Lab 01: Measurement, Uncertainty, and Data Analysis Techniques

Lab 01 – Experiment 1: Measurement, Experimental Uncertainty, Unit Conversion, & Data Analysis Click here to enter Your First and Last Name. Partners’ Names: ........... Enter Partners’ First and Last Names . Instructor: ............................................................ Albert Owino Course: ........................................................................ PHY 120 Conducted On: ............................. Monday 11 September 2023 Due: .............................................. Monday 18 September 2023

Double Click to E nter Last Name Here. 2 Click here to enter Your First and Last Name. Partners’ Names: Click here to enter Your Partners’ First and Last Name. Instructor: Albert Owino Course: PHY 120 Conducted On: 11 September 2023 Due: 18 September 2023 Lab 01 – Experiment 1: Measurement, Experimental Uncertainty, Unit Conversion, & Data Analysis Pre-Laboratory Assignment: 1. Read over the lab material and be prepared to conduct the lab at the session 2. Show up at the specified start time (4:40 pm) stated at the recitation session just prior to the lab Materials:  Lab handout  Pencil  Calculator  Printer Paper, or other Paper for note taking Purpose: The objective of this introductory experiment is to learn to use common instruments for measuring linear dimensions and mass and to perform simple calculations using the measurements, including conversion of units. Also types of experimental uncertainty will be examined, along with some methods of error and data analysis. Particular emphasis will be given to the treatment of significant figures and the reporting of percent error. Theory: Most measurements of linear dimensions are carried out in the laboratory course with the help of  a meter stick,  Vernier caliper,  balance, or a  micrometer

Double Click to E nter Last Name Here. 3 The micrometer is the most precise of these instruments, allowing reading to 0.01 mm (0.001 cm). However, it is limited to measuring distances less than about 25 mm (2.5 cm). The Vernier caliper is capable of measuring lengths as large as about 12 cm. However, it is less precise than the micrometer, allowing readings to 0.02 mm (0.002 cm). For lengths greater than 12 cm, a meter stick is used. The meter sticks are divided into 100 cm, each of which is divided into 10 mm. These sticks allow readings to 1 mm (0.1 cm) and, for some measurements, estimates to tenths of a millimeter. We will usually measure mass by using a triple beam balance, which has three moveable standard masses. The balance allows direct readings to a precision of 0.01 g, and estimates to 0.001 g. There is no such thing as a perfect measurement. All measurements have errors and uncertainties. Understanding possible errors is an important issue in any experimental science. The conclusions we derive from the data, and especially the strength of those conclusions, will depend on how we treat the uncertainties. Let’s consider the example: You measure two values 2.4 and 1.5. From theory, the expected value is 2.2, so the value 2.4 almost agrees, whereas 1.5 is far off. But if you take into account the uncertainties (i.e. the interval in which your result is expected to lie), neither may be far off. For experimental uncertainties of 0.1 and 1.0, respectively, your two measured values may be expressed 2.4 ± 0.1 and 1.5 ± 1.0. The expected value falls within the range of the second measurement but not of the first. This first laboratory experiment deals with the important subject. The issues are important to arrive at good judgements in any field in which it is necessary to understand not just numerical results, but the uncertainties associated with them.

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Double Click to E nter Last Name Here. 4 Types of Uncertainties Experimental error may be classified into two types of uncertainty: (1) Random or Statistical Error (also called indeterminate error) and (2) Systematic Error (also called determinate error). Random Errors: Result from unknown and unpredictable variations in taking measurement data. Some examples of conditions in which random errors may occur are due to unpredictable temperature variations, unpredictable pressure variations, or voltage fluctuations. Systematic Errors: Are due to using a particular measurement instrument or measurement technique. For example, a very common systematic error occurs when not using a calibrated instrument. A common form of systematic error due to technique is called parallax error. This results from the user reading an instrument at an angle resulting in a reading that is consistently high or consistently low. Accuracy vs. Precision: Although these terms are often used interchangeably, they have different meanings. The accuracy of a measurement is determined by how close it comes to the “true” or accepted value. Precision refers to the agreement among two or more repeated measurements – or the “spread” of the measurement values. It is sometimes referred to as the ”reproducibility” of the measurement. This can be easily illustrated by a dart board.

Double Click to E nter Last Name Here. 5 Significant Figures The degree of uncertainty of a number read from an instrument depends on the fineness of its measuring scale and the quality of the instrument. When reading the value from a calibrated scale, only a certain number of figures or digits can properly be read. This depends on the least count of the instrument scale, which is the smallest subdivision on the measurement scale. For example, the least count on a meter stick is usually the mm scale. The least significant digit in a measurement depends on the smallest unit which can be measured using the measuring instrument (least count). The precision of a measurement can then be estimated by the number of significant digits with which the measurement is reported. In general, ay measurement is reported to a precision (estimated fraction) equal to 1/10 of the smallest graduation on the measuring instrument, and the precision of the measurement is said to be 1/10 of the smallest graduation. For example, a measurement of length using a meter stick with 1-mm graduations will be reported with a precision of ±0.1 mm. A measurement of volume using a graduated cylinder with 1-mL graduations will be reported with a precision of ±0.1 mL. Significant Figures 0.00003400 Zeros after nonzero numbers in a decimal are significant All nonzero numbers are significant Zeros are not significant after decimal before non-zero numbers

Double Click to E nter Last Name Here. 6 The following rules apply for calculations: 1) When multiplying / dividing quantities, leave as many significant figures in the answer as there are in the quantity with the least number of significant figures. An example is as follows: 4094.1945 = 38.65 × 105.93 The final answer, limited to four significant figures is: 4094. The first digit dropped is 1 in the tenths place (of the .1945). If we round the tenths place first, it will go to 4094.2 since there is a 9 in the hundredths place. Since the rounded decimal value is .2 in the tenths place, we do not round up to the next integer and simply leave the value at 4094. 2) When adding/subtracting quantities, leave the same number of decimal places (rounded) in the answer as there are in the quantity with the least number of decimal places. An example is as follows: 5.61 = 1.2 + 4.41 The final answer, limited to the tenths place is: 5.6 Percent Error Percent error (sometimes referred to a fractional difference) measures the accuracy of a measurement by the difference between a measured or experimental value E and a true or accepted value A . The percent error is calculated from the following equation: Percent Error = | E − A | A × 100% Percent Difference IT is sometimes helpful to compare the results of two measurements when there isn’t a known or accepted value. Percent difference measures the precision of two measurements by the difference between the measured or experimental value E1 and E2 expressed as a fraction of the average of the two values. Percent Difference = | E 2 − E 1 | [ ( E 2 + E 1 ) 2 ] × 100%

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Double Click to E nter Last Name Here. 7 Measurement Instruments: Basic Measurement Principles  Always determine when “zero” is on the instrument and account for any offset.  Always determine the units of the instrument’s measurements.  Don’t let your eye fool you. Double check the reading.  Confirm that your measurement makes sense, based on an experimental sense of size. Vernier Caliper (Reference YouTube Video Entitled “How to Read a Metric Vernier Caliper”) https://www.youtube.com/watch?v=vkPlzmalvN4 The Vernier caliper consists of a rule with a fixed main scale and moveable jaw with a sliding “Vernier” scale. To measure the width of an object, the object is placed between the calipers’ jaws. The sliding jaw is then moved unit the object is gripped firmly between the jaws. In the case of this figure, the smallest measurement on the main scale is 0.1 cm (1 mm). The Vernier scale can read 0.02 mm. So using both scales, the width can be read to the nearest 0.002 cm (or 0.02 mm). To measure the width, you read the top and bottom scale as follows:

Double Click to E nter Last Name Here. 8 (1) Find where the 0 mark of the Vernier scale lines up on the main scale. (in this case between 2.1 and 2.2 cm) So, the first reading is 2.1 cm (21 mm). (2) Find the mark on the Vernier scale that most closely lines up with one of the marks on the main scale. Here, 0.30 mm and 0.34 mm are very close, but 0.32 mm lines up best with one of the marks on the main scale. This value is the number of hundredths of centimeters (or tenths of millimeters). So, the second reading is 0.032 cm (0.32 mm). (3) Add the two values together to get the total reading: 2.1 cm + 0.032 cm = 2.132 cm (21.32 mm) Micrometers Caliper (Reference YouTube Video Entitled “How to Read a Metric Micrometer”) https://www.youtube.com/watch?v=StBc56ZifMs The micrometer is more accurate than a Vernier caliper because it can measure to within thousandths of a millimeter. It is particularly helpful in measuring diameters of thin wires and thicknesses of thin sheets. The micrometer caliper is used to make very fine measurements beyond the hundredths of a centimeter. As its name implies, distances are measured to 0.000 001 m or 1 x 10 -6 m (recall the SI prefix for an order of magnitude of 10 -6 is micro, Lower case Greek Letter mu: μ ) which is equal to 0.0001 cm. This device uses the uniformity in the spacing of threads on a bolt. If a nut is threaded on the bolt and the bolt is rotated one complete evolution, the end of the bolt will have moved a linear distance equal to the width of a thread. If instead of a nut, we attach a rotating scale as well as place a calibrated line (also called the fixed scale) along the length of the bolt, then it becomes possible to measure small fractions of a rotation (and small fractions of the width of a thread). As shown in the figure, the basic parts of a micrometer are labeled. The object to be measured is placed between the anvil and the spindle. Turn the thimble until the object fits snugly. Do not force the turning of the thimble, since this may damage the very delicate threads on the spindle located inside the thimble. Some micrometer calipers have a ratchet, which helps protect the instrument by not allowing the timbale to turn when forced.

Double Click to E nter Last Name Here. 9 The axial main scale on the sleeve is graduated in millimeters and it also have graduations in halves of millimeters, which are indicated by the lower set of graduations on the sleeve. The threads on the spindle are made so it takes two complete turns of the thimble for the spindle to move precisely one millimeter. The head (rotating scale) is divided into fifty equal divisions – each division indicating 0.01 mm, which is the precision of the instrument. Since our eye can still estimate another decimal place between marks on the rotating scale (or 0.001 mm, which is 0.000 001 m), this device is called a micrometer. Rules for reading a micrometer in millimeters: (1) Completely close the measurement ends of the micrometer and check to see if the zero line on the sleeve lines up with the zero line of the thimble. If it does not line up then there is a zero reading error. If the zero line of the thimble lies below the zero line of the sleeve, then there is a positive zero reading error. In order to obtain a correct micrometer measurement, the positive zero reading error must be subtracted from the micrometer measurement value obtained in steps 2-5 below. If the zero line of the thimble lies above the zero line of the sleeve, then there is a negative zero reading error. To obtain a correct micrometer measurement, the negative zero reading error must be added to the micrometer measurement value obtained in steps 2-5 below. Corrected reading = actual reading – zero reading

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Double Click to E nter Last Name Here. 10 (2) Find the while number of mm in the measurement by counting the number of mm graduations on the sleeve to the left of the head. (3) Find the decimal part of the measurement by reading the graduation on the Head (rotating scale) that is most nearly in line with the centerline on the sleeve, and multiply this reading by 0.01 mm. If the head is at or immediately to the right of the half mm graduation, add 0.50 mm to the reading on the rotating scale. (4) Estimate one more decimal place. (5) Add the numbers found in steps above. The following is an example on how to read the micrometer (assumes that there is no zero- reading error).

Double Click to E nter Last Name Here. 11 Laboratory Balance (Reference YouTube Video Entitled “Dialogram Tutorial”) https://www.youtube.com/watch?v=2UISTrwUNco Zeroing the Balance Slide all the poises to their respective zero positions. Turn the dial to the zero position. You can now zero the balance by rotating the knurled balance compensator knob at the left end of the beam until the pointer on the right lines up with the zero. (Note how quickly the magnetic damping brings the pointer to rest). Whenever the balance is moved to a new location, the balance should be checked and zeroed again, if necessary. Determining the Mass of a sample Place the specimen on the pan of the balance. Proceed as follows for the fastest method of determining the mass of the specimen or sample: (1) Move the 200 gram poise on the rear beam to the first notch which causes the pointer to drop then move it back one notch, causing the pointer to rise. (2) Repeat this procedure with the 100 gram poise. (3) Turn the knob slowly until the pointer is exactly centered on the zero indicator. (4) The weight of the specimen is the sum of the values of both poise positions, the dial position and the Vernier reading.

Double Click to E nter Last Name Here. 12 For specimens less than 100 grams, omit step 1. For specimens less than 10 grams, omit steps 1 and 2. Note: With a little practice, you will become proficient in learning exactly how fast to turn the dial to come exactly to balance position the first time. Reading the Vernier Each graduation on the dial has a value of 0.1 grams. A Vernier adjacent to the dial breaks down these values in 0.01 gram increments. To read the Vernier, read the nearest gram value to the right of the zero Vernier graduation. Add to that the Vernier graduation value at the Vernier line which most closely lines up with any of the other dial graduations. Note: Pages 3 – 12 may be delete out of your final document that you prepare for submission in order to save you from printing excess pages.

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Double Click to E nter Last Name Here. 13 Equipment ☐ Meter Stick ☐ Vernier Caliper ☐ Micrometer Caliper ☐ Laboratory Balance ☐ Printer Paper ☐ Pencil ☐ Lab Handout Procedure This lab was provided by Professor Arthur Eidelson and is based on Wilson & Hernandez-Hall, Physics Laboratory Experiments, 8 th Edition Experiment 3 1.) Determine the least count and estimate fraction for the four measuring instruments listed in Data Table 1. Data Table 1 (use correct units) Instrument Least Count Estimated Fraction Meter Stick 1mm 0.1 mm Vernier Caliper Micrometer Caliper Laboratory Balance 2.) Using the following micrometer figure, determine the zero reading and record it here: (Positive or Negative Zero Reading Error) Circle the Correct Answer. Zero Reading: for micrometer

Double Click to E nter Last Name Here. 14 3) Using the micrometer figures below, determine four measurements of a single page of printer paper (be sure to account for the zero reading correction determined above). Record your measurements in Data Table 2 with correct units. Then compute the average thickness of a single page and place your answer with correct units and number of significant figures in Data Table 2.

Double Click to E nter Last Name Here. 15 4) With the micrometer, you took four thickness measurements of 20 pages (not corrected for zero reading) as follows: Measurement Number Thickness ( mm ) 1 21.25 2 19.83 3 20.00 4 19.98 Record your measurements in Data Table 2 with correct units (be sure to account for the zero reading correction determined above). Also, calculate the average thickness of your 20 page measurements and place your answer with correct units and number of significant figures in: Data Table 2. Reading Thickness of a Single Page (units ) Thickness of 20 Pages (units ) Average Page Thickness (units ) Thickness of Unknown Number of Pages (units ) 1 2 3 4 5 Average 5) Next, divide each of your four 20 page thickness readings by 20 and record with the correct units and number of significant figures in the column titled “Average Page Thickness” in Data Table 2. Then compute the average of each of your entries in the “Average Page Thickness” column in Data Table 2. 6) Using the Vernier caliper figures below determine the four measurements of the total thickness of the Unknown Number of Pages. Record your measurements with the correct units in Data Table 2. Also, calculate the average thickness of your Unknown Number of Pages and place your answer with the correct units and number of significant figures in Data Table 2.

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Double Click to E nter Last Name Here. 18 7) Using the average entry in your “Average Page Thickness” column and your average entry in your “Thickness of Unknown Number of Pages” column compute the experimental number of pages (sheets of paper). For example, if the average thickness entry in your “Average Page Thickness” column per page is 0.160 mm and the average entry in your “Thickness of Unknown Number of Pages” column 75 mm, the experimental number of pages is: Written as: 75mm / (0.160 mm/page) = 469 pages Which may also can be written as: 469 pages = 75 mm ( 0.160 mm 1 page ) = 75 mm× 1 page 0.160 mm Computed experimental number of pages (using the average entry in your “Average Page Thickness” column and your average entry in your “Thickness of Unknown Number of Pages”) is: pages The actual number of pages in your Unknown Number of Pages is 20. Computer your percent error (show calculations taking into account significant figures) using your experimental number of pages above and the actual value of 20 pages. Percent Error % 8) Using the average entry in your “Thickness of Single Page” column and your average entry in your “Thickness of Unknown Number of Pages” column compute the experimental number of pages (sheets of paper). For example, if the average thickness entry in your “Thickness of Single Page” column is 0.140 mm and the average entry in your “Thickness of Unknown Number of Pages” column is 75 mm, the experimental number of pages in your Mystery Number of pages is: Written as: 75mm / (0.140 mm/page) = 536 pages Which may also can be written as: 536 pages = 75 mm ( 0.140 mm 1 page ) = 75 mm× 1 page 0.140 mm

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Double Click to E nter Last Name Here. 19 Compute the experimental number of pages (using the average entry in your “Thickness of Single Page” column and your average entry in your “Thickness of Unknown Number of Pages) is: pages The actual number of pages in your Unknown Number of Pages is 20. Computer your percent error (show calculations taking into account significant figures) using your experimental number of pages above and the actual value of 20 pages. Percent Error % In your report answer the following questions: 1) Explain probable source of error in the experimental determination of your number of printer paper sheets? Identify your errors as random or systematic? Click here to enter text, and type a response to the question. 2) Should using the Average Page Thickness of 20 sheets or using the Average Page Thickness of a single page in determination of the number of pages in your unknown pile by more accurate? Which was more accurate in your case and why? Click here to enter text, and type a response to the question. 3) A right rectangular pyramid is machined from a solid block of aluminum. The base of the pyramid has a length of 38.32 mm and a width of 102.45 mm. The height of the pyramid is 0.127 cm. Find the mass of the pyramid if the density of aluminum is 2.7 g/cm 3 . (Use the correct number of significant figures in your calculation Density = Mass / Volume). Click here to enter text, and type a response to the question. 4) In an experiment to measure the acceleration due to gravity, two values, 9.50 m/s 2 and 9.90 m/s 2 are determined. The acceptable value of acceleration due to gravity is: 9.80 m/s 2 . Find the percent difference of the two measurements (use the correct number of significant figures in your calculation).

Double Click to E nter Last Name Here. 20 Click here to enter text, and type a response to the question. 5) Write the product of 2.10 × 0.5896 with the correct number of significant figures (show all steps) Click here to enter text, and type a response to the question. 6) Write the different of 9.954 – 0.3109 with the correct number of significant figures (show all steps). Click here to enter text, and type a response to the question. 7) Write the sum of 1.586 + 2.31 with the correct number of significant figures (show all steps). Click here to enter text, and type a response to the question. 8) Write the quotient of 16.15 / 2.7 with the correct number of significant figures (show all steps). Click here to enter text, and type a response to the question.

Double Click to E nter Last Name Here. 21 Final Page: Checklist for Laboratory Reports (If an item does not apply write NA next to the item in the Comment here ) ☐ Each page labeled with last name and page number .................... Comment here ☐ Date laboratory performed ............................................................ Comment here ☐ Date laboratory due ....................................................................... Comment here ☐ Lab partner names (First and Last names spelled correctly) ......... Comment here ☐ Data points on graphs enclosed with circles ................................. Comment here ☐ Label graphs with appropriate title ................................................ Comment here ☐ Label X & Y axes with appropriate labels and units ..................... Comment here ☐ Label data tables with appropriate title ......................................... Comment here ☐ Label each row or column including units .................................... Comment here ☐ Answer ALL questions assigned .................................................. Comment here ☐ Show all calculations taking into account Significant Figures ...... Comment here ☐ Attach original data sheets (scans or photographs) ....................... Comment here ☐ Copy your original data to your report in a tabular form with labeled columns (including units) ............. Comment here ☐ Attached this completed checklist with the Laboratory Report .... Comment here This checklist was courtesy of Professor Eidelson

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PHY120_Fall2023_Lab01_writeup

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