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Measurement Introduction In this experiment, you will become more familiar with the metric system of measurement, gain experience with the instruments and techniques used to make some common measurements, learn how to record measurements and evaluate their precision and accuracy, and find out how to use measured quantities in calculations. Equipment and Materials 1. Meter sticks (2) 5. 250-mL Beaker 9. Ice 2. Vernier caliper 6. Timer 10. Penny; nickel 3. Decigram balance 7. 25-mL Graduated cylinder 11. Steel ball 4. Centigram balance 8. Digital thermometer 12. Flask, Test tube Important Concepts The first step in understanding our physical environment is observation, and measurement is the most sophisticated way of making most observations. A measurement is a comparison of a physical quantity, such as length or mass, with a standard, such as a ruler or the weights on a balance. For example, you could tell which object is heavier than another by lifting each, but to tell how much heavier, a measurement using an appropriate instrument would be needed. For centuries, better and better measuring devices have been developed, and the progress of science has gone hand in hand with these developments. All of us, not just scientists, are involved with measurements continually. Examples are when we deal with the area of a room’s floor, the temperature inside the house, the level of humidity on a hot day, the distance to a destination, the amount of sugar to add when baking a cake, and what time to set the VCR to record a TV program. Systems of Measurement Two systems of measurement are in use in the world. The English System is used in everyday life in Burma, Liberia, and the United States. People in the rest of the world use the metric system. In scientific work the world over, the system used is an updated version of the metric system called the International System, abbreviated SI for its French name Systeme Internationale d’Unites. Table 1 shows, for both systems, the five quantities most commonly measured in the laboratory. The metric system was developed in France at the time of the French Revolution (the 1790s), although its roots come from a decimal system of measurement proposed by a Frenchman named Gabriel Moulton about 1670. The modification called the SI was introduced in 1960, and refinements are still made occasionally. The reason that the metric system is so popular is simple: It is easier to learn and easier to use than is the English System. It is easier to learn because of its consistency; for example, knowing that a kilogram equals 1000 grams enables you to know that a kilometer equals 1000 m. (On the other hand, in the English System, knowing that a pound equals 16 ounces tells you nothing about the number of inches in a yard.) The SI is easier to use because the multiples and submultiples are based on powers of 10. For example, changing 57 kilometers to meters involves only moving the decimal point three places to the right (or just 3 3 adding 10 ) to get 57000 m (or 57 x 10 m). However, in the English System, changing 57 miles to yards would involve multiplying 57 mi by 1760 yd/mi, which is certainly a more difficult and error-prone process. Common metric prefixes include the following: Prefix Symbol Value -3 Mili m 10 -2 Cenit c 10 3 Kilo k 10 6 Mega M 10 Physical Quantity Metric Unit and English Unit and Relationship of Metric Abbreviation Abbreviation Unit to English Unit Length meter m yard yd 1 m = 1.1 yd 1 Mass kilogram kg pound lb 1 kg = 2.2 lb Volume (fluid) liter L quart qt 1 L = 1.1 qt Time second s second sec 1 s = 1 sec 2 ˚ ° ° ° Temperature Celsius C Fahrenheit F C = F – 32 1.8 Table 1: five Physical Quantities Commonly Measured in the Laboratory Reading an Instrument and Recording the Value Instruments allow you to obtain more exact information about an object or phenomenon than would be possible with the unaided senses. For example, imagine trying to determine the width of a human hair without a very precise instrument such as a micrometer. Basically, instruments extend and refine our senses. Ordinarily, you read an instrument and record the measurement to the limit of the instrument’s precision. The precision of an instrument refers to its limit of reproducibility of results, which is generally determined by the smallest divisions that can be read directly on the scale of the instrument. (If the distance between the smallest markings is large enough, an estimation to a half or even a tenth of the smallest marking can be made for some instruments.) For example, a decigram balance allows you to measure mass to the nearest 0.1 g; while with a centigram balance, you may measure to the nearest 0.01 g. Thus, the centigram balance has the greater precision. nd In this experiment, you will use a meter stick (precision 1/32 inch on the English side, 0.05 cm on the metric side), a vernier caliper (precision 0.01 cm), a decigram balance (precision 0.1 g), a centigram balance (precision 0.01g), a 25-mL graduated cylinder (precision 0.1 mL), a timer (precision 0.1 s), and a digital thermometer (precision 0.1°). When making a measurement, it is often helpful to repeat the measurement several times to ensure that the reading is reproducible. A reading that is abnormally high or low compared to the instrument’s precision is suspect and is usually discarded. The readings that are close may be averaged, but the average should never be stated more precisely than the precision of the instrument. For example, averaging 47.5 g, 47.5 g, and 47.6 g gives 47.5 g, not 47.53 g. Averaging 47.5 g, 47.5 g, and 47.7 g gives 47.6 g. It is also a good idea to ask someone else to confirm your measurement, particularly if you lack experience with the instrument. 1 The pound is actually a unit of weight, not mass; however, the mass unit (the slug) is seldom used. As long as we are making mass measurements at Earth’s surface, we can say that 1 kg of mass is equal to 2.2 lb of weight. 2 The SI unit of temperature is the Kelvin, K. The term ‘degrees’ is not used. K = °C + 273. When recording any measurement, always attach the correct unit; for example, record 5.62 cm, not just 5.62. However, when results are recorded in table form and the unit is given at the top of the column, to reduce clutter the unit is not added to the numbers in the column. On tests, homework, lab data sheets, and so forth, it is always a good idea to show the setup for any calculation. The setup is the equation rearranged for the unknown and having all numbers and units plugged 3 in. For example, the setup for a density problem might be: D = 15.9 g/7.6 cm . Precision and Accuracy The terms precision and accuracy are often confused; the difference can be illustrated by a target shooting analogy. Assuming the same number of significant figures in the measurements, a series of measurements on an object have poor precision if the values are far apart (Figure 1a), and good precision if they are close to one another (Figure 1b and 1c).They have poor accuracy if they are not close to the correct or accepted value (Figure 1a and 1b) and good accuracy if they are close to the accepted value (Figure 1c). (Accepted values are those determined by highly skilled persons using the best equipment and materials under the best conditions.) As you can see, measurements showing a high degree of precision do not always reflect a high degree of accuracy. However, a high degree of accuracy implies that a high degree of precision has been obtained. For example, consider an object with an accepted mass of 5.02 g. A measurement of 5.0 g would never be as precise as one of 5.03 g, because the former implies an uncertainty in the tenths place, while the latter implies an uncertainty in the hundredths place. Three measurements of 5.03 g, 5.01 g, and 5.02 grams would be highly precise and also highly accurate. Three measurements of 5.20 g, 5.22 g, and 5.21 g would be highly precise but not very accurate, while three measurements of 5.21 g, 5.10 g, and 5.15 g would be neither very precise nor very accurate. (a) low precision, low accuracy (b) high precision, low accuracy (c) high precision, high accuracy Figure 1: A target shooting analogy illustrates the difference in precision and accuracy. Errors in Measurements Making a measurement always involves some uncertainty, called experimental error, due to human factors and to the limited precision of the instrument used. Errors are divided into two categories: systematic and random. Systematic errors are due to some inherent error in the method of measurement or in the instrument’s manufacture; such errors do not affect the precision, but do affect the accuracy. For example, suppose you do a direct weighing on a decigram balance that is not zeroed and is weighing 0.1 g too heavy. Three measurements might give mass readings of 45.5 g, 45.6 g, and 45.5 g. The precision is good, but every value is 0.1 g too large. Another example of a systematic error would be using a meter stick to measure the length of a 5.0 cm object while consistently putting one end of the object at the 1.0 cm mark but forgetting to subtract the 1.0 and recording 6.0 cm, 5.9 cm, and 6.0 cm as the lengths obtained in three trials. A third example would be a graduated cylinder manufactured in such a way that the first graduation line, marked 3.0 mL, is actually 2.9 mL rather than a 3.0 mL; every measurement made with that graduated cylinder will be 0.1 mL high. Random errors are those that occur despite the best efforts of the experimenter and manufacturer. They cluster around the correct answer, some being a little too high and some being a little too low. For example, air currents, slight temperature fluctuations, or balance insensitivity might cause you to get masses of 34.67 g, 34.68 g, and 34.66 g when making three separate weighings on a centigram balance. Mass measurements of such an object are expected to differ slightly in the hundredths place when a centigram balance is used. So, don’t be disturbed if the results of two measurements aren’t exactly the same. However, if the two measurements are so far apart that the difference can’t be attributed to a reasonable amount of random error, you have just cause for concern. For example, if you weigh an object on a decigram balance and get 23.5 g and you partner gets 23.6 g, there is no reason to think that you have a problem, but if you get 23.5 g and your partner gets 22.0 g, something is definitely wrong, and the situation needs to be investigated and the difference resolved. Making Calculations from Measured Quantities The significant figures (sometimes called significant digits) of a quantity are all the digits in a measurement that are certain plus one that is estimated. When measurements are added or subtracted, the answer can contain no more decimal places than the measurement with the least number of decimal places. When measurements are multiplied or divided, the answer can contain no more significant figures than the value with the fewest significant figures. Rules for counting significant figures: a. The most significant digit is the leftmost nonzero digit. In other words, zeros at the left are never significant. b. If there is no decimal point explicitly given, the rightmost nonzero digit is the least significant digit. c. If a decimal point is explicitly given, the rightmost digit is the least significant digit, regardless of whether it is zero or nonzero. d. The number of significant digits is found by counting the places from the most significant to the least significant digit. Note that zeros can cause some confusion when counting significant figures. To clear this confusion, write potentially ambiguous values in scientific notation. Example – How many significant figures does 8000 have? Solution – By the above method 8000 should have one significant figure. Example – How can you report the value 8000 to have two significant figures? Solution – Rewrite 8000 as 8.0 x 10 3. Example – 9.001 cm + 2.1 cm = 11.101 cm, but is reported as 11.1 cm. Example – 9.001 cm x 2.1 cm = 18.9021 cm2, but is reported as 19 cm2. When making calculations using measured quantities, always follow the rules of significant figures (sometimes called significant digits). These rules must be mastered before this experiment is performed. Understand, and remember, that the precision of a given instrument normally determines how many digits to put in the recorded measurement, but the significant figure rules determine the number of significant figures to put in an answer calculated from measured quantities. Evaluating a Measured Quantity or Calculated Answer If there is an accepted value for the measured quantity or calculated answer, the percentage error is often determined. To calculate a percentage error, find the absolute difference between your experimental value and the accepted value. (Absolute difference means subtract the smaller number from the larger so that the answer will always be positive.) Then, divide by the accepted value to get an idea of how large the error is in comparison to the size of the accepted value. (An error of 1 g when the accepted value is 10 g is a 10% error, but an error of 1 g when the accepted value is 100 g is only a 1% error.) Lastly, multiply by 100 to get the result in percentage form. (See Equation 1.) Note that the number of significant figures in the % error will depend on the number of significant figures obtained when the difference in the experimental and accepted values is found in the first step of the calculation. Equation 1: % Error = │ Accepted value – Experimental value │ x 100% Accepted value If there is no accepted answer, you may wish to determine the percentage difference among the results. To compare two results, use Equation 2; for three r more results, use the difference between the two value s farthest apart. st nd Equation 2: % Difference = │1 exp. value – 2 exp. value │ x 100% Average value Common Instruments Used for Measurement Meter Stick. You are familiar with the use of a ruler like a meter stick to measure lengths, except for one minor point. When using a ruler to measure the length of a small object, experts say it is best to line up the 1 mark with one end of the object, since often the ends of rulers are inaccurate. Of course, you must subtract the 1 from the number you read at the other end of the ruler. Since such a measurement always involves the alignment of two ends, both of which alignments might be a little off, it is difficult to measure with a meter stick to a greater precision than the nearest 0.1 cm, although some advocate estimating to the nearest 0.05 cm, as you will try to do. The Vernier Caliper. There are two scales to consider when reading Vernier calipers. The fixed, or main, scale gives the first two digits of the reading (ones place and tenths place), while the sliding, or vernier, scale gives the third digit (hundredths place) in cm. To read the first two digits, simply see where the zero point (reference line) of the sliding scale lies on the fixed scale. In the example above, the reference line gives the first two digits 1.6. The third digit is found by matching the lines on the sliding scale with the lines on the fixed scale. The fifth line on the sliding scale matches the best with a line on the fixed scale. Thus, the third digit is 5. This gives us a reading of 1.6 cm + 0.05 cm or 1.65 cm. Visit http://members.shaw.ca/ron.blond/Vern.APPLET/index.html for an applet that further instructs reading vernier calipers. Figure 2. Vernier Calipers Decigram and Centigram Balances. Measurements of mass (or as we commonly say, “weight”) will be made using either a decigram balance (precision 0.1 g) or a centigram balance (precision 0.01 g). Graduated Cylinder. Graduated cylinders, used to measure liquid volume, come in all sizes. The one you will use most has a capacity of 25 ml. The first line (or graduation) is the 3.0 mL mark, and each mark upward from there represents an additional 0.5 mL. Such a graduated cylinder may be read to the nearest 0.1 mL. Mercury Thermometer. The most common mercury thermometer has a range from -10 °C to 110 °C. It is graduated (marked off) in degrees, and is usually read to the nearest degree or 0.5 degree. Such a thermometer measures the temperature of the medium into which its bulb is placed. Thus, mercury thermometers do not “shake down”, and attempts to do so often break the thermometer. Digital Thermometer. Modern technology has given us inexpensive thermometers that give a digital readout to the nearest 0.1 degree (or better) when the probe is inserted into the medium whose temperature is to be measured. However, the greatest advantage of a digital thermometer is probably not its increased precision or ease of reading. The advantage is that you don’t have to worry about breaking the thermometer and getting poisonous mercury all over the place or cutting yourself on the broken glass. Examination of the digital thermometer shows its ease of operation. Timer. It has been only a very few centuries since the best instruments for measuring time were “hourglasses” that dropped sand or water through a restricted opening between two compartments. Now, -9 atomic clocks measure time to a precision of nanoseconds (nearest 10 seconds). The timer that you will use has a digital readout and self-evident procedure for operations. We have discussed only the most common measuring instruments used to make the most common types of measurements. Even so, a remarkable amount of information can be gathered just with those discussed, and the principles behind the use of these few also apply to many others. Instructions Part A. Length 1. Use the centimeter side of the meter stick to measure and record the length of your desk. Place the meter stick on its edge to avoid parallax. Estimate to the nearest 0.05 cm. Also record the value your partner obtains. (Anytime during this experiment that you and your partner get results that disagree substantially, figure out why. If you can’t, consult the instructor.) 2. Check your vernier caliper to see if any adjustment is needed for the zero value. Record any correction value, indicating whether it should be added or subtracted. Use the vernier caliper to find the diameter and height of a nickel to the nearest 0.01 cm. Remember to adjust your readings on the nickel for any zero error. Answer the Part A questions. Part B. Mass 1. The instructor will have a decigram balance and a centigram balance set up in the lab. Record the total mass reading shown on the beams of each balance. You must become proficient at reading the balance correctly. 2. Zero your decigram balance and use it to find the mass of a penny and then of a nickel to the nearest 0.1 g. Both you and your partner do this independently. 3. Repeat Instruction 1, but use your centigram balance and measure to the nearest 0.01 g. 4. Use your centigram balance to find the mass of a dry flask. Have your partner weigh it, also. 5. Fill a 100 x 13 mm test tube with water and pour it carefully into the flask. Weigh the flask and contents. Have your partner weigh it, also. Answer the Part B questions. Part C. Liquid Volume 1. The instructor will have three graduated cylinders set out in the lab. Find the volume of water that each shows. Estimate to the nearest 0.1 mL. 2. Use your 25-mL graduated cylinder to find the volume of water that your test tube will hold. (Fill the test tube and pour the water into the graduated cylinder.) Have your partner check the reading. Part D. Time 1. Obtain two meter sticks without metal clips on their ends. Prepare a track down which a steel ball can roll by placing the two sticks parallel so that there is a groove between them about 0.5 cm in width. Elevate the zero ends of the sticks about 2 cm by placing some object beneath them. Ready the timer. 2. Hold the steel ball between your thumb and finger so that it is resting in the groove with its back end even with the zero mark on the meter sticks. Release the ball and simultaneously start the timer. Stop the timer when the back end of the ball passes the 100 cm mark. (This should take 4 or 5 seconds.) Record the time to the nearest 0.1 s. 3. Repeat Instruction 2 twice, then let your partner do three trials with the same setup. Answer the Part D questions. Part E. Temperature 1. Obtain a 250-mL beaker about half full of room temperature water. Place the probe of the digital thermometer into the beaker. Record (to the nearest decidegree) the temperature of the water. 2. Dump out the room temperature water and fill the beaker about three fourths full of crushed ice. Add water until a thick slurry of ice and water is obtained. Place the probe of the digital thermometer into the beaker. Stir with a glass stirring rod, while watching the temperature. Record (to the nearest decidegree) the lowest temperature to which the mixture falls during five minutes of stirring (plenty of ice should still be present). Prelab Sheet for “Measurement Lab” Section ______ Your Name __________________ 1. For what does SI stand? 2. Give the abbreviation and meaning for the following metric prefixes: kilo- _____ ______ centi- ____ _______ deci- _____ _______ milli- ____ ______ 3. Give the metric abbreviation for: gram _____ second ______ liter _____ kelvin ______ 4. Make the following conversions. (a) 31 m = _____ km (b) 5.6 g = ______ cg (c) 82 °F = ______ °C 5. Suppose you made four length measurements on an object and found 4.65 cm, 4.85 cm, 4.84 cm, and 4.85cm. What would you report as the length? 6. Always attach the unit when recording a measurement except when____________________________ ______________________________________________________________________________ 7. Joe reports 19.2 g as the mass of a steel ball. Steve reports 19.22 g. Who apparently used the more precise instrument? 8. Al, Bo, and Cy make a series of time measurements (below) on a pendulum with an accepted value of 10.1 s for the period. Whose results show: high precision, low accuracy _____________ high precision, high accuracy __________; low precision, low accuracy ____________ Al: 10.8 s, 10.8 s, 10.7 s Bo: 11.2 s, 10.8 s, 9.7 s Cy: 10.1 s, 10.0 s, 10.2 s 3 3 3 9. The accepted value for the density of aluminum is 2.70 g/cm . Ann gets 2.75 g/cm , Joan gets 2.87 g/cm . (a) Find the percentage difference between the two students’ results. _________________ (b) Find Ann’s percentage error. _________________ (c) Find Joan’s percentage error. _________________ 10. You and your partner measure the length of an object with the same vernier caliper. You get 3.39 cm; your partner measures 3.29 cm. Is the discrepancy probably due to a random error or a systematic error? __________________ Is the discrepancy cause for concern? _____________ 11. Adding the two measured quantities 14.8 cm and 0.57 cm gives a total length of ______________. 3 12. How many significant figures in the answer when dividing 15.77 g by 8.2 cm ? ______________ Data Sheet for “Measurement Lab” Partner’s Name Your Name A. Length Your Value Partner’s Value Length of table (centimeters side of stick) ________________ __________________ Caliper zero value _________________ ___________________ Diameter of nickel (adjusted) __________________ ___________________ Height of nickel (adjusted) ___________________ ____________________ 1. Convert your line length in centimeters to: meters ________ kilometers ______ 2. Assuming that a nickel is a cylinder, calculate its volume. Use the equation V = πd h 2 4 (The number of significant figures in the answer will be determined by the measured quantities only.) setup: Answer: __________ B. Mass Reading on instructor’s: decigram balance __________ centigram balance ___________ Your Value Partner’s Value Mass of penny (dg balance) ________________ ________________ Mass of nickel (dg balance) _________________ _________________ Mass of nickel (cg balance) __________________ __________________ Mass of flask (cg balance) ___________________ ___________________ Mass of flask + water ___________________ ____________________ Mass of water ___________________ _____________________ 1. How many significant figures in the mass of the nickel when measured on the decigram balance? _______ On the centigram balance? _______ Which balance gave you the more precise value? _________________ 2. When finding the mass of the nickel, it was important that the balance be zeroed. Yet, the balance did not necessarily have to be zeroed when the mass of the water was being found. Explain the difference: C. Liquid Volume Reading on instructor’s: 100-mL graduated cylinder ________________ 25-mL graduated cylinder _________________ 50-mL graduated cylinder __________________ Volume of 100 x 13 mm test tube: Your Value _____________ Partner’s Value ___________ D. Time Distance the ball travels _______________ Trial Time (seconds) Time (seconds) Your value Partner’s value 1 ________________ ______________ 2 _________________ _______________ 3 __________________ _______________ average __________________ ________________ 1. Precision is a function of both the number of significant figures in the measurements and of how close together the values are. Discuss whether the six value of time obtained show good precision or not. 2. There are many possible sources of systematic error in the time measurement. Describe two. E. Temperature Your Value Partner’s Value Room temperature water ____________ ______________ Freezing point of water _____________ _______________ F. Calculations from Measured Quantities Follow the rules for significant figures!!! 1. Add the mass of the penny found from the decigram balance to the mass of the nickel found from the centigram balance to find what total mass of metal you have. Setup: Answer: ______________ 2. Use the volume of the nickel calculated in Part A, and the mass of the nickel measured on the centigram balance in Part B, to calculate the density of the metal alloy. Setup: Answer: ________________ 3. Find the percentage difference between your answer for the density and your partner’s answer. Setup: Answer: _________________ 3 4. A “nickel” is an alloy of 75% copper and 25% nickel. Its accepted density is 8.90 g/cm . Calculate your percentage error. Setup: Answer: _________________ 5. Given the precision of your measurements, you may be surprised at the size of the percentage error. Closely examine the nickel by sight and touch, and consider your assumptions. Explain the source of the majority of the error. d 6. Average speed (v) equals distance traveled divided by time, v . Calculate the average speed t of the ball as it rolled down the track in Part D. Setup: Answer: ____________________