G8-Lab 3

Lab #2: Lab Kit Intro – p. 1 Physics 210 – Lab Kit Introduction OBJECTIVE In this lab we will explore the Physics Lab Kit, practice taking basic measurements, and use significant figures to describe limitations on measurement accuracy based on the instrument used to take a measurement. EQUIPMENT • Physics Lab Kit • Circular Objects • Computer THEORY Much of laboratory work is based around taking measurements, and then analyzing those measurements to reach conclusions about the world around us. The precision of any experiment is limited by how precise our measurements are, and this precision is usually determined by the measuring instrument we use. It’s important to use “the right tool for the job”: a ruler would be fine to measure the width of your hand, for example, but it wouldn’t be good for measuring the width of a bridge, or the width of a hair. You will therefore learn about the tools available in the Physics Lab Kit, make simple measurements and determine the corresponding precision. In scientific measurements, it is virtually impossible to get an exact numerical value for something. If we measure a box, and say that it is 12 inches wide, it is unlikely that the box is exactly 12 inches wide, down to the size of an atom. If we had a microscope, we would find that its length differs slightly, one way or the other, from our measurement. So how close is our measurement? Are we accurate to within an inch? To within a millimeter? To within a nanometer? How do we know? This sort of inaccuracy will carry through other calculations as well: if we’re unsure on the exact width of the box, we can’t be perfectly sure of the volume of the box, either, since that depends on the (uncertain) width. When analyzing numbers (either measurements or calculations from measurements), there are two things to consider: 1) The number of decimal places 2) The number of significant figures Decimal Places are a convenient way for us to keep track of how precise a measurement is, and how that precision carries through our calculations. If there is some inaccuracy in our measurement, we can’t write it out to too many decimal places. Effectively, we will round off our measurements to the nearest value we can be confident in, with the understanding that we can’t make any measurements more precise than the lowest decimal place that we’ve measured. If we carry these rounding rules through our calculations, it will show us how far we’ve had to

Lab #2: Lab Kit Intro – p. 2 round other values as well, and demonstrates the limitations on precision for all our calculations. “Significant Figures” (“Sig. Figs.”) are digits that are a meaningful part of the number, rather than just a place-holder. For example, “312” and “0.000000312” both only have three significant figures (the “3”, “1” and “2”), with the zeroes just serving as place-holders in the second number, to show us where the decimal place is. To determine the number of significant figures in a number, follow these rules: SIGNIFICANT FIGURE RULES: • ALL non-zero digits (1,2,3,4,5,6,7,8,9) are ALWAYS significant • ALL zeroes between significant figures are also significant • Leading zeroes (zeroes to the left of all other non-zero digits) are never significant • Zeroes to the right of non-zero digits are significant if they are also to the right of the decimal place “Decimal Places” describe how many digits are to the right of the decimal place, whether they are significant or not. For example, “0.001” has three decimal places, whereas “100” has none. EXAMPLES: Number # of Significant Figures # of Decimal Places 48,923 5 0 3.967 4 3 900.06 5 2 0.0004 1 4 8.1000 5 4 501.040 6 3 In some cases, a number’s significant figures can be ambiguous. For example, if we counted a football field to be 100 yards long to within a yard, we would want all three of those digits to be significant; to say that it has only one significant figure would imply we could only measure it to the nearest hundred yards! The easiest way to show this is to underline the right-most significant figure; for example: 100 yards (three significant figures) 100 yards (two significant figures) 100 yards (one significant figure) Underlining a zero like this shows that it is indeed a significant figure. This is only necessary for numbers that happen to end in zero, and haven’t been measured to any decimal places.

Lab #2: Lab Kit Intro – p. 4 Relative uncertainty = (absolute uncertainty ÷ best estimate) × 100% When you measure various quantities, you may end up having to add/subtract them or multiply/divide them together. Below are the rules to follow to find the uncertainty of the result. To illustrate them, we will use two measurements: (7.4 ± 0.1 cm) and (3.1 ± 0.2 cm). ABSOLUTE VS. RELATIVE UNCERTAINTIES The uncertainty mentioned above is also called the absolute uncertainty. The relative uncertainty gives the uncertainty as a percentage of the original value, and can be calculated as follows: ADDING AND SUBTRACTING UNCERTAINTIES The total uncertainty when adding/subtracting measurements together is the sum of the absolute uncertainties. Example: (7.4 ± 0.1 cm) − (3.1 ± 0.2 cm) = (7.4 − 3.1) ± (0.1 + 0.2) cm = 4.3 ± 0.3 cm MULTIPLYING AND DIVIDING UNCERTAINTIES The total uncertainty when multiplying/dividing measurements together is the sum of the relative uncertainties. Example: (7.4 ± 0.1 cm) ÷ (3.1 ± 0.2 cm) = (7.4 cm ± 1.4%) ÷ (3.1 cm ± 6.5%) = (7.4 ÷ 3.1) ± (1.4% + 6.5%) = 2.4 ± 7.9% MULTIPLYING BY A CONSTANT If you multiply a number with an uncertainty by a constant, its absolute uncertainty does not change. PROCEDURE: Part I: Length Measurements Download and open the spreadsheet file PH210_Measurements_FA20.xls . Find 10 circular objects in the kit or around you. Identify each object and measure its radius and circumference, and enter your findings into the appropriate cells on the “Part I” tab of the spreadsheet. For each measurement, specify the measuring tool, as well as its precision. You have several tools to your disposal. You will need to choose the best tool given the size of the object. You need to consider the degree of precision of the tool. If you are measuring small circles, then you need to use equipment that is more precise to avoid a greater potential for error! Group spokesperson only: Have each of your teammates tell you what they found for their largest and smallest objects and enter the data on the “Group Results” tab on the spreadsheet.

Lab #2: Lab Kit Intro – p. 5 Q1: If we were to convert from centimeters to meters, would we change the number of significant figures in our measurements? The number of decimal places? Explain. In your spreadsheet, and make a graph of Circumference vs Radius. Make sure to include a title and to label the axes, according to the guidelines you used in the previous lab. Use the LINEST function to determine the slope and uncertainty on the slope. Group spokesperson only: Have each of your teammates tell you what they found for their slopes and uncertainties and enter the data on the “Part I - Group Results” tab on the spreadsheet. Q2: Expected value for the slope: Measured slope: Standard Error of the slope: 95% confidence interval: lower bound: upper bound: Does the theoretical value fall within the 95% confidence interval? What can you say about your experiment vs. theory? Explain.

Lab #2: Lab Kit Intro – p. 7 Q5: Using the data in Table 5 and 6 , calculate the volume and density of the balls, with corresponding uncertainties. Show your work here, and report the results in Tables 7 and 8 . Table 4: ½ Meter Stick Density Calculation Physical Quantity Calculated Value Uncertainty Volume (cm 3 ) Density (g/cm 3 ) We will now repeat the procedure for the wooden and steel balls in the kit. Measure the masses and radii of the two balls. Make sure to recalibrate the spring scale as needed. Report the values in Tables 5 and 6 . Group spokesperson only: Have each of your teammates tell you what they found for their density values and enter the data on the “Part II - Group Results” tab on the spreadsheet. Table 5: Steel Ball Measurements Physical Quantity Measurement Uncertainty Mass (g) Radius (cm) Table 6: Wooden Ball Measurements Physical Quantity Measurement Uncertainty Mass (g) Radius (cm) Table 7: Steel Ball Density Calculation Physical Quantity Calculated Value Uncertainty Volume (cm 3 ) Density (g/cm 3 ) Table 8: Wooden Ball Density Calculation Physical Quantity Calculated Value Uncertainty Volume (cm 3 ) Density (g/cm 3 )

Lab #2: Lab Kit Intro – p. 8 Final Questions: Q6: What kind of wood if the ½ Meter Stick likely made of? You may use the following as a reference https://www.engineeringtoolbox.com/wood-density-d_40.html Q7: The given density of steel ranges between 7.75 and 8.05 g/cm 3 . Was your measured value in that range? Discuss your result here. Q8: You can also try to guess what the wooden ball is made of . You may have found some challenges when making that measurement. Discuss the accuracy of your measurement.