouse gases that can lead to climate change. The proposed location of the power station is off a stretch of scenic coastline that attracts a lot of tourism. She claims that her evidence suggests that if built in this location, the power station would produce sufficient electricity to entirely replace a nearby coal-fired power station that uses non-renewable fossil fuels and produces greenhouse gases. Discuss the economic, social and environmental factors that need to be considered by decision-makers considering her proposal. [6 marks] * The quality of your extended response will be assessed in this question. No. Borrowing your friend’s homework to ‘check’ it is not peer review... Hopefully these pages have given you a nice intro to the scientific process, e.g. what scientists do to provide you with ‘facts’. You need to understand this, as you’re expected to know how science works yourself — for the exam and for life. Topic 1 — Working as a Physicist 4 Quantities and Units Learning Physics is a lot like building a house — both involve drinking a lot of tea. Also, both have important foundations — if you skip this stuff everything else is likely to go a bit wrong. So, here goes brick-laying 101... A Physical Quantity has both a Numerical Value and a Unit 1) Every time you measure or calculate a quantity you need to give the units. 2) There are seven base quantities from which all the other quantities you’ll meet can be derived. These other quantities are called derived quantities. 3) The Système International (SI) includes a set of base units for the seven base quantities. The base quantities and their SI base units are: Quantity SI base unit mass kilogram, kg Kilograms are a bit odd — they’re length metre, m the only SI unit with a scaling time second, s prefix (see the next page). current ampere, A You’re more likely to see temperature kelvin, K temperatures given in °C. amount of a substance mole, mol luminous intensity candela, cd 4) The units for derived quantities can be derived from these base units. | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | || Remembering how SI derived units || | | E.g. force is a derived quantity, with the SI derived unit of newtons, N. | | | | | | | | || 1 N is defined as equivalent to 1 kgms–2. are defined will help you make | | | | | | | sure the other quantities in your 5) The derived quantities and SI derived units you’ll need will be covered equations are in the right units. throughout the book and you need to remember them. | | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 6) You also need to have a rough idea of the size of each SI base unit and SI derived unit in this book, so that you can estimate quantities using them. Example: A man drops a ball from the first floor window of his house. It takes 1 s to hit the ground. Estimate the average speed of the ball. A first floor window of a normal house is about 6 m above the ground, so s = 6 m. speed = distance ÷ time = 6 m ÷ 1 s = 6 ms–1 You Can Work Out Derived Units Mathematically The units in any equation must always be the same on both sides. You can use this rule to work out some of the simpler SI derived units, like speed: Example: Show that the SI derived unit for speed is ms–1 (speed = distance ÷ time). Distance is a length, so its SI base unit is the metre, m. The base unit of time is the second, s. To find the unit for speed, just put the units for distance and time into the equation for speed: m ÷ s = ms–1 Some SI derived units have special names, like the newton. You can work out what they’re equivalent to in SI base units using the same method as above. Example: Charge, Q, is measured in coulombs, C, and given by the equation charge = current × time. What is one coulomb equivalent to in SI base units? The SI base unit for current is the ampere, A, and the SI base unit for time is the second, s. Charge = current × time, so 1 C = 1 A × 1 s = 1 As Topic 1 — Working as a Physicist  5 Quantities and Units Prefixes Let You Scale Units Physical quantities come in a huge range of sizes. Prefixes are scaling factors that let you write numbers across this range without having to put everything in standard form. These are the most common prefixes you’ll come across: femto pico nano micro milli centi mega giga tera prefix deci (d) kilo (k) (f) (p) (n) (μ) (m) (c) (M) (G) (T) multiple 0.001 0.01 0.1 1000 1 × 10–15 1 × 10–12 1 × 10–9 1 × 10–6 1 × 106 1 × 109 1 × 1012 of unit (1 × 10–3) (1 × 10–2) (1 × 10–1) (1 × 103) If you’re a bit uncertain about moving directly between these scaling factors, then convert quantities into the standard unit before you do anything else with them: Example 1: Convert 1869 picometres into nanometres. First, convert the value to metres: 1869 pm = 1869 × 10–12 m Then divide by 1 × 10–9 to convert to nanometres: 1869 × 10–12 ÷ 1 × 10–9 = 1.869 nm Or, you can convert between prefixes directly: | | | | | | | | | | | | | | | | | | | | | | | | || | || | | | | | | | | | | | It’s really easy to get muddled up when | | | | | | | | | Example 2: Convert 0.247 megawatts into kilowatts. | | | | | | | | | | | | | | | | || you’re converting between prefixes. The rule 1 MW = 1 × 106 W and 1 kW = 1 × 103 W is, if you’re moving to the right in the table So the scaling factor to move between MW and kW is: above, your number should get smaller, | | | | | | | | | | and if you’re moving to the left the number (1 × 106) ÷ (1 × 103) = 1 × 103. should get larger. If your answer doesn’t So 0.247 MW = 0.247 × 1 × 103 = 247 kW match the rule, you’ve made a mistake. | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Don’t use these prefixes in the middle of calculations — they’ll change the units of your final answer, and could get you in a mess. You should generally convert to the units you need to use before you do any calculations, or once you’ve got a final answer. Practice Questions Q1 What is meant by a base quantity and a derived quantity? Q2 What is the SI unit of mass? Q3 What is meant by an SI base unit and an SI derived unit? Q4 What is: a) 20 000 W in kilowatts b) 2 × 10–6 W in milliwatts c) 1.23 × 107 W in gigawatts? Aliona preferred scaling frozen waterfalls. Exam Question Q1 The density, r, of a material gives its mass per unit volume. It is given by r = m /V, where m = mass and V = volume. a) Express the units of density in terms of SI base units. [1 mark] b) Calculate the density of a cube of mass 9.8 g, and side length 11 mm. Give your answer in the units stated in part a). [2 marks] c) A bath tub is filled with water. Given that the density of water is approximately 1000 kg m–3, estimate the mass of the water in the bath tub. [2 marks] What’s the SI base unit for boring... Not the most exciting pair of pages these, I’ll admit, but it’s important that you have the basics down, or else you’re leaving yourself open to simple little mistakes that’ll cost you marks. So make sure you’ve memorised all the SI base units in the table, then try and write down all the prefixes and their scaling factors. If you don’t get them all first time, keep trying until you can. Remember, you need to know the units for every derived quantity you meet in this book, too. Topic 1 — Working as a Physicist 6 Planning and Implementing Science is all about getting good evidence to support (or disprove) your theories, so scientists need to be able to spot a badly designed experiment, interpret the results of an experiment or study, and design their own experiments too... You Might have to Design an Experiment to Answer a Question 1) You might be asked to design a physics experiment to investigate something or answer a question. 2) It could be a lab experiment that you’ve seen before, a new experiment you aren’t familiar with, or it could be something applied, like deciding which building material is best for a particular job. 3) Whatever you’re asked, you’ll be able to use the physics you know and the skills covered on the next few pages to figure out the best way to do the investigation. A Variable is Anything that has the Potential to Change in an Experiment 1) First, you need to identify your The independent variable is the thing you change. independent and dependent variables: The dependent variable is the thing you measure. Example 1: If you’re investigating how changing the potential difference across a component affects the current through it, the independent variable is the potential difference, and the dependent variable is the current. 2) Apart from the independent and dependent variables, all other variables should stay the same during your experiment. If not, you can’t tell whether or not the independent variable is responsible for any changes in your dependent variable, so your results won’t be valid (p.12). This is known as controlling variables. It might be worth measuring control variables that are likely to change during your experiment to check that they really are under control. Example 2: If you’re investigating the value of Example 1 (continued): In the example acceleration due to gravity by dropping an object and above, you need to use the same circuit timing its fall, draughts in the room could really mess up components, and to keep the temperature your results. Picking an object that is more resistant to of the apparatus constant — e.g. by letting being blown about (like a ball-bearing) will help make the circuit cool down between readings. your results more precise (p.12). Select Appropriate Apparatus and Techniques 1) You need to think about what units your measurements of the independent and dependent variables are likely to be in before you begin (e.g. millimetres or metres, milliseconds or hours). 2) Think about the range you plan on taking measurements over too — e.g. if you’re measuring the effect of increasing the force on a spring, you need to know whether you should increase the force in steps of 1 newton, 10 newtons or 100 newtons. Sometimes, you’ll be able to estimate what effect changing your independent variable will have, or sometimes a pilot experiment might help. | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || | | | 3) Considering your measurements before you start will also help you choose There’s a whole range of apparatus and | | | | | | | | | | || the most appropriate apparatus and techniques for the experiment: techniques that could come up in your | | | | | | | | exam. Make sure you know how to use. Example: all the ones you’ve come across in class | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || If you’re measuring the length of a spring that you’re applying a force to, you might need a ruler. If you’re measuring the diameter of a wire, you’d be better off with a set of callipers. If the wire doesn’t stretch easily, you may need to use a very long wire to get an extension that’s big enough to measure, so you might need to use a pulley like in the Young modulus experiment on p.60. If you’re measuring a time interval, you could use a stopwatch. If the time is really short (for example if you’re investigating acceleration due to gravity), you might need something more sensitive, like light gates. 4) Whatever apparatus and techniques you use, make sure you use them correctly. E.g. if you’re measuring a length, make sure your eye is level with the ruler when you take the measurement. 5) While you’re planning, you should also think about the risks involved in your experiment and how to manage them — e.g. if you’re investigating a material that might snap, wear safety goggles to protect your eyes. Topic 1 — Working as a Physicist  7 Planning 1862_RG_MainHead and Implementing Figure Out how to Record your Data Before you Start Before you get going, you’ll need a data table to record your results in. 1) It should include space for your independent variable and your Current / A dependent variable. You should specify the units in the headers, Trial Trial Trial P.d. / V Average not within the table itself. 1 2 3 2) The readings of the independent variable should be 1.00 0.052 0.047 0.050 0.050 taken at evenly spaced intervals. 1.50 0.079 0.075 0.077 0.077 3) Your table will need enough room for repeated measurements. You 2.00 0.303 0.098 0.097... should aim to repeat each measurement at least three times. Taking 2.50 0.129 0.125 0.130... repeat measurements can reduce the effect of random errors in your 3.00 0.149 0.151 0.145... results (see p.12) and makes spotting anomalous results, like this one,............... much easier. If there’s no way to take repeat readings, then you should increase the total number of readings that you take. 4) There should be space in your table for any data processing you need to do, e.g. calculating an average from repeated measurements, or calculating speed from measurements of distance and time. 5) Most of the time, your data will be quantitative (i.e. you’ll be recording numerical values). Occasionally, you may have to deal with qualitative data (data that can be observed but not measured with a numerical value). It’s still best to record this kind of data in a table, to keep your results organised, but the layout may be a little different. You Could be Asked to Evaluate An Experimental Design If you need to evaluate an experimental design, whether it’s your own or someone else’s, you need to think about these sorts of things: Does the experiment actually test what it sets out to test? Is the method clear enough for someone else to follow? Apart from the independent and dependent variables, is everything else going to be properly controlled? Are the apparatus and techniques appropriate for what’s being measured? Will they be used correctly? Are enough repeated measurements going to be taken? Greta was paying Is the experiment going to be conducted safely, for those involved and those nearby? the price for not Are there any other ethical considerations? For example, does the experiment produce planning her experiment properly. harmful waste products? Is the apparatus being used in a way that won’t damage it? Practice Questions Q1 What is meant by the term independent variable? What is a dependent variable? Q2 Why do you need to plan to control all of the other variables in an experiment? Q3 What do you need to consider when selecting your apparatus? Q4 Why should you take repeated measurements in an experiment? Exam Question Q1 A student is investigating the effect of the light level on the resistance of an LDR (light-dependent resistor). The student connects the LDR to a power supply, and measures the resistance of the LDR at various distances from a light source using a multimeter. a) State the independent and dependent variables for this experiment. [1 mark] b) State two variables that the student needs to control in order to ensure his results are valid. [2 marks] The best-planned experiments of mice and men......often get top marks. The details of planning and carrying out an experiment will vary a lot depending on what you’re investigating, but if all this stuff is wedged in your brain you shouldn’t go far wrong, so make sure you’ve got it learned. Topic 1 — Working as a Physicist 8 Analysing Results You’ve planned an experiment, and you’ve got some results (or you’ve been given some in your exam). Now it’s time to look into them a bit more closely... Do any Calculations You Need to First 1) Before you calculate anything, check for any anomalous results. If there’s something in the results that’s clearly wrong, then don’t include it in your calculations — it’ll just muck everything up. Be careful though, you should only exclude an anomalous result if you have good reason to think it’s wrong, e.g. it looks like a decimal point is in the wrong place, or you suspect that one of the control variables changed. And you should talk about any anomalous results when you’re evaluating the experiment (pages 12-13). 2) For most experiments, you’ll at least need to calculate the mean (average) of some repeated measurements: | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || | | || In class, you could use a spreadsheet to process || | | | | | || sum of your repeated measurements | | | | | | || mean (average) of your data (and plot graphs), but it’s important that = a measurement number of repeats taken you know how to do it by hand for the exam. | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || 3) Calculate any quantities that you’re interested in that you haven’t directly measured (e.g. pressure, speed). You should try to give any values you calculate to the same number of significant figures as the data value with the fewest significant figures in your calculation. If you give your result to too many significant figures, you’re saying your final result is more precise than it actually is (see p.12). Present Your Results on a Graph | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || If you need to use your graph to measure | | | | | | | | || | | | | | | | | | | do Make sure you know how to plot a graph of your results: something, select axes that will let you this easil y (e.g. by mea surin g the grad ient 1) Usually, the independent variable goes on the x-axis and the or the intercept , see the next page). | | | | | | | | | | | | | | | | | | | | | dependent variable goes on the y-axis. Both axes should be labelled || | | | | | | | | | | | | | | | | | clearly, with the quantity and units. The scales used should be sensible (i.e. they should go up in sensible steps, and should spread the data out over the full graph rather than bunching it up in a corner). 0.35 2) Plot your points using 0.30 a sharp pencil, to make sure they’re as 0.25 accurate as possible. Current / A 0.20 3) Draw a line of best fit for your 0.15 results. Around half the data 0.10 points should be above the line, and half should be below it (you 0.05 should ignore anomalous results). Depending on the data, the line 0.00 might be straight, or curved. 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Potential difference / V Graphs can Show Different Kinds of Correlation | || | | | | | | | | | | | | | | | | | | | | | | Remember, correlation does not | | | | | || || | | | | | | | | | | | | necessarily mean cause — p.3. The correlation describes the relationship between the variables. Data can show: | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Positive correlation: Negative correlation: No correlation: As one variable As one variable No relationship increases the increases the between the other increases. other decreases. variables. Topic 1 — Working as a Physicist  9 Analysing Results You Might Need to Find a Gradient or Intercept If the line of best fit is straight, then the graph is linear. This means a change in one always leads to a change in the other. The line of best fit for a linear graph has the equation: Where m is the gradient of the y = mx + c line and c is the y-intercept. If the line of best fit goes through the origin (c is 0), you can | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || || | | || | | | || say the variables are directly proportional to each other: yµx µ just means ‘is directly proportional to’. | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Example: This graph shows displacement against time for a motorbike travelling west. Find the bike’s velocity. 1200 For a displacement-time graph, the gradient gives the 1000 velocity (as velocity = displacement ÷ time). Displacement / m 800 Dy = 1000 – 400 = 600 m || | | | | | | | | | | | | | | | | | | | 600 | | | | | || | | | Dx = 56 – 16 = 40 s D means ‘change in’. 400 || | | | | | | | | | | | | | | | | | | | | The y intercept is 160. 200 displacement Dy velocity =  = = 600 ÷ 40 = 15 ms–1 west This means the bike’s time Dx 0 displacement was 0 20 40 60 160 m at time 0 s. Time / s If a graph has a curved line of best fit, you can find the gradient of a given point on the line by drawing a tangent to the curve (see page 23). It’s sometimes helpful to choose axes that turn a curved graph into a straight one: Example: If you plot pressure Pressure The gradient is: Pressure For a given force, the graph against 1 ÷ area, the of pressure applied against        pressure ÷ (1 ÷ area) graph looks like this: the area that the force is = pressure × area applied over looks like this: || | | | | | | | | | | | | | | | | | | | | | 1/Area = force applied (p.64) Area || | | | pressure = force ÷ area. || | || | | | | | | | | | | | | | | | | | | | | | | | Practice Questions Q1 Describe what you should do with anomalous results when processing data. Q2 How do you calculate an average of repeated results? Q3 Sketch a graph showing a negative correlation. Exam Question Q1 An engineer is investigating the performance of a prototype car with a new kind of environmentally-friendly engine. The data below shows the speed of the car, going from stationary to over 70 kilometres per hour. (In this question, you may use the formula: acceleration = change in speed ÷ time taken to change speed.) Time / s 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Speed / km per hour 0 3 8 22 40 54 66 70 a) Draw a graph showing speed against time for this data. [4 marks] b) State the times, to the nearest second, between which the graph is linear. [1 mark] c) Using the graph, calculate the maximum acceleration of the car. [4 marks] My level of boredom is proportional to the time I’ve spent on this page... This stuff can get a bit fiddly, especially measuring the gradient of a curved line, but for the most part it’s not too bad, and you should have seen a lot of it before. So dust off your pencil sharpener, and get to work... Topic 1 — Working as a Physicist 10 Measurements 1862_RG_MainHead and Uncertainties There are errors and uncertainties in every measurement. You need to be aware of them. All Results have Some Uncertainty || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || | | | | | || The smallest measuring interval on an | | | | | | | 1) Every measurement you take has an experimental uncertainty. instrument gives you its resolution. | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || 2) The smallest uncertainty you can have in a measurement is the uncertainty due to the resolution of your equipment. It’s ± half of one division on the measuring instrument used. E.g. using a thermometer with a scale where each division represents 2 °C, a measurement of 30 °C will at best be measured to be 30 ± 1 °C. And that’s without taking into account any other errors that might be in your measurement. 3) The ± sign gives you the range in which the true length (the one you’d really like to know) probably lies. 30 ± 0.5 cm tells you the true length is very likely to lie in the range of 29.5 to 30.5 cm. The maximum difference between your value and the true value (here 0.5 cm) is sometimes called the margin of error. 4) There are two measures of uncertainty you need to know about: Absolute uncertainty — the total uncertainty for a measurement. Percentage uncertainty — the uncertainty given as a percentage of the measurement. Measuring larger values reduces the percentage uncertainty. 5) An uncertainty should also include a level of confidence or probability, to indicate how likely the true value is to lie in the interval. The more variation there is in your results, and the fewer repeats you have done, the less confident you can be. 6) The uncertainty on a mean (see p.8) of repeated results is the largest difference between the mean and any of the values used to calculate it. So if you take repeated measurements of a current and get values of 0.1 A, 0.4 A and 0.4 A, the mean current is 0.3 A, and the uncertainty on the mean is 0.3 – 0.1 = ± 0.2 A. You can estimate the uncertainty on the mean as half the range — e.g. (0.4 – 0.1) ÷ 2 = 0.15 A. 7) If no uncertainty is given for a value, the assumed uncertainty is half the increment of the last significant figure that the value is given to. E.g. 2.0 is given to 2 significant figures, so you would assume an uncertainty of 0.05. You should always assume the largest amount of uncertainty when doing an experiment. Sometimes You Need to Combine Uncertainties You have to combine the uncertainties of different measured values to find the uncertainty of a calculated result: Adding or Subtracting Data — ADD the Absolute Uncertainties Example: A wire is stretched from 4.3 ± 0.1 cm to 5.5 ± 0.1 cm. Calculate the extension of the wire. 1) First subtract the lengths without the uncertainty values: 5.5 – 4.3 = 1.2 cm 2) Then find the total uncertainty by adding the individual absolute uncertainties: 0.1 + 0.1 = 0.2 cm So, the extension of the wire is 1.2 ± 0.2 cm. Multiplying or Dividing Data — ADD the Percentage Uncertainties Example: A force of 15 N ± 3% is applied to a stationary object which has a mass of 6.0 ± 0.3 kg. Calculate the acceleration of the object and state the percentage uncertainty in this value. 1) First calculate the acceleration without uncertainty: a = F ÷ m = 15 ÷ 6.0 = 2.5 ms–2 2) Next, calculate the percentage uncertainty in the mass: % uncertainty in m = 0.3 # 100 = 5% 6.0 3) Add the percentage uncertainties in the force and mass values to find the total uncertainty in the acceleration: Total uncertainty = 3% + 5% = 8% So, the acceleration = 2.5 ms­–2 ± 8% Raising to a Power — MULTIPLY the Percentage Uncertainty by the Power Example: The radius of a circle is r = 40 cm ± 2.5%. Find the percentage uncertainty in the circle’s area (pr2). The radius is raised to the power of 2 to calculate the area. So, the percentage uncertainty is 2.5% × 2 = 5% Topic 1 — Working as a Physicist  11 Measurements 1862_RG_MainHead and Uncertainties Error Bars Show the Uncertainty of Individual Points This is an ‘error box’. The true value of the 1) Most of the time, you work out the data point could lie uncertainty in your final result using the 7.0 anywhere in this area. uncertainty in each measurement you make. 6.0 2) When you’re plotting a graph, you can The error bars extend 1 square right displacement / m 5.0 show the uncertainty in each measurement and left for each measurement, and by using error bars to show the 4.0 1 square up and down. This gives an range the point is likely to lie in. uncertainty of ± 0.1 s in each time 3.0 measurement, and ± 0.2 m in each 3) You can have error bars for just one displacement measurement. variable (see below), or both the 2.0 | | | | | | | | | | | | | | | | | | | | | | | || dependent and the independent variable. || 1.0 Your line of best fit (p.8) | | | | | | | | || | | | | | | | | || Error bars on both variables give you an should always go through ‘error box’ for each point. all of the error bars (see 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 time / s | | | |below) or boxes. || | | | | | | | | | | | | | | | | | | | | | You Can Calculate the Uncertainty of Final Results from a Line of Best Fit Normally when you draw a graph you’ll want to find the gradient or intercept 700 (p.9). For example, you can calculate k, the force constant of the object being 600 stretched, from the gradient of the graph on the right — here it’s about 500 force / N 20 000 Nm–1. You can find the uncertainty in that value by using worst lines: 400 1) Draw lines of best fit which have the maximum and minimum possible 300 slopes for the data and which should go through all of the error bars (see the pink and blue lines on the right). These are the worst lines for your data. 200 If your data has errors in both the independent and dependent variables, then 100 the worst lines should go through the corners of the error boxes. 0 5 10 15 20 25 30 35 2) Calculate the gradients of the worst lines. The blue line’s gradient is extension / mm about 21 000 Nm–1 and the pink line’s gradient is about 19 000 Nm–1. When the force is 0 N the 3) The uncertainty in the gradient is given by half the difference between the extension is 0 mm — this is a worst gradients — here it’s 1000 Nm–1. So this is the uncertainty in the value of measurement with no uncertainty. the force constant. For this object, the force constant is 20 000 ± 1000 Nm–1 (or 20 000 Nm–1 ± 5%). 4) Similarly, the uncertainty in the y-intercept is just half the difference between the worst intercepts (although there’s no uncertainty here since the worst lines both go through the origin). Practice Questions Q1 What is meant by experimental uncertainty? How does it relate to an instrument’s resolution? Q2 What are the rules for combining uncertainties? Q3 What are worst lines? How could you use them to find the uncertainty in the intercept of a graph? Exam Question Q1 A student is investigating the acceleration of a remote controlled car. The car has an initial velocity of 0.52 ± 0.02 ms–1 and accelerates to 0.94 ± 0.02 ms–1 over an interval of 2.5 ± 0.5 s. a) Calculate the percentage uncertainty in the car’s initial speed. [1 mark] b) Calculate the percentage uncertainty in the car’s final speed. [1 mark] c) Calculate the car’s average acceleration over this interval. Include the absolute uncertainty of the result in your answer. (acceleration = change in velocity ÷ time taken). [4 marks] My percentage uncertainty about these pages is 0.99%... Uncertainties are a bit of a pain, but they’re really important. Learn the rules for combining uncertainties, and make sure you know how to draw those pesky error bars, and what to do with them when you’ve got them. Topic 1 — Working as a Physicist 12 Evaluating and Concluding Once you’ve drawn your graphs and analysed your results, you need to think about your conclusions. Evaluate the Quality of Your Results Before you draw any conclusions, you should think about the quality of the results — if the quality’s not great you won’t be able to have much confidence in your conclusion. Good results are valid, precise, repeatable, reproducible and accurate. Quality control — also important in construction. 1) Valid measurements measure what they’re supposed to be measuring. If you haven’t controlled all the variables (p.6) then your results won’t be valid, because you won’t just be testing the effect of the independent variable. 2) You can say your results are precise if the range that your repeated data is spread over is small. The precision of a result is only influenced by random errors (see below). 3) Results are repeatable if you can repeat an experiment multiple times and get the same results. You can measure the repeatability of your results by looking at their precision, and by comparing them to the results of other students in your class who have used exactly the same method and equipment. 4) For a result to be reproducible, it needs to be obtained by different experimenters, using different equipment and different methods. Testing whether a result is reproducible is a better test of its quality than testing whether it is repeatable, as it’s less likely that the same systematic errors could have affected both methods (see below). 5) An accurate result is one that’s really close to the true value. You often can’t know the accuracy of a result, since you can never know the true value of what you’re measuring. The exception to this is if you’re measuring a known constant, like g, which has been tested many times, and is known to a good degree of certainty. In cases like this, you can assess how accurate your results are by comparing them to this known value. One way of doing this is by calculating the percentage difference — the difference between your result and the true value, expressed as a percentage of the true value. You’ll Need to Think About Random and Systematic Errors An error is the difference between your measured value and the true value of whatever you’re trying to measure. There are two different kinds of error that could affect the quality of your results — random errors and systematic errors. You should think about these when you’re planning your experiment (so you can minimise them) and when you’re evaluating your results. Systematic Errors 1) Systematic errors (including zero errors) are the same every time you repeat the experiment (they shift all the values by the same amount). They may be caused by the equipment you’re using or how it’s set-up, e.g. not lining up a ruler correctly when measuring the extension of a spring. 2) Systematic errors are really hard to spot, and they affect the accuracy of your results, but not the precision — so it can look like you’ve measured something really well, even if your results are actually completely off. Systematic errors might show up when you compare results with other people in your class who have done the same experiment if they’re caused by a mistake you made, or when you compare your results with someone else who used a different method if there is something wrong with the procedure itself (i.e. when you check the reproducibility of your results). 3) It’s always worth checking your apparatus at the start of an experiment, e.g. measure a few known masses to check that a mass balance is calibrated properly to make systematic errors less likely. Random Errors 1) Random errors vary — they’re what make the results a bit different each time you repeat an experiment. For example, if you measured the length of a wire 20 times, the chances are you’d get a slightly different value each time, e.g. due to your head being in a slightly different position when reading the scale. It could be that you just can’t keep controlled variables (p.6) exactly the same throughout the experiment. 2) Unlike systematic errors, you can at least generally tell when random errors are there, as they affect your precision. 3) Using apparatus with a better resolution (p.10) can reduce the size of random errors in individual measurements. 4) Doing more repeats can also reduce the effect of random errors on calculated results like the mean (p.8). Topic 1 — Working as a Physicist  13 Evaluating and Concluding | | | | | | | | | | | | | | | | | | | | | | || Draw Conclusions that Your Results Support || Your conclusion can’t | | | | | | | | | | | | | | | | | | be more precise than 1) A conclusion explains what the data shows. the margin given by the | | | | You can only draw a valid conclusion if you have valid data supporting it. uncertainty in your results. | | | | | | | | | | | | | | | | | | | | | | | | || 2) Your conclusion should be limited to the circumstances you’ve tested it under — if you’ve been investigating how the current flowing through a resistor changes with the potential difference across it, and have only used potential differences between 0 and 6 V, you can’t claim to know what would happen if you used a potential difference of 100 V, or if you used a different resistor. 3) You also need to think about how much you can believe your conclusion, by evaluating the quality of your results (see previous page). If you can’t believe your results, you can’t form a strong conclusion. Think About how the Experiment Could be Improved Having collected the data, is there anything you think should have been done differently? Were there any limitations to your method? 1) If the results aren’t valid, could you change the experiment to fix this, e.g. by changing the data you’re collecting? 2) If the results aren’t accurate, what could have caused this? If there are systematic errors in your results, what could you do to prevent them? 3) Are there any changes you could make to the apparatus or procedure that would make the results more precise? There are some simple ways to reduce random errors or their effects (see p.12), including using the most appropriate equipment, (e.g. swapping a millimetre ruler for a micrometer to measure the diameter of a wire) and increasing the number of repeats. You can also use a computer to collect data — e.g. using light gates to measure a time interval rather than a stopwatch. This makes results more precise by reducing human error. 4) Are there any other ways you could have reduced the errors in the measurements? Practice Questions Q1 What is a valid result? Q2 What is the difference between a repeatable result and a reproducible result? Q3 What is the difference between saying the results of an experiment are precise and saying that they are accurate? Q4 Give two examples of possible sources of random error and one example of a possible source of systematic error in an experiment. Which kind of error won’t affect the precision of the results? Q5 What should you think about when you are trying to improve an experimental design? Exam Question Q1 A student is investigating how the speed of a falling object is affected by 4.0 how long it has been falling for. He drops an object from heights between 10 cm and 60 cm and measures its speed at the end of its fall, and the time speed / ms–1 3.0 the fall takes, using light gates. He plots a graph of the final speed of the 2.0 object against the time it took to fall, as shown on the left. 1.0 a) Identify the anomalous result. [1 mark] 0.0 b) The student concludes that the speed of any falling object is 0.0 0.1 0.2 0.3 0.4 always proportional to the time it has been falling for. time / s Explain whether or not the results support this conclusion. [2 marks] In conclusion, Physics causes headaches... Valid, precise, repeatable, reproducible and accurate... you’d think they all mean the same thing, but they really don’t. Make sure you know the difference, and are careful about which one you use, or you’ll be throwing marks away. Topic 1 — Working as a Physicist 14 Topic 2 — Mechanics Scalars and Vectors And now time to draw some lovely triangles. Please, don’t all thank me at once... Scalars Only Have Size, but Vectors Have Size and Direction 1) A scalar has no direction — it’s just an amount of Scalars Vectors something, like the mass of a sack of meaty dog food. 2) A vector has magnitude (size) and direction — like the speed mass, time, energy displacement, force, and direction of next door’s cat running away. temperature, velocity, acceleration, 3) Force, velocity and momentum are all vectors — you need to length, speed momentum know which way they’re going as well as how big they are. Here are some of the common scalars and vectors that you’ll come across in your exams: 4) Vectors are drawn as arrows (to show their direction) with their size written | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Sometimes vectors are printed in bold, || | | | | | | | next to them (see Example 1 below). In the exam, you might see quantities | | | | | | | | e.g. v, but it’s quite hard to handwrite written with arrows above them, e.g. v , to show that they’re vectors. in bold, so the arrow is used too. || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || You can Add Vectors to Find the Resultant 1) Adding two or more vectors is called finding the resultant of them. 2) You should always start by drawing a diagram. Draw the vectors ‘tip to tail’. If you’re doing a vector subtraction, draw the vector you’re subtracting with the same magnitude but pointing in the opposite direction. 3) If the vectors are at right angles to each other, then you can use Pythagoras and trigonometry to find the resultant. 4) If the vectors aren’t at right angles, you may need to draw a scale diagram. Example 1: Jemima goes for a walk. She walks 3.0 m north and 4.0 m east. She has walked 7.0 m but she isn’t 7.0 m from her starting point. Find the magnitude and direction of her displacement. First, draw the vectors tip-to-tail. Then draw a line from the tail 4.0 m Jemima’s ‘displacement’ gives of the first vector to the tip of the last vector to give the resultant: her position relative to her Because the vectors are at right angles, you 3.0 m starting point, see p.16. get the magnitude of the resultant using Pythagoras: resultant vector, R R2 = 3.02 + 4.02 = 25.0 So R = 5.0 m 4.0 m Now find the bearing of Jemima’s new position from her original position. You use the triangle again, but this time you need to use trigonometry. 3.0 m  resultant You know the opposite and the adjacent sides, so you can use: vector = 5.0 m Jemima tan q = 4.0 / 3.0 So q = 053° (to 2 s.f.) || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || | | | | | || A bearing is just an angle measured clockwise from the | | | | | | | Example 2: A van is accelerating north, with a resultant force of north line, represented by three digits, e.g. 10° = 010°. 510 N. A wind begins to blow on a bearing of 150º. | | | | | | | | || || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | It exerts a force of 200 N (to 2 s.f.) on the van. North What is the new resultant force acting on the van? The vectors aren’t at right angles, so you need to do a scale drawing. 150º Pick a sensible scale. Here, 1 cm = 100 N seems good. Wind: 2.0 cm = 200 N Using a really sharp pencil, draw the initial resultant force on the van. As the van is going north, this should be a 5.1 cm long line going straight up. Van: 5.1 cm = 510 N The force of the wind acts on a bearing of 150º, so add this to your diagram. Using the same scale, this vector has a length of 2.0 cm. New resultant force: Then you can draw on the new resultant force and measure its length. 3.5 cm = 350 N Measure the angle carefully to get the bearing. 17º The resultant force has a magnitude of 350 N (to 2 s.f.), acting on a bearing of 017º (to 2 s.f.). Topic 2 — Mechanics  15 Scalars 1862_RG_MainHead and Vectors It’s Useful to Split a Vector into Horizontal and Vertical Components This is the opposite of finding the resultant — it’s called resolving. You start from the resultant vector and split it into two components at right angles to each other. You’re basically working backwards from Example 1 on the last page. Resolving a vector v into horizontal and vertical components: vertical You get the horizontal...and the vertical component vh like this: vh component vv like this: cos q = vh / v sin q = vv / v vv v vv | | | | | | | | | | | | | | | | | | | | | | | | | | | || || | | | | || Where q is the angle vh = v cos q vv = v sin q from the horizontal.  vh | | | | | | | | | | | | | | | | | | | | || horizontal Example: Charley’s amazing floating home is travelling at a speed of 5.0 ms–1 at an angle of 60° (to 2 s.f.) up from the horizontal. Find the