Physics and Measurement

Transcription

1 Physics and Measurement Eternity is nothing else but God Himself. Hence God is not called eternal, as if He were in any way measured; but the idea of measurement is there taken according to the apprehension of our mind alone. Aquinas I:Q10, A2 Introduction and Pre-Lab Experiments rely on measurements, but no human measurement is perfectly accurate. The quality of a measurement is stated in terms of accuracy and precision. This laboratory exercise will focus on the proper use laboratory equipment as well as concepts of accuracy, precision, and uncertainty in taking measurements. Accuracy and Precision The Accuracy of a measurement indicates how close that measurement is to the true value. It is subject to the reliability of the measuring device. If a student uses an uncalibrated scale to measure mass, the measurements may not be accurate if the instrument has become damaged. Precision refers to how reproducible is a set of measurements. A precise instrument gives the same (or acceptably close) reading each time the same sample is measured. In Figure 1, see how CO2 Monitor 1 was precise but not accurate. On the other hand, CO2 Monitor 3 gave both a precise and an accurate measurement. Precision sometimes refers to the number of divisions between numerical values. In Fig 2, the lower ruler is more precise than the upper With the upper ruler, there are 2 significant digits. The 4 is certain and the 3 is estimated. With the lower ruler, there are 3 significant digits. The 4.3 is certain and the 4 is estimated. ruler. Before measuring, study the tool which you will use to determine how you will report the data. The number of significant figures reported in the final measurement includes all the digits that are certain, plus a last digit that is estimated. This practice applies to all measurements, including the ones you will be performing in this lab Catholic Initiatives in Math and Science, LLC All Rights Reserved 1

2 Calculations of Uncertainty We often need a mathematical way to discuss how different a measurement is from the expected or true value. Percent Error (% Error, % Difference) expresses the difference between the measured experimental value and the known, theoretical or true value. In Figure 1, the Percent Error would have helped determine whether Monitor 1 or Monitor 3 gave true CO2 values. Error = experimental value theoretical value % Error = experimental value theoretical value theoretical value x 100% Note absolute value sign in numerator. Although %Error reveals how far data deviate from true values, it does not give any indication of the scatter. In Figure 1, the Percent Error of Monitors 2 and 3 would be essentially the same since the mean of the data does not differ significantly from the true value. Therefore, a different calculation is needed. Percent Variation (% Var) measures how far data is dispersed from the mean of the data. Whereas Percent Variation is not helpful in determining the accuracy of Monitor 1, it is helpful in determining the precision of Monitor 2. Percent Variation is obtained from the following equations: Standard Deviation = range highest value lowest value = N # samples in the set % Variation = Standard Deviation Mean (or Average) x 100% Measuring Mass The mass of an object indicates the amount of matter present in that object. In the metric system, the SI mass unit is the gram (g). The scale provided in your kit with which you will measure mass is a Spring Scale (See an example of a spring scale in Fig 3). There are a few important notes on your spring scale: The scale is divided in Newtons on the left and grams on the right. You will record all information in grams. Examine the divisions between the numerical markings on your scale. Determine the value of each division on the grams side. Before making any measurement, adjust the metal tab to the zero setting as shown Catholic Initiatives in Math and Science, LLC All Rights Reserved 2

3 Note the hook on the lower end. This hook is for either hanging a mass with a hook, a paperclip used as a hook or a weigh boat. Make a weigh boat: A container used for weighing samples o The body may be constructed from plastic cups, small paper plates (dessert size), or even from aluminum foil o The sides may be constructed from wire, string, or yarn o Choose durable materials as they will be used for the rest of the class Making measurements with a hanging mass: o Make sure metal tab is set to zero o Hang object from hook and allow spring to stabilize o Read and record the measurement Making measurements with weigh boat: o Make sure metal tab is set to zero o Place empty weigh boat on hook and record initial reading o Place object in/on the weigh boat and record the final reading o Calculate actual mass of object by subtracting the initial reading from the final reading Measuring Volume The volume of a substance measures the amount of space it occupies. The liter (L) is the unit for volume in the metric system. Prefixes are used to express smaller volumes, such as milliliters (ml), and are typically the volumes measured by your graduated cylinders. A meniscus is the curved upper surface of a nonturbulent liquid in a container. If the liquid wets the walls of the container the upper surface is concave, otherwise the upper surface is convex. The liquids you will be determining volumes on in biology lab may have convex or concave menisci. The picture (right) shows the correct way to read the measurement of the liquid with a concave surface. Place cylinder on steady surface View the meniscus at eye level Take the reading from the bottom of the meniscus (if concave as shown) Take the reading from the top of a convex meniscus Record the number and units, using the appropriate number of significant figures 2015 Catholic Initiatives in Math and Science, LLC All Rights Reserved 3

4 Learning Objectives: Demonstrate proficiency in the use of numbers and units and relate them to physical concepts and calculations Perform measurements with the appropriate precision and acceptable accuracy Explore accuracy, precision and uncertainty with length, mass and volume measurements Materials Required: From Physics Kit Protractor Spring Scale 200g mass 100g mass Graduated Cylinder, 100mL Student Supplied Ruler (marked to mm) US $1 bill (or $5 Canadian) Plastic cup or small plate Wire, yarn or string Scissors or sharp knife 1/3 rd measuring cup for liquids Safety Perform this lab in a clear area. Falling objects may cause serious injury. Experiment: Precision and Accuracy of Length Measurements 1. Gather the following materials Protractor from kit Obtain a metric ruler marked in millimeters Obtain a US dollar bill a crisp, new bill is best 2. Measure length with two instruments Theoretical height of US dollar bill is 6.63 cm (2.61 inches) [Canadian students may perform a similar study using a $5.00 banknote which has a height of cm] Use the protractor and then the metric ruler to measure in centimeters the height of the bill Be as accurate and precise as you can, paying attention to the number of significant digits Record the measurements in Table 1A Calculate Percent Error, showing calculations in the table 3. Measure height multiple times Have 5 different people measure the height of the dollar bill with the protractor (in cm) [If 5 persons are not available, measure at 5 different times during the day] As above, be accurate and precise and record the measurement in Table 1B Calculate the Mean, %Error, Std Dev, % Variation Record the calculations and results in Table 1B 4. Continue to next part; afterwards perform Data Analysis and Conclusions Experiment: Accuracy with Mass Measurements 1. Gather materials From kit, gather the spring scale and the 200 g mass Obtain a small plastic cup and 3 pieces of wire, string or yarn 2015 Catholic Initiatives in Math and Science, LLC All Rights Reserved 4

5 2. Construct weigh boat to be used in future labs Follow the instructions in the Pre-Lab (Figure 4A) to construct a weigh boat Thread and tie 3 pieces of string or wire through holes made with scissors Make a loop to hang from the Spring Scale 3. Measure mass Hang Spring Scale from a stable hanger, ensuring that masses will not hit an obstacle Set the metal tab to zero Carefully, place the 100 g mass on the Spring Scale Allow the motion to stop Record the measurement to the correct number of decimal places Record in Table 2 Repeat the step above with the 200 g mass Calculate %Error, showing calculations and results in Table 2 4. Continue to next part; afterwards perform Data Analysis and Conclusions Experiment: Accuracy and Precision with Volume Measurements 1. Gather materials From kit, obtain the 100mL graduated cylinder Obtain a 1/3 rd measuring cup for liquids 2. Accuracy in volume measurement Fill the 1/3 rd liquid measuring cup to the indicated level Carefully, pour all the liquid into graduated cylinder (use of funnel will help) Accurately measure the volume, paying attention to the number of significant digits Record the measurement in Table 3A 3. Calculate Calculate %Error, using the graduated cylinder as Theoretical volume Record results and calculations in Table 3A 5. Precision in volume measurements times Have 5 different people repeat Step 2 [If 5 persons are not available, repeat Step 2 at 5 different times during the day] As above, be accurate and precise and record the measurements in Table 3B Calculate the Mean, %Error, Std Dev, % Variation Record the calculations and results in Table 3B 6. Perform Data Analysis and Conclusions 2015 Catholic Initiatives in Math and Science, LLC All Rights Reserved 5

6 Lab Report for: NAME: Protractor Ruler Table 1A Accuracy Measuring Length of Object Experimental Theoretical %Error Show % Calculations Table 1B Precision Measuring Length (with Protractor) Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial # Mean %Error Std. Dev. % Variation *Use the empty space to show your calculations. 100g 200g Volume (ml) Trial # Mean %Error Std. Dev. % Variation Table 2 Accuracy Measuring Mass Experimental Theoretical %Error Show % Calculations Table 3A Accuracy Measuring Volume Experimental Theoretical %Error Show % Calculations Table 3B Precision Measuring Volume Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 *Use the empty space to show your calculations. Data Analysis and Conclusions 1. Explain the difference between accuracy and precision when making measurements? 2015 Catholic Initiatives in Math and Science, LLC All Rights Reserved 6

7 2. Which calculation(s) performed today was/were helpful in evaluating accuracy? Which was/were not? 3. Which calculation(s) performed today was/were helpful in evaluating precision? Which was/were not? 4. MEASURING LENGTH a. Did the protractor or the ruler give a more accurate answer for the height of the currency? Explain with your data. b. The height of the currency was measured 5 times with the protractor. Did repeating the measurements have any effect on the accuracy? Explain. c. What kind of precision was there between the 5 measurements? Explain with your data. d. The standard deviation may be combined with the mean to give an expression of the level of uncertainty. Typically, it is used in the form of: Mean +/- Std. Dev. For example, in Figure 1, Monitor 2 was accurate but not precise with a mean and std dev of 650 ppm +/- 200 ppm. In the table below, organize your data and express your results by: i. Stating the theoretical height for the government banknote ii. Express your results from Table 1B as Mean +/- Std. Dev. iii. If the government used your mean +/- std dev in their quality control department, what could be the SHORTEST bills allowed? iv. What could be the TALLEST bills allowed? 2015 Catholic Initiatives in Math and Science, LLC All Rights Reserved 7

8 5. Percent Error; Mean and Standard Deviation; Percent Variation: Consider an instrument in a hospital tests blood and then reports an Ebola virus titer value and then interprets it as positive or negative result. Using this example, explain how all three calculations are needed to give a more accurate evaluation of data so that critical information may not be concealed. 6. Precision of instruments: To how many decimal places must you record measurements: a. For volume in the 100mL graduated cylinder? b. For length with the protractor? c. For length with a mm-ruler? d. For the spring scale? 7. Four CO2 monitoring instruments were tested with a standard (known CO2 = 537ppm) Data were collected, and the mean/std dev were collected for each instrument. For each data set, tell whether the instrument demonstrates high accuracy/high precision; high accuracy/low precision; low accuracy/high precision, or; low accuracy/low precision. a. 538ppm +/- 250ppm b. 120ppm +/- 0.3ppm c. 538ppm +/- 1.2ppm d. 725ppm +/- 129ppm 8. A team of 5 students performed the same experiment that you performed and reported in Table 1B. Each measured the height of a US dollar bill (theoretical height of 6.63 cm). Their calculated Percent Error = 0.15% Their calculated Percent Variation = 47% a. Evaluate their Accuracy. Explain b. Evaluate their Precision. Explain c. What could they do to improve? 2015 Catholic Initiatives in Math and Science, LLC All Rights Reserved 8