Experiment : Measurements and Error Analysis 1 Measurements and Error Analysis Introduction: [Two students per group. There should not be more than one group of three students.] All experiments require some form of measurements. In engineering, measurements are taken to determine if the product that was built will perform as designed. A medical professional must measure the patent s temperature, height, blood pressure, brain wave, body fat, etc. Anyone can make a measurement, but most are not competent at making measurements; good measurement skills require lots of practice. In addition one must understand the limits of measurements. Objectives: 1. Become competent at making measurements.. Understand the accuracy limits of any measurements. 3. Understand how error analysis is used in assessing results. MATERIALS: There are several materials needed, so get the materials as they are needed. Part I: Distance 1. Measure length using the Meter Stick.. Determine the inherent percent error in the Meter Stick. 3. Measure the area and volume of the lab table with the calculated error analysis. Materials: Meter Stick (one for each student), Pre-cut unknown length of string (one string for four students). 1. Since we will be using the meter stick a lot in the lab, we will need to determine its accuracy. Follow the procedure exactly even though I encourage you to come up with your own techniques, this is not the time to invent your own technique. This part has to be done individually because if your lab partner helps you out in any way, they inadvertently end up biasing the results. Measure the length of the string. The string is longer than your meter stick, but do not fold the string into half! Just lay out the string straight on the table and measure the length of the string. (Remember you have to do everything by yourself, your lab partner cannot even hold the string straight for you). Since the string is longer than the meter stick, you will have to rearrange the meter stick to make two readings. (Each time you need to rearrange the ruler, some error is introduced. We are trying to quantify that error). Record your length in units of cm L=. (Also record your string length into Excel at the front table). If your length seems to be slightly different from the others, don t worry about it. Do not let the other values bias your own value.
Experiment : Measurements and Error Analysis As you have observed, everyone s measurement of the string length varied. So, if someone uses the meter stick and measures.7 meters, how confident are you that the actual measurement was.7 meters?.. (This part will be done by the instructor or the lab tech). The average length of the string is (units ) The standard deviation of the ruler is (units ) The percent error of the ruler is Background of the Standard Deviation: The standard deviation is determined by the following equation: N ( x i x) i= 1 where σ is the standard deviation, x i is one measurement, x is the average of σ = N 1 the measurements, and N is the number or measurements made. Example: x 1 = 5, x = 7, and x 3 = 4 5 + 7 + 4 The average is x = = 5. 33 3 The standard deviation is ( 5 5.33) + ( 7 5.33) + ( 3 5.33) σ = = 1.53. 3 1 So we express the final value as 5.33 ± 1.53 or 5 ±. The significance of the standard deviation is that the person who makes the measurement is 68% confident that the true value lies within one standard deviation. Lab Drill: These are the numbers, 1,, 3, 4, and 5. Find the average and the standard deviation. Average = Standard Deviation = Show your calculations:
Experiment : Measurements and Error Analysis 3 Note: if you had more than 5 measurements, the standard deviation calculation could become tedious, thus an easier way is to use the STAT function on your calculator or Excel. Whenever, we use a meter stick in this lab, the value you record will always be formatted in the following way: Measured Length ± uncertainty. As an example, let's say you measured the length of a rod to be 84.00 cm. Let's also say the percent error of the ruler you used to measure the length of the rod was.6%, then 84.00 cm 0.06 =.18 cm. Thus you will record the length of the rod to be (84.00 ±.18) cm or (84.00 ±.6%) cm. 4. Calculate the area of your lab table with the associated error analysis: Measure the width of the lab table: ± (units ) Measure the length of the lab table: ± (units ) The Area equation is A = WL. The systematic error is obtained from the equation, df σf = i dx σ i however, when you multiply or divide numbers, there is a shortcut method we can use to find the propagated error. σ σ σ = WL W L + A W L where W is the width, L is the Length, σ is the uncertainty in W the width, and σ L is the uncertainty in the length. 5. Calculate the volume of your lab table with the associated error analysis: Measure the height of the lab table: ± (units ) (Assume the table is a solid block and do not worry about the table indentation or the knee hole). Volume equation is V = The propagated error equation is σ V = Volume of the table = ± (units )
Experiment : Measurements and Error Analysis 4 Part II. Mass. 1. Measure the mass of objects.. Determine the inherent percent error in the electronic balance. 3. Use mass to determine the number of sheets of paper. Materials: Ohaus electronic balance mass scale, some masses, stacks of paper. 1. Briefly define what mass measures:.. Is the electronic balance measuring mass or weight (Explain)? 3. What is the smallest increment of mass that the electronic balance can measured. 4. Measure the mass of the following five known masses, 10 g, 50 g, 100 g, 500 g, and 1000 g. Table 1: Comparing the recorded mass value to the measured mass value. Known Mass Value (grams) 10 50 100 500 1000 Measured Mass Value (grams) Percent Error known measured %error = known Use the largest Percent Error that you found to be the Percent Error in the Ohaus electronic balance mass scale. The percent error in the Ohaus electronic balance mass scale is.
Experiment : Measurements and Error Analysis 5 5. Measuring the mass of a sheet of paper: Mass of one sheet of paper = ± (units ) Mass of 50 sheets of paper= ± (units ) Mass of unknown stack of paper = ± (units ) Number of sheets in the unknown stack = ± Part III: Time So far, we looked at two different ways of determining the percent error of measuring devices. Can you or your group think of another? In this part of the lab, you will not have a written procedure. Your goal is to determine the percent error for the stop watch. 1. Develop a procedure to measure the inherent percent error in the stop watch.. Take some measurements using a stop watch. Materials: Stop watch and Experimental: Briefly describe your procedure: Results: Find the standard deviation for the stop watch and convert it to a percentage.
Experiment : Measurements and Error Analysis 6 Part IV: Speed 1. Estimate the distance of your walking stride. Determine your average walking speed with the error analysis. Materials: Stop watch, ruler, and masking tape. 1. Use masking tape to mark the distance of 10 meters.. Walk 10 meters at your regular pace and count how many steps you took. Number of steps in 10 meters is ± 3. Determine the time it takes you to walk 10 meters. Time it took to walk 10 meters ± (units ) 4. Determine your average walking speed. Average walking speed is ± (units ) Part V: Measuring distances you are physically unable to get to: Sometimes it is necessary to measure the distance to an object you cannot physically measure; for example, the width of a canyon, the distance to the moon, or the distance to a nearby star. 1. Determine the length of the table using an indirect method. Let s say you needed to measure the distance across a canyon that you cannot physically measure using a ruler. One method is to establish a baseline line and use the ratio of similar triangles shown in the sketch y = x The similar triangle states that the ratio, so if you were unable to measure b, but you can measure y, x, and a, then you can determine b. Measure the length of the table using this method; show all calculations. b a Length of table ± (units )
Experiment : Measurements and Error Analysis 7 Scenario: 1. Class competition (extra credit to the person who gets closets to the actual amount). Use the measurement techniques covered in lab, or develop your own measurement technique to determine the number of pinto beans that are contained inside the plastic jar. Show all reasoning and calculations. Answered that were guessed -- correct or not -- will not count. Number of pinto beans ±. The instructor will give your group an unknown distance to measure. Your group needs to be within 5% of the actual distance. One method is to use the method shown below. Show a sample calculation of your ±σ unknown. Or your group can critically think of another method to measure that unknown distance. [Note: you cannot physically measure the distance by walking to the unknown distance; pretend there is a canyon between you and the unknown distance]. x ± σ x y ± σ y Unknown Trials Baseline ± σ baseline distance ± σ unknown 1 3 4 5 Average Unknown distance ±