Physics 201_Laboratory Exercise (2)-1-1

Physics 201 (College Physics-1) Laboratory Exercise (2) Experimental Measurements: Precision, Accuracy, and Plotting Graph Equipment needed: Rod or other linear object less than 1 m in length, Four meter-long measuring stick with calibrations of meter, decimeter, centimeter and millimeter respectively, Pencil and ruler, Hand calculator, Three sheets of Cartesian graph paper. Introduction: All experimental data have associated uncertainties that ultimately limit the conclusions one can make. Uncertainty is a part of the experimental process, no matter how hard one tries to minimize it. Thus, it is important to express uncertainty clearly when giving experimental results. Laboratory investigation involves taking measurements of Physical quantities. Taking any measurements always involves some experimental uncertainties or error. For example, Taking several independent measurements of length of an object by two people is highly unlikely that both of them will come up with exactly the same results. So several questions arises: 1. Whose data are better? 2. How does one express the degree of uncertainties or error in measurements? 3. How do you compare your experimental results with the accepted value? 4. How does one graphically analyze and report experimental data? Objective: In this introductory study different types of experimental uncertainties will be analyzed along with some methods of data analysis. After performing the experiment and analyzing the data you should be able to (1) Categorize the type of experimental uncertainty or error (2) Distinguish between measurement accuracy and precision (3) Define the term what is mean by least account and significant figures. (4) Express experimental results and uncertainty in appropriate numerical (5) Represent measurement data in graphical form Theory: Laboratory experiments involve taking measurements of physical quantities. No measurement of any physical quantity is ever perfectly accurate, except possibly the counting of objects. The discrepancy between the measured value and the true value of the quantity may arise from different sources. To obtain an experimental result with an estimate of the degree of uncertainty in the measurements, you need to know the types of errors, the ways to reduce the errors, and how to treat the data properly. In this lab we will study about: a. Types of Experimental Uncertainties b. Accuracy and Precision c. Least count and Significant Figure d. Computations with measured values e. Expressing experimental uncertainties f. Graphical representation of the data a) Types of Experimental Uncertainties: Classified as being two types i. Random or Statistical Error or Indeterminate error

Random or Statistical Error or Indeterminate error results from unknown and unpredictable variations that arise in all experimental measurements situation. Example : Unpredictable fluctuations in temperature or line voltage, Mechanical vibrations of an experimental setup and unbiased estimates of reading by observer i. Systematic or Determinate error. Systematic or Determinate error is associated with particular measurement instruments or technique, such as improperly calibrated instruments or bias on the part of the observer. Systematic implies that the same magnitude and sign of experimental uncertainties are obtained when the measurements are reaped several times. Determinate means that the magnitude and signs of the uncertainties can be determined if the error is identified. Example: An improperly zeroed instrument such as an ammeter, a faulty instrument such as thermometer, Personal error, such as using a wrong constant in calculation or always taking a high or low reading of a scale division. b) Accuracy and Precision: In experimental measurements there is an important distinction between accuracy and precision. The accuracy of a measurement signifies how close it comes to the true value, i.e., how correct it is. Example: If one arrow hits exactly in the center of the target, it is highly accurate. Obtaining greater accuracy for an experimental value depends in general on minimizing systematic and random error The precision refers to the agreement-among--repeated-measurements, i.e., the "spread" of the measurements, or how close together they are. Example: If six arrows hit within one centimeter of each other at one side of the target, the shooting is highly precise but not accurate. High precision does not necessarily imply high accuracy, but high precision is necessary to obtain high accuracy. c) Least Count (LC) and Significant Figure (SF): Exact numbers and measured numbers: In general there are exact numbers and measured numbers. Such as 100 used in calculating percentage and 2 in 2πr are exact numbers. Measured numbers are those obtained from measurements instruments and generally involve some error or uncertainties. Least Count (LC) : Least count is the smallest subdivision on the measurement scale. This is the unit of the smallest reading that can be made without estimating Example: The least count of a meter stick is usually 1 millimeter. Significant Figures (SF) : sometimes called significant digits of a measured value include all the numbers that can be read directly from the instrument scale plus one doubtful or estimated number- the fractional part of the least count smallest division. Example: The length of the rod is estimated to be 2.6 and 2.7 cm. The estimated fraction is taken to be 4/10 of the least count. So the doubtful figure is 4, giving 2.64 cm with three significant figures. The greater is the number of significant figures, the greater the reliability of the measurement numbers. Zeros and decimal points must be properly dealt with in determining the number of significant figures. Example:

Elements required in the graph: Graph should have the following elements i. Each axis labeled with the quantity plotted ii. The units of the quantity plotted iii. The title of the graph on the graph paper iv. Your name and date. Straight-line graph: Two quantities (x and y) are often linearly related; that is, there is an algebraic relation of the form y = mx + b, where m and b are constant. Slope of the graph: The value of ‘m’ in the algebraic relationship is called the slope and is equal to the ratio of the intervals Δy/Δx, any set of intervals may be used to determine the slope of a straight-line graph. Intercepts: The ‘b’ in the algebraic relationship is called the y-intercept and is equal to the value of the y coordinate where the graph line intercepts Y axis Experimental procedure: Part 1: Given the meter-length sticks calibrated in meters, decimeters, centimeters, and millimeters, respectively. Use the sticks to measure the length of the object provided and record the appropriate numbers of significant figures. (a) express the least counts and measurements. (b) from the given actual length calculate the % error in length. Part 2: Express some of the numbers in 3 significant figures (the first column in normal notation and the second column in scientific notation. Part 3: Calculate fractional, Percent error, and percent difference, plot some graphs in linear form and measure slope. Also, calculate the % error from the given slope INCLUDE EVERYTHING after this page in your report c Data tables: Table 1: Express least count and % error calculation (show your calculation in a space below) Type of object Object’s length in m/dm/cm/mm Least count Actual length in m/dm/cm/mm % error Cylindrical rod 0.5 m 0.25 m 0.5 m 25% Cylindrical rod 4.5 dm 1 dm 4.5 dm 350% Cylindrical rod 45.7 cm 1 cm 45.7 cm 4470% Cylindrical rod 457 mm 1 mm 457 mm 45600% Table 2: Express the number in table in three significant numbers Numbers 3 significant fig. Numbers 3 significant fig.( In scientific notation) 0.524 0.524 52800 5.28 x 10^4

15.08 15.1 0.060 60.0 x 10^-3 1444 1440 82,453 8.25 x 10^4 .0254 0.0254 0.00010 10.0 x 10^-5 83,909 83,900 2700,000,000 2.70 x 10^9 Table 3: Graphical analysis of data Time t Distance (m) Y Ave t 2 (s) Y 1 Y 2 Y 3 Y 4 Y 5 0 0 0 0 0 0 0 0 0.50 1.0 1.4 1.1 1.4 1.5 1.28 0.25 0.75 2.6 3.2 2.8 2.5 3.1 2.84 0.5625 1.00 4.8 4.4 5.1 4.7 4.8 4.76 1 1.25 8.2 7.9 7.5 8.1 7.4 7.82 1.5625 NOTE: For both questions 1 and 2 show your calculation in the space provided below Question (1): In an experiment to determine the value of π, a cylinder is measured to have an average value of 4.25 cm for its diameter and an average value of 13.39 cm for its circumference. What is the experimental value of π to the correct number of significant figures? Experimental value of π: 13.39/4.25= 3.1506 If the accepted value of π is 3.1416, what are the fractional error and percent error of the experimental value found in previous problem? Fractional error: (3.15-3.1416)/3.1416 Percent error: 2.674% Question (2): In an experiment to measure the acceleration due to gravity, two values 9.96 m/s 2 and 9.72 m/s 2 are determined. (i) Find the percentage difference of the measurements, (ii) percent error of each measurement E 1 and E 2 , and (iv) the percent error of their mean (accepted value of g is 9.8 m/s 2 ) i. Percent difference: 9.96- 9.72/9.96 x 100 = 2.40% i. Percent error of E 1 : 9.96-9.8= 1.6%. i. Percent error of E 2 : 9.8-9.72= 0.82% i. Percent error of mean: 9.84-9.8/9.8 x 100 = 0.408%

Q1: Do percent error and percent difference give indication of accuracy or precision? Discuss each. Percent error indicates accuracy because it compares the actual value to the experimental value. Percent difference gives indication of precision because it compares the experimental values to one another. Q2: What determines how many figures are significant in a measurement values? The amount of significant figures in a number depends on the number of significant figures in the least count of the instrument scale. Q3: What is graphical analysis? Plot a typical straight line graph of F = kX where F=force, X=displacement, what is the slope? Graphical analysis is the studying of graphs to obtain a conclusion about the relationship between two variables. The slope of the graph between force and displacement would be acceleration. Conclusion and summary: The average of the experimental values is usually “better” or more accurate data than any one experimental point would be. Degrees of uncertainty and errors in measurements can be expressed by calculating the random or statistical error and the systematic or determinate error. Experimental results can be compared to the accepted value by calculating the percent error. The slope of a graph can be used to determine the relationship between two variables and conduct graphical analysis.