Laboratory Guidelines 1 Instrumentation Being both quantitative and experimental, physics is basically a science of measurement. A great deal of effort has been expended over the centuries improving the accuracy with which the fundamental quantities of length, mass, time, and charge can be measured. It is important that the appropriate instrument be used when measuring. Ordinarily, a rough comparison with a numerical scale, taken at a glance and given in round numbers, is adequate.
Increasing precision, though, requires a more accurate scale read to a fraction of its smallest division. The “least mount” of an instrument is the smallest division that is marked on the scale. This is the smallest quantity that can be read directly without estimating fractions of a division. Even at the limit of an instrument’s precision, however, accidental errors? which cannot be eliminated?still occur. These errors result in a distribution of results when a series of seemingly identical measurements are made.
The best value, known as the most probable value, is the arithmetic mean or average of the measurements. Other errors, characteristic of all instruments, are known as systematic errors. These can be minimized by improving the equipment and by asking precautions when using it. 1. 1 Length Measurement Three instruments will be available in this class for length measurements: meter sticks, for example), the fernier caliper, and the micrometer caliper. 1. 1. 1 The Meter Stick A meter stick, by definition, is 1 meter (m) long.
Its scale is divided, and numbered, into 100 centimeters (CM). Each centimeter, in turn, is divided into 10 millimeters. Thus 1 CM = 10 -2 m, and 1 mm = 10-1 CM = 10-3 m. When measuring a length with a meter stick, different regions along the scale should be used for the series of measurements resulting in an average value. This way, non-uniformities resulting from the meter stick manufacturing process will tend to cancel out and so reduce systematic errors. The ends of the stick, too, should be avoided, because these may be worn down and not give a true reading.
Another error which arises in the reading of the scale is introduced by the positioning of the eyes, an effect known as parallax. Uncertainty due to this effect can be reduced by arranging the scale on the stick as close to the object being measured as possible. 1. 1. 2 The Fernier Caliper A fernier is a small auxiliary scale that slides along the main scale. It allows more accurate estimates of fractional parts of the smallest division on the main scale. On a fernier caliper, the main scale, divided into centimeters and millimeters, is engraved on the fixed part of the instrument.
The fernier scale, engraved on the movable Jaw, has ten divisions 9 that cover the same spatial interval as nine divisions on the main scale: each fernier division is 10 1 a ruler (one- or two- the length of a main scale division. In the case of a fernier caliper, the fernier division length is 0. 9 mm. [See Figure 1. ] 01 01 0. 05 1. 23 Figure 1: Examples of fernier caliper readings. To measure length with a fernier aliped, close the Jaws on the object and read the main scale at the position indicated by the zero-line of the fernier.
The fractional part of a main-scale division is obtained from the first fernier division to coincide with a main scale line. Several examples are shown in Figure 1 If the zero-lines of the main and fernier scales do not coincide when the Jaws are closed, all measurements will be systematically shifted. The magnitude of this shift, called the zero reading or zero correction, should be noted and recorded, so that length measurements made with the fernier caliper can be corrected, thereby removing the systematic error. . 1. The Micrometer Caliper A micrometer caliper is an instrument that allows direct readings to one hundredth of a millimeter and estimations to one thousandth of a millimeter or one millionth of a meter (and, hence, its name). It is essentially a carefully machined screw housed in a strong frame. To measure objects, place them between the end of the screw and the projecting end of the frame (the anvil). The screw is advanced or retracted by rotating a thimble on which is engraved a circular scale. The thimble thus moves along the barrel of the frame which contains the screw and on which is engraved a longitudinal call divided in millimeters.
The pitch of the screw is 0. 5 mm, so that a complete revolution of the thimble moves the screw 0. 5 mm. The scale on the thimble has 50 divisions, so 1 that a turn of one division is 50 of 0. 5 mm, or 0. 01 mm. Advance the screw until the object is gripped gently. Do not force the screw. A micrometer caliper is a delicate instrument. To read a micrometer caliper, note the position of the edge of the thimble along the longitudinal scale and the position of the axial line on the circular scale. The first scale gives the measurement to the nearest whole division; he second scale gives the fractional part.
It takes two revolutions to advance one full millimeter, so note carefully whether you are on the first or second half of a millimeter. The result is the sum of the two scales. Two examples are shown in Figure 2. As with the fernier caliper, the zero reading may not be exactly zero. A zero error should be checked for and recorded, and measurements should be appropriately corrected. 2 200 15 10 5045 0. 307 CM 300 25 20 15 10 5 0. 367 CM Figure 2: Examples of micrometer readings. 1. 2 Mass Measurement Three kinds of instruments will be available to determine mass: a digital scale and woo types of balances.
The operation of the first instrument is trivial, and so will not be explained here. Please understand that with each of these instruments we are really comparing weights, not masses, but the proportionality of weight and mass allows the instruments to be calibrated for mass. 1. 2. 1 The Equal-Arm Balance The equal-arm balance NAS two trays on opposite sides to a pivot. The total mass placed on one tray required to balance the object on the other gives the mass of the object. Most equal-arm balances have a slider, as well, that can move along a scale and allow or greater precision than the smallest calibrated mass available.
Typically, this scale has 0. 5 g divisions. 1. 2. 2 The Triple-Beam Balance The triple-beam balance, so-called because of its three slider scales, can be read to 0. 1 gang estimated to half that. With an object on the tray, the masses of the different scales are slid to notches until balanced. Get close with the larger masses first and then fine-adjust with the smallest slider. 3 1. 3 Time Measurement Time measurements in this course will be made either with a computer or with a stop watch. This first is out of your control. 1. 3.
The Stop Watch The stop watches you will use in class have a time range of from hours to hundredths of a second. There are two buttons at the top: a stop/start button and a reset button. The operation of these should be evident, although once the watch is reset, the reset button also starts the watch (but doesn’t stop it). Please be aware of this feature. 1. 4 Charge Measurements The magnitude of charge is among the most difficult measurements to make. Instead a number of indirect measurements are undertaken to understand electric phenomena. These measurements are most often carried out with a digital millimeter. . 4. The Digital Millimeter The digital millimeters available for laboratory exercises have pushbutton or knob control to select five AC and dc voltage ranges, five AC and dc current ranges, and six resistance ranges. The ranges of accuracy are 100 microvolt to 1200 volts AC and dc, 100 ampersands to 1. 999 amperes AC and dc, and 100 millions to 19. 99 messages. To perform a DC voltage measurement, select the DC function and choose a range maximum from one of 200 millionth or 2, 20, 200, or 1200 volts. Be sure the input connections used are V-Q and COMMON.
The same is true for AC voltage, regarding range and inputs, but the CAP unction button should be selected. For DC current choose DC MA (for DC millimeters), while for AC current choose AC MA. Your choices for largest current are 200 micrometers or 2, 20, 200, or 2000 millimeters. Check that the input are connected to MA and COMMON. There are two choices for resistance measurement: Kilos (K) and Memos (MGM). The input connectors are the same as when measuring voltage, namely V-n and COMMON. The range switches do not function with the Memos function, but one of the range buttons must be set.
The maximum settings for Kilo readings are 2000 or 2, 20, 200, or quick. Recording Data When performing an experiment, record all required original observations as soon as they are made. By “original observations” is meant what you actually see, not quantities found by calculation. For example, suppose you want to know the stretch to a coiled spring as caused by an added weigh NT. You must read a scale boot bettor and after the weight is added and then subtract one reading from the other to get the desired result. The proper scientific procedure is to record both readings as seen.
Errors in calculations can be checked only if the original readings are on record. 4 All data should be recorded with units. If several measurements are made of the same physical quantity, the data should be recorded in a table with the units reported in the column heading. 3 Errors and Uncertainties Any measurement has an associated uncertainty. In general, if two or more measurements are made of the same physical quantity, they differ. The difference between a measurement and the true value of the physical quantity is the error in the measurement.
The true value lies somewhere within a range of values near the measured value. This range of values is called the uncertainty of the measurement. Therefore the error is likely to be less than or equal to the uncertainty. The experimenter’s goal is to reduce the error as much as possible while recognizing and specifying the appropriate uncertainty. The term “error” as used in science should not be confused with “mistake. ” Errors are inherent in measurements and cannot be avoided. Mistakes can, and should, be avoided. 3. 1 Types of Errors There are two types of errors: random and systematic.
A systematic error usually repeats itself with the same sign. That is, all the measurements are either too large or too small. A poorly calibrated instrument, such as a meter stick, voltmeter, or hermetic, will introduce systematic errors. Other types of systematic errors are also possible. For instance, if you tend to always hold your head to one side of a scale, you may introduce a systematic parallax error. Random errors are associated with fluctuations in repeated measurements of the same physical quantity. These errors are Just as likely to be positive as negative.
Therefore the effects of random errors tend to cancel when you average the results of repeated measurements. For this reason the average of repeated measurements is assumed to be less uncertain than a single measurement. Human error” is not a class of scientific error, and the term should never be used. 3. 2 Precision and Accuracy The precision of a set of measurements of the same quantity refers to the amount by which they differ. If the measurements are very close together the precision is high, while the precision is low if they scatter over a wide range.
Accuracy refers to how close the measured value is to the true value. It is possible for a high precision measurement to have low accuracy if a systematic error is present. As an example, consider a game of darts between two students, Ryan and Ashley. They each throw here darts resulting in the distributions shown in Figure 1 Ryan’s darts snow high precision because they are close together, but low accuracy because they are far from the bull’s eye. On the other hand, Ashley darts have lower precision than Ryan’s because they are farther apart, but have higher accuracy because they are scattered around the bull’s eye. Ryan Ashley Figure 3: The results of one round in a game of darts between Ashley and Ryan. Ryan’s distribution has higher precession but lower accuracy than Ashley. 3. 3 Significant Figures A laboratory worker must learn to determine how many figures in any measurement r calculation are reliable, or “significant” (that is, have physical meaning), and should avoid making long calculations using figures which he/she could not possibly claim to know. All sure figures plus one estimated figure are considered significant.
The measured diameter of a circle, for example, might be recorded to four significant figures, the fourth figure being in doubt, since it is an estimated fraction of the smallest division on the measuring apparatus. How this doubtful fourth figure affects the accuracy of the computed area can be seen from the following example. Assume or example that the diameter of the circle has been measured as 0. 5264 CM, with the last digit being in doubt as indicated by the line under it. When this number is squared the result will contain eight digits, of which the last five are doubtful.
Only one of the five doubtful digits should be retained, yielding a four-digit number as the final result. In the sample calculation shown below, each doubtful figure has a short line under it. Of course, each figure obtained from the use of a doubtful figure will itself be doubtful. The result of this calculation should be recorded as 0. 2771 CM , including the doubtful fourth figure. The zero to the left of the decimal point is often used to emphasize that no significant figures precede the decimal point. This zero is not itself a significant figure. ) (0. 5264 CM)2 = 0. 27709696 CM = 0. 771 CM Very large or very small numbers should always be recorded using scientific notation. For example, the proper way to record the area calculated above in m 2 is 2. 771 x 10-5 mm . All of the figures, including the zeros, are significant in a measurement properly recorded using scientific notation. There are several rules that must be kept in mind when determining the appropriate number of significant figures to record in porting a result: 6 1 . In multiplication and division, a calculated result should contain the same number of significant figures as the least that were used in the calculation. . In addition and subtraction, do not carry a result beyond the first column that contains a doubtful figure. 3. Zeros that serve only to locate the decimal point are not significant. 4 Expressing Uncle artiness An experimental determination of the quantity x that has a mean (or average) value x and an uncertainty Ax should be expressed as x = x В± Ax. (1) For example, suppose you measure the diameter, D, of a circle with a meter stick and get 3. 42 CM. The “2” is a doubtful digit because it was obtained by estimating between the smallest divisions (mm) on the scale of the meter stick.
In the absence of additional information, the measurement must be assumed to have an uncertainty of at least 0. 01 CM and the measurement would be recorded as D = 3. 42 В± 0. 01 CM. (2) In a later section we will discuss more sophisticated ways of finding uncertainty from multiple measurements off physical quantity. 3. 5 Propagation of Uncertainties Often in the laboratory you will use a measured quantity to calculate another quantity and you will have to determine the error associated with the calculated annuity due to the error in the measured quantity.
The uncertainty in the calculated quantity can be determined using calculus. In general, if a physical quantity f is a function of x, f = f (x), then the uncertainty Oaf in f is defined as M = UDF Ax, DXL (4) (3) where Ax is the uncertainty in x. This may be extended to physical quantities that are functions of several variables. If a physical quantity f is a function of two variables x and y, for example, then f = f (x, y), UDF (x, y) = UDF UDF DXL + day, DXL Day (5) (6) 7 and UDF UDF Ax + Ay. (7) Day In many cases the uncertainties Ax and Ay will be uncorrelated ND CEQ. 7) will yield an overestimate of the uncertainty in f because Ax and Ay can compensate for each other. Therefore, it is usually best to add the terms in quadrate and determine the uncertainty in f using the expression Oaf = Oaf = UDF DXL 2 2 (AY)2 . The relative uncertainty is the ratio of the magnitude of the uncertainty to the magnitude of the quantity (e. G. Oaf If). The percent uncertainty is Just the relative uncertainty x 100%. As an example, suppose you measure the diameter, D, of a circle 2 00 + 0 02 CM and you want to calculate the area to the circle. First, notice that he relative uncertainty in the diameter is 0. 1 and the percent uncertainty is 1%. The area of the circle is 1 D 2 1 = DID 2 = 11(2. 00 CM)2 = 3. 14 CM . 244 To find the uncertainty in the area, we apply the technique described above: dad 1 = DID id 2 1 dad = Tided 2 or 1 1 LA = = TIT(2. O = 0. 06 corn . 22 so, the final result for the area would be recorded as A = nor 2 = A = (3. 14 В± 0. 06) CM . (9) This gives us a relative uncertainty in the area of about 0. 02 or a percent uncertainty of about 2%. 3. 6 Statistical Analysis Any measurement is an intelligent estimation of the true value of the quantity being measured.
To arrive at a “best value” we often make several measurements of the same quantity and then analyze these measurements statistically. The results of such an analysis can be represented in several ways. Those in which we are most interested in this course are the the mean, standard deviation, and standard deviation of the mean. 3. 6. 1 Mean The mean is the sum off number of measurements of a quantity divided by the number of such measurements, which is just the arithmetic mean or the so-called average. For a group of N measurements, xi , the mean, x, is defined by xi x= . 14) N It represents the most probable value of he measurement. 8 3. 6. 2 Standard Deviation The standard deviation, o, is a measure of the spread of the data about the mean value. The equation for calculating the standard deviation is (15) where xi are the individual measurements, x is the mean, and N is the total number of measurements. 3. 6. 3 Standard Deviation of the Mean A related quantity is the standard deviation of the mean, mom , defined as o mom = v’ N where OSI the standard deviation and N is the number of measurements.
If the distribution of the measurements is normal (I. E. Gaussian), then there is a 68% chance that the true value falls within the interval x В± mom . While few measurements sets are truly normal distributions, this is a reasonable approximation for bell-shaped distributions. In this course we will assume that the uncertainty in a quantity determined from a set of measurements is the standard deviation of the mean, mom , and that the measured value should be reported as x В± mom . 3. 6. Standard Deviation of the Mean of a Function Some physical quantities are not measured directly, but rather are computed from other quantities which are directly measured. The following is a general prescription for calculating the standard deviation of the computed quantity room the standard deviations to the measured quantities. Let z be a donation to x and y, z = f (x, y), (16) where x and y are directly measurable. Suppose x and y have been measured several times each, and that their mean values, x and y, and standard deviations of the meaner, o mix and mom , have been computed.
The mean value of z can be calculated using z = f (x, y), and the best estimate of the standard deviation of z is computed from OMG = The final result should be reported as The extension of this procedure to functions of more than two variables is straightforward. 9 z = z В± OMG . Adz 2 2 OMG (17) 2 2 mom . (18) 19) As an example, suppose we have made several measurements of the length I and radius r of a cylinder and we want to calculate the volume, V . Four measurements of the length yielded 16. 45, 15. 95, 17. 02, and 16. 00 CM, and six measurements of the radius gave 3. 9, 3. 17, 3. 19, 3. 62, 3. 43, and 3. 28 CM. Calculations of the mean and standard deviation of the length measurements yield I = 16. 4 CM and mol = 0. 1 CM, and for the radius measurements we get r = 3. 33 CM and o Mr. = 0. 03 CM. We can calculate the average volume using the formula for the volume of a cylinder V = Tell 1 = 571 CM = 5. 71 x 102 CM . The standard deviation of the volume can be calculated using move = where and so that ova = (34. CACM )2 (0. 1 CM)2 + (comma )2 (0. CACM)2 = 1 1 CM . (24) Thus, the final result for the volume is V = (5. 7 В± 0. 1) x 102 CM . 25) DVD Del 2 2 mol + (20) 22 norm , (21) 2 = 34. 8 corn , VA (22) (23) = enroll = 343 corn , 4 Comparing Measured and Accepted or Predicted Values Often one wishes to compare the value of a quantity determined in the laboratory Witt the best known or “accepted value” to the quantity obtained through repeated determinations by a number of investigators or with a value predicted by theory. A initiative way to do this is to calculate the percent difference. The % difference is calculated by subtracting the accepted value from your value, dividing by the accepted value, and multiplying by 100.
If your value is greater than the accepted value, the % difference will be positive. If your value is less than the accepted value, the % difference will be negative. The % difference between two values in a case where neither is an accepted value can be calculated by choosing one as the accepted value. It is also often appropriate to state whether the accepted or predicted value lies within the uncertainty of your measured value. If not, there may be a systematic uncertainty present that has not been taken into account. 5 Lab Reports Formal lab reports should follow a standard format with several clearly marked sections.
The first page should contain the title, your name and the names of your lab partners, and the date. The 10 rest of the report should have the following sections, in order: Abstract, Introduction, Experimental Procedure, Results, Discussions/Conclusions, and References. You should assume that your readers are at the same level in physics as you, but that they have not read your lab manual or handouts. 5. 1 Sections of the Lab Report . 1. 1 Abstract Summarize your experiment in one short paragraph. State the purpose, the experimental method, and the result.
Be concise-for example, often Just giving the name of the method is sufficient. If you have made a quantitative measurement of some quantity, state the result of the measurement, with uncertainty. Remember to include the appropriate units on any results that you present. The abstract should be self-contained and there should be no references to figures or tables in the main body of the report. 5. 1. 2 Introduction Discuss any relevant theory and/or motivation for the experiment. This section serves primarily to help the reader understand the significance of the experiment and all the issues that are later addressed.
The main questions to be addressed in this section of the report are “Why are you doing this experiment? ” and “What do you hope to find? ” If the experiment is designed to test a particular physical theory discussed in class, you should describe both the general theory and the particular prediction you’re attempting to check in the Introduction. This section is often the most difficult to write, and you may want to try writing this section last, since you want to be sure to introduce any important concepts that are needed for your discussion in later sections. 5. 1. Experimental Procedure Describe your experimental set-up (drawings are usually needed) and the method used. Do not Just restructure the instructions in the lab handout, and do not assume that your reader has read the lab handout. Make sure you describe the apparatus before referring to parts of it. A Procedure section which starts out “We moved the cart back and forth on the track and recorded the position with the sensor” will be incomprehensible to a reader who was not in your class. You need to tell the reader that the apparatus consisted to a cart, a track, and a sensor, and also what kind of cart, track, and sensor you used.
Including a sketch of the apparatus is not sufficient description; you must also describe the apparatus briefly in words. Do not include unimportant details, such as where a particular switch is, what combination of keystrokes and mouse-clicks you use to do something in Science Workshop, or how you line up your head to see that something is aligned. In particular, you do not need to include steps like “Then we made a graph of position vs.. Mime and printed the graph out”- inserting the graph of position vs.. Time in the lab report is sufficient.
It’s also not necessary to identify the software packages used to generate graphs and tables, though it is important to identify the software used to acquire data. For example, “We recorded the position as a function of time using the Science 11 Workshop package” is important, while “We entered numbers into a data table in Microsoft Excel” is not. Explain what you did in the course of the experiment, but don’t write the procedure section like an instruction manual. Use the most direct ascriptions possible, writing in past tense and active voice.
For example, it’s better to write “We measured the length of the track using a meter stick” than “The length of the track was measured with a meter stick” or “Use a meter stick to measure the length of the track. ” Include measurements that are related to the limitations of the experiment, such as a measure of the amount of error in a particular measurement. You will discuss these errors in more detail in the Results section, but how you measure them and what you measure them to be is relevant to the procedure and so needs to be discussed in this section.
They belong in this section because they give a quantitative measure of the accuracy and reliability of the results. 5. 1. 4 Results The Results section is where you present your data and calculations, and is the meat of your report. Begin by presenting the raw data. Numerical data should be listed in a table and the table referred to in the text; graphical data (for example, position vs.. Time plots from Science Workshop) should be presented as figures and referred to in the text. Be sure to include uncertainties in any measured quantities. After presenting the raw data, discuss any calculations that you made from that data.
If there are results of calculations that would be best presented in a table, make sure they are clearly distinguishable from the raw data, either by putting the processed data in a separate table, or by clearly labeling the columns. If there are results that would be best presented in a figure, label the figure clearly, and be sure to refer to it in the text. Be sure to label the tables and figures and to refer to them in the text by name (e. G. “Figure 1 “Table 2”). Don’t include a figure without discussing it in the text. Explain the relevance of the figure, and what it tells you about the experiment.
Be sure to include an uncertainty with every measurement. In general the discussion of the measurement of the uncertainty (that is, how you determine the value of the uncertainty) is given in the experimental procedure section, but you still need to give the uncertainty (after a plus-minus sign) when presenting the results. Say “We measured the mass of the cart to be 1. 02 В± 0. 03 keg,” not “The mass of the cart was 1. 02. ” Discuss your errors in this section. Discuss the sources of error, both random and systematic, and how the errors affect your results. Do not put off the discussion