AS Physics 9702 unit 1: Standards and their Measurements 1 UNIT 1 - STANDARDS AND THEIR MEASUREMENT: This unit includes topic 1 and 2 from the CIE syllabus for AS course. Units of Measurement: Measuring any physical quantity means comparing it with a standard to determine its relationship with the standard. This standard is called unit, for example one metre, one kilogram etc. All the physical quantities are expressed in terms of i) some number and ii) some units. For example distance between two points is 200 kilometres where 200 is the number and kilometre is the unit. Base and derived units: There are seven units which are regarded as basic or base units. They are called SI (Systems International) base units. These are the fundamental units from which all the other units are derived. The units that are derived from base units are called derived units. They are the combination of base units. For example unit of force is newton (N) which is derived from kg.m.s -2, unit of pressure is pascal (Pa) which is derived from kg.m.s -2.m -2 or kg.m -1.s -2 etc. Multiple and submultiples of the units: 1 Quantity Unit Unit Symbol Length (l) metre m (small) Mass(m) kilogram kg (small k and g) Time (t or T) second s (small) Temperature (T) kelvin K (capital) Current (I) ampere A (capital) Amount of substance mole mol (small) Intensity of light (I) candela Cd (capital C small d) When units are too long or too small to express one can get the multiple or sub-multiple as desired. For example, 1 kilometre = 1000 metres. The word kilo means one thousand. The table shows the multiple and submultiples of units, their symbols and prefixes. Prefix Symbol Multiple Tera T 10 12 Giga G 10 9 Mega M 10 6 kilo k 10 3 deci d 10-1 centi c 10-2 milli m 10-3 micro µ 10-6 nano n 10-9 pico p 10-12 femto f 10-15 atto a 10-18 1 Different physical quantities estimation exercise in the class
2 Prepared by Faisal Jaffer, revised on Oct 2011 Measurement of physical quantities base units: 2 Length or distance (l or x): The SI unit of length or distance is metre and metre rule can be used to measure it. Other instruments that can be used to measure small distances between the two points are vernier calliper and micrometer screw gauge. The precision of length measurement may be increased by using these devices. 3 Measuring instrument Metre rule Vernier calliper Smallest possible measurement 0.1 cm 0.01 cm Micrometer screw gauge 0.001cm Parallax error is the error caused by improper placement of eye or view angle when recording certain measurement. The error can be avoided by placing eye parallel to the point where the measurement is to be taken. Vernier Calliper: It is a more precise length measuring instrument. It can measure the depth inside a cylinder, internal or external diameter. When there is no object in between the jaws and the zero mark of the vernier scale does not coincide with the zero mark of main scale then vernier calliper has zero error. If the zero error is +ve then subtract it from final reading or if the zero is ve then add it to the final reading. 2 http://physics.nist.gov/cuu/units/meter.html 3 http://www.phy.uct.ac.za/courses/c1lab/vernier1.html
AS Physics 9702 unit 1: Standards and their Measurements 3 Screw Gauge: 4 It is used to measure very small dimension for example diameter of a copper wire or thickness of a paper. If the zero error is positive than subtract it from the final reading and if the zero error is negative then add in the final reading. No zero error positive zero error Negative zero error +0.02mm -0.04mm Exercise: 1.1 Solve exercise from the website: http://www.technologystudent.com/equip1/vernier3.htm http://www.technologystudent.com/pdf2/micromt1.pdf Mass (m): It is the measure of the quantity of matter in an object and its unit is kilogram (kg) in SI systems. It depends on the number of molecules in the matter and their masses and does not depend on gravity. Therefore an object would have same mass on the Earth and on the Moon but different weight because of the difference of gravity. Mass of an object can be measured by using spring balance (newton-metre), top pan balance or lever balance. 4 http://www.technologystudent.com/equip1/microm1.htm
4 Prepared by Faisal Jaffer, revised on Oct 2011 Time (t): It is the duration of an event. The unit of time is second (s) Clock and stopwatches are used to determine event or to measure duration of an event. When the event start the time is considered to be t=0. Temperature (T): The SI unit of temperature is kelvin (not degree kelvin or o K). Temperature is a degree of hotness or coldness of any substance that can be measured using a thermometer. It is also a measure of how fast the atoms and molecules of a substance are moving. Different type thermometers are used to measure the temperature in different situations. For example liquid in glass thermometers are used to measure temperatures in the range of -78 o C to 500 o C, however thermocouple thermometer is used of wide range of temperature from -250 o C to 1500 o C. Other units of temperature are degree Celsius ( o C) and degree Fahrenheit ( o F). Note: 1 o F = 32 + 1.8 x o C and 1 K = 273+ o C Heat is the energy that flows from the object of higher temperature (hotter) to the object of lower temperature (colder). Current (I): In electrical terms the rate of flow of charges is called current. The unit of current is ampere, A. It is expressed as I =. where I is the current, Q is the charge in coulombs and t is the time in seconds. Ammeter is used to measure the current in the circuit. It is always connected in series in the loop in which the current is to be measured.
AS Physics 9702 unit 1: Standards and their Measurements 5 Homogeneous Equation: An equation is said to be homogeneous when each term in the equation is expressed in its base units. This can be use to prove that the different forms of equations have same base units. Following are some examples that show the concept: Example 1: work done = F. s work done = m. a. s joules = kg. m. s 2. m joules = kg. s 2. m 2 Exercise: 1.2 Example 2: p = F A ρgh = ma A kg m 3. m. kg. m. s 2 s 2. m = m 2 kg. s 2. m 1 = kg. s 2. m 1 Example 3: V = E q V = J A. s V = kg. s 2. m 2 A. s V = kg. s 3. m 2. A 1 a) Solve assignment # 1 from the website http://www.freewebs.com/faisalj/as/physics%20assignment%201%202%20%203.pdf b) Solve the following questions from past papers 1. May/June 2010 paper 21 2. May/June 2009, Paper 22, question 1(a) 3. Oct/Nov 2008, Paper 2, question 1 4. Oct/Nov 2007, Paper 1, question 1 5. May/June 2002, Paper 1, question 1 6. May/June 2001, Paper 2 question 1(b) Calibration: Calibration is the process of comparing a measuring instrument with a measurement called standard to establish the relationship between the values indicated by the instrument and those of the standard, for example calibrating the thermocouple thermometer with the voltmeter. The calibration is achieved by plotting a graph between the change in temperature and the corresponding value on the voltmeter at regular intervals and plotting a graph. The best fit curve is use to place the appropriate Celsius scale correspond to the voltages on the voltmeter which can be used to measure temperature. Exercise: 1.3 Solve Q1 of May/June 2007 of past paper AS 9702/2
6 Prepared by Faisal Jaffer, revised on Oct 2011 Estimation and order of magnitude: In physics we guess many things. When we think carefully about our guess and use all our knowledge and common sense, we are making estimation. By estimation we guess the order of magnitude of any physical constant. Order of magnitude is the number raised as a power of ten. In the following Size of an atom Size of a nucleus Size of an electron Mass of electron Mass of proton or neutron table there are some common order of magnitude used in A-level course. 10-9 m 10-15 m 3 10-15 m 10-31 kg 10-27 kg Exercise: 1.4 Estimate the following quantities and then measure them. Compare the estimated value with the measured value. Measuring Quantity Estimated Value Measured Value 1. height of bench 2. length of room 3. length of pencil 4. diameter of pencil 5. diameter of pencil lead 6. volume of brick 7. volume of liquid in drinking cup 8. mass of brick 9. mass of a person 10. mass of a nail 11. time between heartbeats 12. period of pendulum 13. volume of your head 14. mass of your ruler 15. thickness of protractor Exercise: 1.5 Solve the following questions from past papers. 1. May/June 2009, Paper 21, question 1(a) 2. May/June 2008, Paper 2, question 1 3. Oct/Nov 2007, Paper 1, question 2 4. May/June 2002, Paper 2, question 1
AS Physics 9702 unit 1: Standards and their Measurements 7 ACCURACY, PRECISION, ERROR AND UNCERTAINTY: 5 Accuracy refers to the agreement between a measurement and the true or correct value which should be known before. For example if a clock strikes twelve when the sun is exactly overhead, the clock is said to be accurate. The measurement of the clock at twelve and the phenomena it is meant to measure that is the sun located at the top are in agreement. Accuracy cannot be found unless the true value is known or simply to say that it is impossible to find absolute accurate value of any measurement. Precision refers to the repeatability of measurement when experiment is performed number of times. It does not require us to know the correct or true value. Hence the precise result means when the all the values are very close to each other. Systematic Error and Random Error: Error refers to the disagreement between a measurement and the true, correct or accepted value. Systematic error in an experiment occurs due to the faulty apparatus such as incorrectly labelled scale, incorrect zero mark on a metre scale or stopwatch running slowly. Repeating the measurement a number of times will have no effect on the results and will only be noticed at the end of the experiment. The only way to eliminate this type of error is to change the apparatus or recalibrate the measuring instrument. Zero error is a systematic error. Random error depends upon how well the experimenter uses the apparatus. The better the experimenter, the smaller the random error will be. The random can be minimized by performing the experiment number of times and taking the average of readings. Random error can be classified as instrumental error and observational error. Instrumental error is due to the lack of sensitivity of the measuring instrument, for example measuring the diameter of a pendulum bob using ruler. Observational error is the error due to experimenter`s reaction time, parallax error or when two objects cannot come close together. Calculating the Error: 6 i) For sum or difference of two quantities say Q a b or Q a b where error in a is a and error in b is b. The error Q in Q is define as simply the sum of the errors in a and b. We can write as ΔQ = Δa + Δb ii) For the product or division of two quantities Q = a b or Q = the error Q in Q is expressed as ΔQ Q = Δa a + Δb b Also same as is true for exponential equation Q = a b n Q Q = a a + n b b 5 See the web link http://scidiv.bcc.ctc.edu/physics/measure&sigfigs/b-acc-prec-unc.html 6 For more detail explaination and example visit the website http://instructor.physics.lsa.umich.edu/iplabs/tutorials/errors/estimate.html
8 Prepared by Faisal Jaffer, revised on Oct 2011 Uncertainty: 7 All measurements are approximations--no measuring device can give perfect measurements without certain experimental uncertainty. By convention, a mass measured to 13.2 g is said to have an absolute uncertainty of 0.1 g and is said to have been measured to the nearest 0.1 g. In other words, we are somewhat uncertain about that last digit it could be a "2"; then again, it could be a "1" or a "3". A mass of 13.20 g indicates an absolute uncertainty of 0.01 g. The uncertainty or absolute uncertainty in a stated measurement is the interval of confidence around the measured value such that the measured value is certain not to lie outside this stated interval. For example your classmate has measured the width of a standard piece of notebook paper and states the result as 24.53 0.08 cm. By stating the uncertainty to be 0.08 cm your classmate is claiming with confidence that every reasonable measurement of this piece of paper by other experimenters will produce a value not less than 24.45 cm and not greater than 24.61 cm. The uncertainty of a measured value can also be presented as a percent or as a simple ratio (the relative uncertainty). Percentage uncertainty = uncertainty measure value 100 or Percentage uncertainty = d d 100 Relative uncertainty is the simple ratio of uncertainty to the measured value. As a ratio of similar quantities, the relative uncertainty has no units. In fact there is no special symbol or notation for the relative uncertainty; it must make it quite clear when reporting relative uncertainty. Exercise: 1.6 Relative uncertainty = uncertainty measured value a) Solve assignment # 2 from the website http://www.freewebs.com/faisalj/as/physics%20assignment%201%202%20%203.pdf b) Find the maximum possible error in the measurement of the force on an object of mass m, traveling at velocity v in a circle of radius r if m=3.5kg 0.1kg, v = 20 m s -1 1m s -1 and r=12.5 m 0.5 m. 2 V c) The power loss P in a resistor is calculated using the formula P. The uncertainty in potential R difference V is 3% and uncertainty in the resistance is R is 2%.. What is the uncertainty in P? d) The mean diameter of the wire is found to be 0.50 0.02 mm. Calculate the percentage uncertainty in i) Diameter ii) The area of cross-section of the wire e) Solve the following questions from past papers 1. May/June 2010, Paper 22, question 1 2. Oct/Nov 2009, Paper 22, question 1 3. May/June 2009, Paper 21, question 1(b) 4. Oct/Nov 2007, Paper 2, question 1 7 How does an uncertainty in a measurement affect the FINAL result? http://www.saburchill.com/physics/chapters/0070.html#1
AS Physics 9702 unit 1: Standards and their Measurements 9 Significant figures: 1. The significant figures express the accuracy with which a physical quantity may be expressed. 2. Greater the number of significant figures obtained when making a measurement, more accurate is the measurement. 3. The significant figures in a measured quantity indicate the number of digits in which we have confidence. The number of significant figures in a result is simply the number of figures that are known with some degree of reliability. The number 13.2 is said to have 3 significant figures. The number 13.20 is said to have 4 significant figures. Rules for deciding the number of significant figures in a measured quantity: 1. All non zero digits are significant: 12.34 g has 4 significant figures, 1.2 g has 2 significant figures. 2. Zeroes between nonzero digits are significant: 1002 kg has 4 significant figures, 3.07 ml has 3 significant figures. 3. Leading zeros to the left of the first non zero digits are not significant; such zeroes merely indicate the position of the decimal point: 05 o C has only 1 significant figure, 0.012 g has 2 significant figures. 4. Trailing zeroes that are also to the right of a decimal point in a number are significant: 0.0230 ml has 3 significant figures, 0.20 g has 2 significant figures. 5. When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant: 190 miles may be 2 or 3 significant figures, 50,600 calories may be 3, 4, or 5 significant figures. The potential ambiguity in the last rule can be avoided by the use of standard exponential, or "scientific," notation. For example, depending on whether the number of significant figures is 3, 4, or 5, we would write 50,600 calories as: 5.06 10 4 calories (3 significant figures) or 5.060 10 4 calories (4 significant figures), or 5.0600 10 4 calories (5 significant figures). Rules for mathematical operations In carrying out calculations, the general rule is that the accuracy of a calculated result is limited by the least accurate measurement involved in the calculation. 1. In addition and subtraction, the result is rounded off so that it has the same number of significant numbers as the least significant measurement. For example, 100 (assume 3 significant figures) + 23.643 (5 significant figures) = 123.643, which should be rounded to 124 (3 significant figures). 2. In multiplication and division, the result should be rounded off so as to have the same number of significant figures as in the component with the least number of significant figures. For example, 3.0 (2 significant figures) 12.60 (4 significant figures) = 37.8000 which should be rounded off to 38 (2 significant figures).
10 Prepared by Faisal Jaffer, revised on Oct 2011 Rules for rounding off numbers 1. If the digit to be dropped is greater than 5, the last retained digit is increased by one. For example, 12.6 is rounded to 13 for two significant figures. 2. If the last digit is equal to 5 than the previous digit will increase by one. For example 47.5 must be rounded off to 48 for two significant figures. 3. If the digit to be dropped is less than 5, the last remaining digit is left as it is. For example, 12.4 is rounded to 12 for two significant figures. Exercise: 1.7 Solve the following questions and give answers in correct number of significant figures: 1. 37.76 + 3.907 + 226.4 = 2. 319.15-32.614 = 3. 104.630 + 27.08362 + 0.61 = 4. 125-0.23 + 4.109 = 5. 2.02 2.5 = 6. 600.0 / 5.2302 = 7. 0.0032 273 = 8. (5.5) 3 = 9. 0.556 (40-32.5) = 10. 45 3.00 = 11. 3.00 x 10 5-1.5 x 10 2 = (Give the exact numerical result, and then express it the correct number of significant figures). 12. What is the average of 0.1707, 0.1713, 0.1720, 0.1704, and 0.1715? Solve exercises on the website: http://scidiv.bellevuecollege.edu/physics/measure&sigfigs/measure&sigfigsintro.html
AS Physics 9702 unit 1: Standards and their Measurements 11 SCALAR AND VECTOR QUANTITIES: Scalar quantity is one that can be described by a single number, including any units, giving its size or magnitude. For example volume of water, temperature etc are scalar quantities. The quantity that deals with both magnitude and direction is called vector quantity. For example velocity, acceleration, force, displacement etc. Vector quantities can be represented by choosing the appropriate scales and drawing the straight line equal in length of the magnitude of vector and in the direction of the vector. In printing and writing vectors can be represented by for example F or F. F or F to represent the vector force or magnitude F towards right. Addition and subtraction of vectors: a) When two vectors pointing in the same direction, that is they are collinear or coplanar, then the sum of two vectors called resultant vector R is simply addition of their magnitudes and direction is same as the direction of two vectors. + = A = 4 cm B = 2 cm R = A + B = 6 cm b) when two coplanar vectors are in opposite direction for example A = 4 cm + B = -2 cm c) When two vectors A and B are perpendicular to each other as expressed in figure: = R = A + B =2 cm then the magnitude of resultant vector R is given by R A 2 B 2 and direction of the vector R is given by A R B θ = tan B A d) When two vectors A and B are not in same direction and resultant vector R does not make a right angle triangle. The resultant vector can be found i) by graphical parallelograms method (you have learn in IGCSE course), ii) by addition of components of each vector, iii) by cosine rule The magnitude of vector R can be expressed by R 2 = A 2 + B 2 (2 AB cos ) iv) by sine rule = = and direction by protractor Where θ, a, and b are the angles opposite to R, A and B. b R A a B
12 Prepared by Faisal Jaffer, revised on Oct 2011 Horizontal and vertical components of a vector: In two dimensions, the components of a vector A are two perpendicular vectors A x and A y that are parallel to the x and y axes respectively. The components A x and A y, when added, convey exactly the same meaning as does the original vector A. In general the components of any vector can be used in place of vector itself in any calculation where it is convenient to do so. The θ is the angle made by the vector with the horizontal. The components of vector A can be expressed as: y A x A A y x A x = A cos θ A y = A sin θ Head to tail method of vector addition: 8 In the following problem as mentioned in the figure you will learn to show vector addition using the head to tail method. Consider to two vectors u and v are acting at a certain point. To find vector u + v slide v along u so that the tail of v is at the head of u, draw the resultant vector, which starts at the tail of u and ends at the head of v. The vector u + v is the resultant vector of vector v and u. The angle between the two vectors is 180 o -75 o =105 o. Exercise: 1.8 a) Solve more problems on the website: http://www.mathworksheetsgo.com/downloads/trigonometry/advanced/vectors/resultant-vector-worksheet.pdf b) Solve question in the following past papers Oct/Nov 2009, Paper 12, question 11 Oct/Nov 2007, Paper 1, question 3 May/June 2002, Paper 1, question 2 Oct/Nov 2001, Paper 2, question 3(b) 8 http://www.mathworksheetsgo.com/downloads/trigonometry/advanced/vectors/resultant-vector-worksheet.pdf