Physics and Chemistry UNIT 1: SCIENCE. THE SCIENTIFIC METHOD. QUANTITIES AND UNITS

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1 UNIT 1: SCIENCE. THE SCIENTIFIC METHOD. QUANTITIES AND UNITS 1. SCIENCE AND THE SCIENTIFIC METHOD A G In this unit we will learn how scientists work, make new discoveries and explain unknown phenomena of nature. All this knowledge is what we call Science. Science is the systematic knowledge of the world gained through observation, experimentation and reasoning and the formulation of laws and theories. But Science has different branches depending on what is being studied. We can divide Science into two major groups: a) Natural Sciencies which study natural phenomena including biological life. b) Social Sciences, which study human behaviour and societies. Natural Sciencies are empirical sciences, which means knowledge must be based on observable phenomena and capable of being tested for its validity with expermients made by E other researchers working under the same conditions. Natural Sciencies include Physics, Chemistry, Biology and Geology. In this subject we will learn what Physics and Chemistry study. 1

2 Physics is an experimental and natural science that studies the composition of matter, its general properties and its physical changes. A physical change of matter is a change that undergoes a substance, a body or an object without transformation into other substance/s, its chemical composition doesn t change. Examples of physical changes are: changes in position, in volume, in form, in temperature or state; mixtures, electrical phenomena Chemistry is an experimental and natural science that studies the composition, the structure and the properties of matter and its chemical changes. A chemical change: of matter is a change that undergoes a substance, a body or an object with transformation into other substance/s, its chemical composition changes (iron oxidation, fruit ripening, cooking, digestion, respiration ). So Physics and Chemistry are two natural sciences which study matter and its properties. The method that these two sciences use to understand matter and its changes is the scientific method. So scientists use the scientific method to know everything about Nature. But the scientific method is not only used by scientists. Any worker, a doctor, a policeman or a technician, use this method to solve their unknown problems. The scientific method is a sequence of steps that the scientist follows to solve a problem. a) The first step is the observation. Scientists observe nature through the senses. Scientific observation consists of receiving knowledge of the outside world through our senses. But our senses have limits. We cannot see a distant object, or a very small object. We need to use technological iinstruments to obtain information from nature. We use telescopes to see distant objects. We can see small objects with the microscope. While human beings can't detect radiations using their natural senses, many technologies exist that help detect and identify radiations, wherever it may be, such as the use of spectrrophotometers. b) The second step is to identify the problem and to make the right question. When a scientist watches nature with his instruments he sometimes finds unknown or unexplained phenomena. Scientists must make the right question to identify clearly the unsolved problem. If I want to study the growth of a plant, it is a complex problem because there are multiple factors that can affect it, the amount of water, oxygen carbon dioxide, light A scientist can t study all the variables that affect plant growth at the same time. Even plant growth is another variable. We must reduce the number of studied variables to two variables. So the right question is not, what affects plant growth? The right question would be: Does the amount of water affect the growth of a plant? We only choose one factor that can affect plant growth. In the question there can only be two variables, the amount of water and the growth of a plant. 2

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4 c) The third step is to obtain the background information. To answer the question a scientist must obtain information about his problem, which can be found in scientific journals, specialized books, in scientific congresses or by an internet research. d) The fourth step is to define the variables of the problem. With this background information we have to identy the variables of the problem.for example, if we are studying the growth of a plant a problem that we can study is the following one: Does the amount of water affect the growth of a plant? There are two variables written in the question: the amount of water (the volume of water) and the plant growth, which can be measured with the length of the trunk or its diameter, or the number of new leaves or new branches. Of course, there are other variables that affect plant growth: the amount of sunlight, oxygen, carbon dioxide... But we want to find the relationship between the amount of water and plant growth. A variable is a quantitative property of an object that can take on different values (length, speed, density, volume ). Now, we must define the variables of our problem: The independent variable is the variable that a scientist deliberately changes during the experiment. In our example is the amount of water. The dependent variable is the variable that responds to the changes in the independent variable. In our case it can be plant growth. The rest of the variables must be controlled, although these variables are changing in nature. But we must controll them to concentrate our study on the amount of water and the growth of the plant. e) The next step is to formulate a hypothesis. With the background information we should answer the question. For example, the amount of water affects plant growth, without water plants die. But this is a possible answer and it must be tested. We call hypothesis an educated guess, a prediction that must be tested with an experiment. It is a tentative statement that proposes a possible explanation to some phenomenon or event. f) Experimentation. Our hypothesis must be tested with an experiment. The experiment must be designed, indicating the procedure and a list of materials. Then, the experiment or experiments are made and numerical data are recorded. These numerical data should be organized into tables and represented in graphs. Graphs represent data visually and it is much easier their analysis to test our hypothesis. g) In the following step of data analysis we will test if our hypothesis is correct or it is not valid. If the hypothesis is not valid, our work hasn t finished. A new hypothesis should be proposed or formulated according to the new knowledge. If the hypothesis is valid, then we should write a report with our conclusions. 4

5 h) Laws and theories. In science, once a hypothesis has been widely accepted, it is called a law, which is a concise verbal or mathematical statement which describes a relationship between the variables of a determined nature phenomenon. (e.g. Newton s second law, F = m a). Theories are general explanations based on a large amount of data. For example, the theory of evolution applies to all living things and is based on wide range of observations. A theory is an explanation of a set of observations and scientific laws of an entire group of related phenomena of nature. It explains how it works, what causes it, and how it behaves. For example, the kinetic theory of matter or Einstein s General Theory of Relativity. Concepts SCIENCE is the systematic knowledge of the world gained through OBSERVATION, EXPERIMENTATION and REASONING and the formulation of LAWS and THEORIES. PHYSICAL CHANGE: Change that undergoes a substance without transformation into other substance/s, its chemical composition doesn t change. Examples of physical changes are: changes in position, in volume, in form, in temperature or state; mixtures, electrical phenomena PHYSICS is an EXPERIMENTAL AND NATURAL SCIENCE that studies the composition of matter, its general properties and its PHYSICAL CHANGES. CHEMICAL CHANGE: Change that undergoes a substance with transformation into other substance/s, its chemical composition changes (iron oxidation, fruit ripening, cooking, digestion, respiration ). CHEMISTRY: EXPERIMENTAL AND NATURAL SCIENCE that studies the composition, the structure and the properties of matter and its chemical changes. SCIENTIFIC METHOD: procedure used in scientific research to solve problems of nature. OBSERVATION: an activity consisting of receiving information or knowledge of natural phenomena of the outside world through the senses, recording of data using scientific instruments. BACKGROUND INFORMATION: knowledge that you have acquired through the research in scientific journals, Internet or books on the topic of the problem. Scientific inventions must be published to be known by the rest of the scientific community. VARIABLE: a quantitative property of an object that can take on different values (length, speed, density, volume ). INDEPENDENT VARIABLE: variable that a scientist deliberately changes during the experiment. DEPENDENT VARIABLE: variable that responds to the changes in the independent variable. CONTROLLED VARIABLES: variables that are constant (don t change) during the experiment. HYPOTHESIS: an educated guess, a prediction that must be tested with an experiment. EXPERIMENTATION: involves carrying out experiments under controlled conditions to make 5

6 observations and collect data to test the hypothesis. You must design the experiment before, indicating the materials and the procedure in steps and include as much detail as possible about measurements and techniques in each step. EXPERIMENTAL DATA ORGANIZATION: The experimental data must be organized the experimental data in tables and graphs for analysis and interpretation. DATA ANALYSIS: analysis of the data to look for patterns to know if there is a relationship between the independent variable and the dependent variable of the experiment. SCIENTIFIC LAW: a concise verbal or mathematical statement which describes a relationship between the variables of a determined nature phenomenon. (Newton s second law, F = m a). THEORY: explanation of a set of observations and scientific laws of an entire group of related phenomena of nature (it explains how it works, what causes it, and how it behaves) (The kinetic theory of matter, Einstein s General Theory of Relativity). 1. Write the steps of the scientific method. Activities 2. Write the names of all the variables that affect plant growth. 3. Does the amount of sunlight affect plant growth? Formulate a hypothesis and design an experiment to test it. Classify the different variables of your experiment. 4. Imagine that you have a lantern and it doesn t work. Formulate a hypothesis and design an experiment to test it. 6

7 2. QUANTITIES AND UNITS The goal of science is to provide an understanding of the physical world by developing theories based on experiments. The basic laws of physics and chemistry involve the measurement of properties of matter and of the all observed phenomena. A quantity is a property of a phenomenon, body or substance that can be quantified or measured. Examples of quantities are mass, volume, temperature, speed). A quality is property of a phenomenon, body or substance that cannot be quantified. Examples of qualities are colour, shape., roughness, toughness To measure a quantity we compare it with a chosen value of that quantity to express its value with a number followed by the chosen unit. A unit is a definite magnitude of a particular quantity, defined and adopted by convention, with which other particular quantities of the same kind are compared to express their value. For example, when we measure a quantity, we always compare it with some reference standard. When we say that a rope is 30 metres long, we mean that it is 30 times as long as a metre stick, which we define to be 1 metre long. The result of a measurement includes a number and the chosen unit for the measurement. The value of a quantity is generally expressed as the product of a number and a unit. quantities. All physical quantities can be classified into two types: base quantities and derived Base quantities are quantities that are common to every object or phenomenon and are assumed to be mutually independent. There are 7 base quantities used in The International System of Units (SI). Each base quantity has its own SI base units. The symbols used for the base quantities and their base units are given as follows: A G BASE QUANTITIES AND THEIR SI BASE UNITS Quantity Symbol Unit Symbol Length l, x, r, metre m Mass m kilogram kg Time t second s Intensity of electric I, i ampere A Temperature T kelvin K Amount of substance n mole mol Luminous intensity I v candela cd Base units can be too large or too small for some measurements, so the base units may be modified by attaching prefixes, which are decimal multiples and submultiples and we obtain E secondary units. These prefixes are given the name of SI prefixes. The prefix names and symbols are listed below: 7

8 DECIMAL MULTIPLES AND SUBMULTIPLES OF SI UNITS. SI PREFIXES Factor Prefix Name Symbol Examples: 10 6 ( ) mega M 10 3 (1.000) kilo k 10 2 (100) hecto h 10 1 (10) deca da 10-1 (0,1) deci d 10-2 (0,01) centi c 10-3 (0,001) mili m 10-6 (0,000001) micro μ Decimetre: 1 dm = 0,1 m. Centimetre: 1 cm = 0,01 m. Hectometre: 1 hm = 100 m. The most common units for the base quantities length, mass and time are listed below: LENGTH UNITS MASS UNITS kg TIME UNITS :10 :10 :10 :10 :10 :10 hg dag g dg cg mg Name Symbol Value (SI Units) Name Symbol Value (SI Units) second s hour h 1 h = 60 min = s minute min 1 min = 60 s day d 1 d = 24 h = s All other quantities different than base quantities are known as derived quantities, quantities that derive from the base quantities by equations. Examples of derived quantities are area, volume, capacity, density, speed... The most common units for the derived quantities area, volume and capacity are listed below: AREA UNITS :100 :100 :100 :100 :100 :100 km 2 hm 2 dam 2 m 2 dm 2 cm 2 mm

9 VOLUME UNITS. The amount of 3-dimensional space an object occupies. :1000 :1000 :1000 :1000 :1000 :1000 km 3 hm 3 dam 3 m 3 dm 3 cm 3 mm CAPACITY UNITS. The amount of substance, liquid or gas, which a container can hold. kl :10 hl :10 dal :10 L :10 dl :10 cl :10 ml CAPACITY AND VOLUME CONVERSION 9

10 3. UNIT CONVERSIONS Units are multiplied and divided just like ordinary algebraic symbols. We can express the same physical quantity in two different units and form an equality. For example, when we say that 1 m = 100 cm we don t mean that the number 1 is equal to the number 60, but rather that 1 m represents the same physical length interval as 100 cm. To find the number on centimetres of 25 m, we write: 100 cm 25 m = / = 1 m / ( 25 m) 2500 cm A Conversion factor is a ratio of units, which expresses a quantity expressed in some unit or units divided by its equal expressed in some different unit or units, such as 1 m/100 cm (or 1 m = 100 cm). The units of the quantity and the conversion factor must be combined properly to give the desired final units. The unit conversion method: 1. Write the quantity you want to change its unit ml 2. Write the equality between the old unit and the new unit. Remember the table of prefixes. 3. Write the conversion factor with the two units properly combined to give the desired final unit. 4. Multiply the quantity by the conversion factor, cancel the old units and express the final result. Example. Convert 0,5 hl to mm 3. 1L = 1000 ml ( 2000 ml) 1000 ml 1 L 1 L 1000 ml ( 2000 ml) = 2 L The unit conversion method can also be used to convert units of derived quantities which are composed of two different units. In this case, you must use a conversion factor for each unit. Example: The maximum speed limit on motorways in Spain is 120 km/h. Express this speed in m/s. km 1000 m 1 h 120 = 33,33 h 1 km 3600 s m s 10

11 4. SCIENTIFIC NOTATION It is a way of writing very large and very small numbers in standard decimal notation, based on powers of the base number 10. The number is written as a product of a number between 1 and 10 and a power of LARGE NUMBERS For example, the following number is written: = 8, The first number, 8,76, is called the coefficient. It must be greater than or equal to 1 and less than 10. form. The second number, 10 7, is called the base. It must always be 10 written in exponent To write a number in scientific notation: a) Find the coefficient: Put the decimal after the first digit and drop the zeroes. b) Find the exponent: Count the number of the places from the decimal to the end of the number. Example: 4.2. SMALL NUMBERS To write a number in scientific notation that is less than 1: a) Find the coefficient: Move the decimal point right until you reach a coefficient greater than 1 but less than 10 and place the decimal point there. b) Find the exponent: count the number of places the decimal point was moved and the exponent is 10 raised to the negative of that number. Examples: Quantity 0, = 8, Earth-Sun distance: m Electric charge of the proton: 0, C Mass of the electron: 0, kg Scientific notation 1, m 1, C 9, kg Activities 5. Define quantity. Write four examples of quantities and four examples of properties which are not quantities. 6. Define unit. E A G 11

12 7. Write the units of mass in a decreasing sequence. 8. Write the units of length in a decreasing sequence. 9. Write the units of area in a decreasing sequence. 10. Write the units of volume in a decreasing sequence. 11. Write the units of capacity in a decreasing sequence. 12. Complete: a) = 6,76 10 b) 6760 = 6,76 10 c) = 5 10 d) = 4,3 10 e) = 2,57 10 f) = 4, Complete: a) 0, = 6,76 10 b) 0, = 8,9 10 c) 0,00005 = 5,0 10 d) 0, = e) 0, = f) 0, = 14. Convert the following units and indicate their corresponding quantity. Write the resulting numbers in decimal and in scientific notations. a. 72 hm to m b. 4 m to mm c. 3,5 m to cm d. 2,3 ml to m 3 e. 5,2 m 2 to mm 2 f. 3 m 3 to dm 3 g. 0,2 m 3 to cm 3 h. 3 dm 3 to dl i cm 3 to L j cl to dm 3 k. 220 dl to dm 3 l. 0,0003 cm 3 to dal m. 0,0045 mm 2 to hm 2 n. 0,0002 kg to mg o. 1 day to s p. 20,456 hl to dl q. 0,0012 hl to dm 3 r. 14,56 mm 3 to dl s. 40,56 g to hg t s to h u. 1,0003 mm 3 to L v. 2,2 cm 2 to m 2 w. 7,2 mm to dam x. 0,2 cm 3 to ml y. 2,34 L to dm 3 z. 220 kl to m Express in metres the result of the following operation: 0,30 km + 3,6 hm m cm. 16. How many ml are there in 1/3 L? 17. How many g are there in 1/4 kg? 18. Convert the following units and indicate their corresponding quantity. Write the resulting numbers in decimal and in scientific notations: a. 55 m to cm b. 3 m 2 to cm 2 c. 2 ml to cm 3 d. 1 Tm to kg e. 40 cm to dm f. 2 mm 2 to dm 2 g. 200 ml to m 3 h. 20 m 3 to dm 3 i. 2 mm to m j. 4 cm 2 to m 2 k. 200 kg to g l. 200 L to mm 3 m. 7 m to mm n. 20 m 3 to dm 3 o g to kg p. 4,67 m 2 to mm 2 q. 5 km to cm r. 20 cm 3 to dm 3 s. 200 g to mg t. 20 dam 3 to mm 3 u mg to kg v. 3 hours to s w. 1 h to s x. 200 hl to km 3 y. 180 s to min. z. 2 days to s 12

13 5. MEASUREMENT AND ERROR All measurements are approximate values not true values. An error is a comparison between the measured value and the true value of a measurement. Errors can be classified into two types: a) Systematic or determined errors, errors that produce a result that differs from the true value by a fixed amount. They can be identified and corrected. These errors result from biases introduced by: 1. Instrumental bias. For example, errors in the calibration of the measuring instruments. 2. Human bias. For example, the person who is measuring might read an instrument incorrectly or might let knowledge of the expected value of a result influence the measurement. 3. Method bias. For example, the person might make an incorrect scale reading because of parallax error. 4. Operative bias. For example, the person knows the measurement method but might make it wrong. b) Random errors are caused by unknown and unpredictable changes in the measurement. These changes may occur in the measuring instruments or in the environmental conditions. They cannot be corrected but the can be minimized by: 1. Making several measurements. 2. Calculating the arithmetic mean. Example. Several measurements of the mass of an object: 2,350 g, 2,352 g, 2,348 g, 2,350 g. Arithmetic mean = 2, , , ,350 4 = 2,350 g An error in a measurement of a physical quantity may be represented by the actual amount of error, or by a ratio comparing the error to the size of the measurement. We use the absolute error and the relative error. value. The absolute error is the difference between the measured value and the true or exact V m is the measured value. V r is the true value. E a = V m V r If E a > 0 the absolute error is positive, and the error in excess. If E a < 0 absolute error is negative, it is a scant measurement. 13

14 The relative error is computed by dividing the absolute error by the true value. It is the error per unit of measured quantity. E = Ea To convert to percentage, multiply by 100: Er (%) = 100 V r E V a r The relative error gives an indication of how good a measurement is relative to the size of the thing being measured. It indicates the accuracy of a measurement. Example: Two students measure two different lengths. Student A measures a real length of 5 m. The result of his measurement is 6 m. Student B measures a real length of 500 m. The result of his measurement is 501 m. Although both students have the same absolute error, +1 m, student B is more precise than student A because he has measured a greater length. His error per measured metre is lower, so is his relative error. Student True Value Measured Value E a E r (%) A 5 m 6 m 1 m (excess) (1/5) 100 = 20 % B 500 m 501 m 1 m (excess) (1/500) 100 = 0,2 % The sensitivity of a measurement instrument is the smallest amount of a quantity that the instrument can detect. For example, ruler 1 has a sensitivity of 1 cm, and ruler 2 has a sensitivity of 1 mm (0,1 cm). The precision or reproducibility of a measurement instrument is the degree to which repeated measurements of the same quantity under unchanged conditions show the same results. This instrument is precise because all measurements are grouped tightly together. r 0,2 0,3 The accuracy of a measurement instrument is the degree of closeness of the measurements of a determined quantity to that quantity s true value. A direct measurement is the determination of the value of a quantity with a measurement instrument. For example, the measurement of the mass of an object with a balance. Uncertainty or margin or error of a measurement is the range of values likely to enclose the true value. It depends on the sensitivity of the measurement instrument. The uncertainty of ruler 1 is 1 cm and the uncertainty of ruler 2 is 0,1 cm. The result of any physical measurement has two essential components: 1. a numerical value giving the best estimate possible of the quantity measured, and 2. the degree of uncertainty associated with this estimated value. 14

15 measurement = best estimate ± uncertainty Ruler 1 L = 17 ± 1 cm (ruler in cm) Ruler 2 L = 17,6 ± 0,1 cm (ruler in mm) Significant figures are the digits that carry meaning contributing to its precision. For example the number 5,423 cm has four significant figures. There are some rules for counting the significant figures in a number: 1. Nonzero integers always count as significant figures (452 3; 2,227 4). 2. Zeros appearing between two significant digits are significant (101,25 5). 3. Leading zeros (zeros that precede all the nonzero digits) are not significant (0,706 3; 0, ). 4. Trailing zeros (zeroes at the right end of the number) in a number containing a decimal point are significant (13,4300 6; 5,030 4; 0, ; 35,0 3; 35,00 4). 5. In scientific notation, all the figures that appear before 10 are significant (1, ). 6. If you perform an arithmetic operation (addition, subtraction, multiplication or division), the number of significant figures of the result mustn t exceed the lowest number of significant figures of the numbers (3, ,531 = 6,0742 6,074; 2,33 2,4 = 5,592 5,6). Examples: Measurement No. of significant figures Measurement No. of significant figures 0, dg kg 1 0,0036 g 2 4, m 3 0, m 4 3,05 hm 3 7,64 cm 3 18,5 s m 3 7,35 s 3 64,01 kg 4 0,220 kg 3 0,00003 m 1 42,05 km 4 2, kg 3 0,075 m mm 4 80,0 s 3 An indirect measurement is the determination of the value of a quantity by calculation. For example, the determination of the density of a solid substance requires the measurement of its mass and its volume followed by a calculation. Rounding is the process of replacing a number by another number of approximately the same value but having fewer digits. If the digit following the last digit to be retained is: 15

16 1. 5 or greater than 5 the last digit should be increased by less than 5 the last digit should be stay the same. 3. It is advisable to round a number only until two decimal places (E r < 1 %). Example: π = 3, , , ,1416 3,142 3,14 Activities 19. We measured with a chronometer the time of the free fall of an object from the roof of a building. We obtained the following measurements: 1,43 s; 1,45 s; 1, 47 s; 1,42 s; 1,46 s; 1,43 s; 1,47 s. a) How long does it take for the object to reach the ground? b) What is the sensitivity of the chronometer? c) What is the absolute error of the last measurement? d) What is the relative error of the last measurement? e) Indicate de number of significant figures in the second measurement. 20. We measured with a balance the mass of an object. We obtained the following measurements: 2,430 g; 2,427 g; 2, 433 g; 2,428 g; 2,432 g. a) What is the mass of the object? b) What is the sensitivity of the balance? c) What is the absolute error of the second measurement? d) What is the relative error of the second measurement? e) Indicate de number of significant figures in the second measurement. 21. The exact mass of an object is 2,350 g. A student measured that mass on a balance which was 2,348 g. a) What is the absolute error of the measurement? Explain the meaning of this result. b) What is the relative error of the second measurement? Explain the meaning of this result. c) Indicate de number of significant figures in the exact mass. d) What is the sensitivity of the balance? 22. The exact mass of an object is 0,70 g. A student measured that mass on a balance and obtained the following results: 0,67 g, 0,72 g and 0,71 g. a) What is the absolute error of the first measurement? Explain the meaning of this result. b) What is the relative error of the first measurement? Explain the meaning of this result. 16

17 c) Indicate de number of significant figures in the exact mass. d) What is the sensitivity of the balance? 23. The exact mass of an object is 2,050 g. A student measured that mass on a balance and obtained the following results: 2,048 g, 2,049 g and 2,051 g. a) What is the absolute error of the third measurement? Explain the meaning of this result. b) What is the relative error of the third measurement? Explain the meaning of this result. c) Indicate de number of significant figures in the exact mass. d) What is the sensitivity of the balance? 6. TABLES AND GRAPHS The experimental data obtained during a experiment must be ordered in a table in order to find a relationship between two quantities, the independent variable and the dependent variable, indicating at the top of the table each quantity and its unit. The next step consists in creating a graph following the next rules: 1. Graphs are done on graph paper. The graph is two-dimensional because it represents the possible relationship between two variables. 2. The independent variable is represented with a horizontal straight line, from a central point in the graph, called the origin. This horizontal line is known as the X-axis and should be labelled with the corresponding quantity what it measures and its units at the end of the line. 3. The dependent variable is represented with a vertical straight line from the origin. This vertical line is known as the Y-axis and should be labelled with the corresponding quantity what it measures and its units at the end of the line. 4. The scale of the axes should be chosen to include all data points and to allow as much room as possible on both axes. 5. Each axis should be evenly divided with plenty of space between divisions. Divisions should be labelled in multiple units of 1, 2, 5, or The experimental data must be plotted in the graph with points corresponding to both coordinates. You mustn t draw lines parallel to the axes to determine the plotted points. 7. It is important to add a title to the top of the graph. The form of the line that connects the plotted data points shows the mathematical relationship between the two variables. There are different relationships between two variables: 1. If the plotted data points roughly form a straight line, use a ruler to draw a line that best represents the data points, which comes closest to all the data. This straight line is represented as a mathematical equation, the straight line equation, which quantify the relationship between the two variables of the experiment. 17

18 The straight line means that the two variables are directly proportional. The equation of a straight line is usually written this way: y = y 0 + m x x and y are the corresponding variables of the experiment. m is the slope or the gradient of the straight line. Its value is constant, and it can be determined choosing two points of the straight line: P 1 (x 1, y 1 ) y P 2 (x 2, y 2 ): (0, 0). y m = x 2 2 y x 1 1 y 0 : It is simply the value of y where the line crosses the Y axis, when x = 0. If the line passes through the origin one of the points, i.e. the point P 1 (x 1, y 1 ), can be the point Example: Study of the relationship between the extension of a spring, Δx, and the weight of an object suspended from it, F. F and Δx: are directly proportional. Equation of the straight line: The slope m is equal to 2. Hooke s Law: F = k Δx Δx = 2 F 18

19 Example: Study of the relationship between the potential difference, V, applied to an electric circuit and its current intensity, I. V (V) I (A) 0 0 2,2 0,1 5,3 0,2 7,2 0,3 10,0 0,4 13,0 0,5 14,4 0,6 18,0 0,7 19,6 0,8 V and I are directly proportional. Relationship between the voltage, V, and the current intensity, I V (V) ,0 0,2 0,4 0,6 0,8 The equation of the straight line is: V = 0+ R I V = R I R is the slope of the straight line, a constant value, which corresponds to the electrical resistance of the circuit. If the plotted data points do not form a straight line but appear to form a curve, we must find the mathematical equation which gives the relationship between the two quantities of the experiment. 2. If the plotted data points form a curve of an equilateral hyperbola the two quantities of the experiment are inversely proportional. The mathematical equation which gives the relationship between the quantities is: k y = y x = k x The product of the two variables is always constant. k is a constant value. Y I (A) X 19

20 Example: Study of the relationship between the pressure of a gas, P, and its volume, V. There is a relationship between P and V: P and V are inversely proportional. is: The equation of an equilateral hyperbola k V = V P = k P The product of the two variables is always constant. k is a constant value. The relationship between P and V can be found much easier if we plot V against 1/P. We will get a straight line passing through the origin and k (the constant) is the slope or the gradient of the graph. V and 1/P are directly proportional. The equation of the straight line is: k V = P 3. If the plotted data points form a curve of a parabola the mathematical relationship between two quantities of the experiment is a second grade equation: y = k x k is a constant value. Example: Study of the free fall of an object. Relationship between height, s, and time, t. The mass, m, of an object doesn t affect its motion during free fall; if two objects of different masses are dropped from the same height at the 2 20 V (L) Y P(atm) Object A m = 10 g s (m) t (s) Object B m = 20 g 20 2 X

21 same time, they hit the ground at the same time However, height affects the time of free fall. The graph s-t for the object A is a parabola which means that the mathematical relationship between height and time is a second grade equation. s = k t 2 k is a constant value. The relationship between s and t can be found much easier if we plot s against t 2. We will get a straight line passing through the origin and k (the constant) is the slope or the gradient of the graph (k = 1/2 g). s (m) t (s) t 2 (s 2 ) s and t 2 are directly proportional. The equation of the straight line is: s = 1/2 g t 2 s (m) t (s) Sustituir Activities 24. During an investigation to find the relationship between the diameter and the circumference of a circle, a group of students chose seven circle-shaped objects of different sizes and measured their diameters (D) and their circumferences (C): D (cm) 2,0 3,0 4,0 5,0 6,0 7,0 C (cm) 6,3 9,4 12,6 15,7 18,8 22,00 a) Identify the independent variable, the dependent variable and the possible controlled variables of this investigation. b) Verify if there is some type of mathematical relationship between variables, the independent variable and the dependent variable (for example, calculating the ratio between each pair of values and observing the obtained results). c) Formulate a hypothesis. 21

22 d) Do the graph on graph paper of the variable diameter and the variable circumference and determine the mathematical equation of the graph that relates both variables. e) Use the obtained mathematical relationship and calculate the circumference of a circleshaped object of 2,0 m of diameter. 25. An investigation has been made on the relationship of the masses of the substances and their volumes. The exposition of the problem is the following one: Does the volume of a substance affect its mass? The volumes and the masses of different marble pieces have been measured to verify if there is dependence between the volume and the mass, and the following values have been obtained: V (cm 3 ) 3,1 4,8 7,6 10,0 14,1 m (g) a) Identify the independent variable, the dependent variable and the possible controlled variables of this investigation. b) Verify if there is some type of mathematical relationship between variables, the independent variable and the dependent variable (for example, calculating the ratio between each pair of values and observing the obtained results). c) Formulate a hypothesis. d) Do the graph on graph paper of the variable mass and the variable volume. Determine the mathematical equation of the graph that relates both variables. e) A sculptor needs a cubical marble block of 2 m of edge to make a sculpture. The people in charge of the quarry want to know the mass of the block for their transport. Use the obtained mathematical relationship and calculate the mass of the cubical marble block. 26. During an investigation to find the relationship between the pressure and volume of an enclosed gas at constant temperature, a glass syringe was filled of air, its opening was sealed so that no air can escape the syringe, and the plunger was slowly pushed decreasing the volume and at the same time measuring the air pressure inside the syringe. The following results were obtained. P (atm) 2,5 3,6 4,2 6,3 8,3 12,5 25 V (cm 3 ) a) Identify the independent variable, the dependent variable and the possible controlled variables of this investigation. b) Verify if there is some type of mathematical relationship between variables, the independent variable and the dependent variable (for example, calculating the ratio 22

23 between each pair of values and observing the obtained results). c) Formulate a hypothesis. d) Do the graph on graph paper of the variable pressure and the variable volume. Do the graph on graph paper of the variable pressure and the variable 1/V. Compare both graphs and try to determine the mathematical equation of the graph that relates both variables. e) Use the obtained mathematical relationship and calculate the pressure of the air inside the syringe in a volume of m Tests were carried out to determine the magnitude of the force of air resistance, F, based on the speed, v, of a body which moves immersed in the air: the body takes different speeds and for each speed the force of air resistance is measured. The following results have been obtained: v (m/s) 2,0 3,5 5,0 6,5 8,0 F (N) 8,4 25,9 52,8 87,4 135,1 a) Identify the independent variable, the dependent variable and the possible controlled variables of this investigation. b) Verify if there is some type of mathematical relationship between variables, the independent variable and the dependent variable (for example, calculating the ratio between each pair of values and observing the obtained results). c) Formulate a hypothesis. d) Do the graph on graph paper of the variable speed and the variable force and explain if there is some type of mathematical relationship between both variables. 28. Science project. Free fall 23

24 7. DICTIONARY science base quantity knowledge base unit reasoning multiple law submultiple theory prefix physical change derived quantity undergo length phenomenon mass phenomena volume physics time matter area chemical change capacity fruit ripening container iron conversion factor scientific method express background information scientific notation independent variable decimal dependent variable power controlled variable base hypothesis coefficient experimentation zero (zeroes) procedure decrease table systematic error graph differ scientific law amount receive bias acquire random error deliberately minimize step arithmetic mean statement absolute error quantity relative error quality accuracy measure sensitivity measurement precision unit reproducibility 24

25 magnitude uncertainty enclose scale leading zeroes plot trailing zeroes roughly significant figures slope rounding weight graph paper directly proportional axis(axes) equilateral hyperbola label inversely proportional 25