Review of Vectors; Introduction to Measurement and Uncertainty Objective: To learn how to use vectors in real life. To learn about experimental uncertainty, and to distinguish between precision and accuracy. Apparatus: Vectors: GPS receiver (or ios/android app), pen, notebook, calculator, a good pair of walking shoes. Uncertainty: Computer, reaction-time software, fast hands. Vectors A vector is a mathematical quantity that contains both a magnitude and a direction. In Physics, vectors represent things such as Velocity, Force, Magnetic Field and many others. A vector differs from a scalar in that a scalar only contains information about the magnitude. For example, a cyclist cruising at 7 meters/second is a scalar quantity, wheras a cyclist crusing at 7 meters/second West is a vector quantity. In this lab you will be represent your position on a map with a vector, and learn how changing your position results in adding vectors. This is an outdoor experiment; you and your lab partner will roam around the campus armed with a GPS receiver (or your smartphone), which will determine your position with the aid of several GPS satellites in orbit around the Earth. Radians and Arc Length Remember from high school geometry that the distance L along a circular arc of radius R an angle θ (in radians) subtends is given by the equation: L=R θ For example, a 45 degree angle ( corresponds to a length of : π/ 4 radians) on a circle of radius 1.0 meter L = (1 m) ( π/ 4 ) = 0.79 m Likewise, a 360 degree( 2π )angle on the same circle covers 6.28 meters. GPS GPS stands for Global Positioning System. It allows you to determine your location anywhere near the surface of the earth in real-time, by using a receiver that compares the time it takes for radio signals from several orbiting satellites to get to it. Imagine you are blindfolded in a large rectangular hall. If you know that you are intially standing facing the North wall of the hall, you could tell that you are closer to the East Wall if an echo from your shout returns sooner than one shouted toward the West Wall. Similarly, the receiver triangulates radio signals from several satellites and gives you a position measurement: Latitude, Longitude, and Elevation.
Latitude, Longitude and Elevation Latitude and longitude are angles that uniquely define points on a sphere. These two values comprise a coordinate scheme that can locate or identify geographic positions on the surfaces of planets such as the earth. Latitude is defined with respect to an equatorial reference plane. This plane passes through the center C of the sphere, and also contains the great circle representing the equator. The latitude of a point P on the surface is defined as the angle that a straight line, passing through both P and C, subtends with respect to the equatorial plane. If P is above the reference plane, the latitude is positive (or northerly); if P is below the reference plane, the latitude is negative (or southerly). Latitude angles can range up to +90 degrees (or 90 degrees north), and down to -90 degrees (or 90 degrees south). Longitude is defined in terms of meridians, which are half-circles running from pole to pole. A reference meridian, called the prime meridian, is selected, and this forms the reference by which longitudes are defined. On the earth, the prime meridian passes through Greenwich, England; for this reason it is also called the Greenwich meridian. The longitude of a point P on the surface is defined as the angle that the plane containing the meridian passing through P subtends with respect to the plane containing the prime meridian. If P is to the east of the prime meridian, the longitude is positive; if P is to the west of the prime meridian, the longitude is negative. Longitude angles can range up to +180 degrees (180 degrees east), and down to -180 degrees (180 degrees west). The +180 and -180 degree longitude meridians coincide directly opposite the prime meridian. It is interesting to note that travelling one degree of arc on the surface of the earth corresponds to travelling L = R θ = (6.4 x 10 6 m) (0.0174 rad) = 112,000 m = 69 mi, where 1 deg (2 π rad/360 deg) = 0.0174 radians
In this lab, the Latitude/Longitude data pair you will obtain for each location corresponds to a point on a two-dimensional plane (the surface of the flat earth). When you have moved to another location (another point), the line drawn from the first location to the second location defines a vector in this plane (on the Earth's flat surface). So your vector calculations will be restricted to two dimensions, not the three dimensions required to define the location of an object far from the surface of the earth, e.g., a satellite. You may find this sample calculation of the distance between two points on Earth useful: DISTANCE FROM CENTER OF PISCATAWAY, NJ TO NEW BRUNSWICK, NJ Piscataway: 40.542374,-74.465332 New Brunswick: 40.489651,-74.452457 Note that your GPS will report to you In Deg-Decimal Minutes. The coordinates above have already been converted from Degrees-Decima. Minutes to Decimal Degrees. Remember that there are 60 Minutes in a Degree, and 60 Seconds in a Minute. You will need to convert your GPS reading to Decimal Degrees. Notice also from the above numbers that latitude (North-South, 40.xxxxxx) changes more than the longitude (East-West, 74.4xxxxx), so it is apparent that most of the motion is North to South (Up to Down) on the map, rather than West to East (Left to Right); see here. So here Y = 40.542374 40.489651=0.05273 and X = -74.465332 - (-74.452457) = -0.012875. The straight line ( as the crow flies ) distance between the two points is then
R= (ΔY ) 2 +(Δ X ) 2 = (0.05273) 2 +( 0.012875) 2 = 0.05428 0 or 0.00095 radians. The distance then of a circular arc formed by this angle, from a distance equal to that of the Earth's radius, is then L=R θ=(6.4 x 10 6 m) x(0.00095 rad)=6063m=3.77 miles which compares favorably with the 5.2 mile estimate for the non- as the crow files curvy car ride, as you can see above. Latitude (y) vs. Longitude (x) map of the United States. Note that we are roughly at (-74, 40). Procedure Vector Here you will walk around campus, stop at certain points, and note down the GPS coordinates of those landmarks. Back at the lab room, you will draw arrows (displacement vectors) between those points and confirm the vectors add mathematically in a two-dimensional plane (the approximately flat earth around campus). Each lab group will be required to leave two forms of ID (Rutgers ID or Driver's License) with your instructor if you borrow the GPS receiver. 0. You can either use the GPS receiver or download a GPS coordinate app to your smartphone; GPS Tour is one such free app for ios; Waypoint is one for Android. 1. Go outside the building where your lab is located. You will need to be outdoors with a
clear view of the sky, or indoors near a window with a clear view of the sky (only if it is raining heavily). 2. Press the red Power button on the GPS unit until it powers on. You should see the message Wait...tracking satellites displayed immediately. After a few minutes, the display will read Ready to Navigate. Accuracy: ft. Wait a few more minutes until this number goes down to around 32 ft. or less. 3. Press the screen change button (under the red Power button) to switch among the screens, until you find the screen that reads Ready to Navigate at the top (below that it looks like a compass dial). If numbers are displayed for Lat and Lon, you are ready to begin your experiment. If you do not see the screen that displays Lat and Lon, press the arrow buttons until it is displayed. Note that the Lat/Lon is displayed in degrees and decimal minutes (DD 0 MM.MMM'). For instance, 40 0 31.392' means 40 degrees and 31.392 minutes of arc. Since there are 60 minutes of arc in a degree, you the above value would be 40.523 0. You will have approximately one hour to walk around the campus and note the coordinates of various landmarks, so pace yourself such that your loop takes you back in time. Consult maps.google.com if you wish. 4. Walk around campus and record coordinates of popular landmarks in your notebook. Be specific about location - Front Entrance (by bus stop), Busch Student Center is much more useful than Busch Student Center. 5. Include five of the following locations and add two of your own. Just make sure you describe the location with enough detail for someone else to find it. Do not repeat any locations: BUSCH Physics & Astronomy West Entrance (facing ARC bldg) Physics & Astronomy East Entrance (facing Lecture Hall) Physics Lecture Hall Front Entrance Middle of Hill Center Engineering Front Entrance SERC Red Statue BPO Entrance (inside) Busch Student Center Front Entrance Busch Dining Hall Front Entrance Ceramic Building Center Three Flag Poles Bus Stop Psychology Statue Golf Field
DOUGLASS Regina Best Heldrich Science Building Voorhees Chapel Mable Smith Douglass Library Main Entrance New Theatre Main Entrance Nicholas Music Center Main Entrance Hickman Hall Main Entrance Loree Gym Main Entrance Carriage House Passion Puddle Floriculture Greenhouses Main Entrance Cooper Dining Hall Entrance Douglass College Center Main Entrance College Hall 6. Press the red Power button on the GPS until it turns off, then go back to your lab room to do the data analysis. Note that your analysis will be performed on vectors whose tail is defined by your initial location, and whose head is defined by your final location - these are vectors in the plane of the "flat" earth. 7. Pick any three locations to be the vertices of a triangle. Draw a vector from location 1 (tail of arrow) to location 2 (head of arrow) and label that vector 12. Do the same for 2 to 3 (23) and 1 to 3 (13). Show numerically that 12 + 23 = 13. Uncertainty A scientist or engineer collects experimental data by taking measurements. You have no doubt collected experimental data in your life perhaps by finding out how much you weigh, how tall you are, or how fast you have run a certain distance. Let's assume that you know your height to be 5'6 (168 cm), your weight 140 lb (64 kg) and your time in the 100 meter dash 12.16 seconds how valid are these numbers? Are you exactly 168 cm tall, and not 168.0625 cm (5'6 1 / 8 ) tall? How does your 140 lb weight as measured on your $20 bathroom scale compare to your weight measured on a calibrated pharmaceutical scale used to weigh the ingredients at a manufacturing plant? To answer these questions, you will need to understand three basic concepts uncertainty, precision and accuracy. You may have used the last two interchangeably in everyday life, but here in the laboratory we will make an important distinction between the two. Precision and accuracy are two different things - you can have data with high precision but low accuracy, and vice versa. Uncertainty: There are two types of uncertainties you will encounter in the lab: Systematic Uncertainties and Random Uncertainties. Systematic uncertainties cause a measurement to be skewed in a certain direction, i.e.,
consistently large or consistently small. For example, weighing yourself repeatedly on a bathroom scale that has has an initial reading of 20 lbs will result in your always appearing to be heavier than you actually are. This form of uncertainty can be removed, once identified in this case you can just zero the scale using the little ridged knob (calibration). Random uncertainties are variations in measurement that linger even after systematic uncertainties have been eliminated. If you weighed yourself on ten bathroom scales around your neighborhood (or weighed yourself ten times on your bathroom scale) over the period of a few minutes, you would probably record ten different values even though you know your weight is essentially not changing. This sort of uncertainty cannot be eliminated but can be reduced by making lots of measurements and averaging. Precision: Measurement A is more precise than Measurement B if the uncertainty in A is less than that of B. If you weighed yourself 10 times on a bathroom scale, the difference between your highest and lowest readings may be as much as 5 pounds. However, at the doctor's office, the same measurement may only yield a difference of perhaps an ounce (1/16 of pound); we say that the latter measurement is more precise, or that a doctor's scale is more precise than a cheap bathroom scale. Precision is affected by random uncertainty. Accuracy: Measurement A is more accurate than Measurement B if the result you get from A is closer to the generally accepted, or standard value, than B is. When you zeroed your bathroom scale, you improved the accuracy of the measurement. Accuracy is affected by systematic uncertainty. Note that you can have a high-precision, low-accuracy measurement, such as a doctor's scale hat hasn't been calibrated. Conversely, you can have a low-precision, high accuracy scale, such as a cheap bathroom scale that happens to yield an average weight close to your true weight.. Below is a representation of the permutations of accuracy and precision:
Can you comment on the precision and accuracy of the darts in the four targets above? MATHEMATICAL DESCRIPTION OF RANDOM UNCERTAINTIES The first concept is the average (arithmetic mean) of a number of measurements: x= N x i i=1 N (1) This formula says: Add the N measurements x 1, x 2, x 3, etc., up to x N. This sum is N written as x i. Now divide by N to get the mean value of x. Another important i =1 statistical parameter is the standard deviation, σ, which is a measure of the spread of the individual data points. A large value for σ implies a large spread in the values whereas a smaller value implies a small or narrow spread. We call (calculated from a set of measurements as shown below) the standard deviation for repeated individual measurements, x i. The standard deviation σ of the mean x of measurements gives the spread in the means of repeated sets of measurements x 1, x 2, x 3,..., x N. Mathematically, these are different. σ= N ( x i x) 2 1 ( N 1) (2a) σ= N ( x i x) 2 1 [ N ( N 1)] (2b) (Some definitions use N for σ instead of N-1 which is more accurate.) A bin plot (histogram) of the number n i of individual measurement differences from the mean (in units of the standard deviation σ ), has a bell-shaped normal distribution.
The area under the normal curve centered around the mean between (x + ) and x ) is 68.3% of the total area under the curve. This means that about two-thirds of the measurements, xi, fall between (x + ) and (x - ). The area between (x + 2 )and (x - 2 ) is 95.4% of the total area. And, ±3 covers 99.7%. The standard deviation is used as a measure of the random uncertainty expected for an individual measurement. We write 15.5 ± 0.2 m for the mean and spread of individual measurements. This implies that the probability is about 66% that the true value lies between 15.3 and 15.7. However, in most science and engineering applications, you will need to calculate x±, the mean and the standard deviation of the mean. To illustrate the procedure we will work out the mean value x and the standard deviation σ of a set of 21 individual data points, and then the predicted uncertainty,, of the set s mean. What does all this mean? In Psychology, someone with an IQ score within 1 of the mean is said to have average intelligence. Those scoring above +1 are considered to have above-average intelligence. Anything over +2 is termed gifted. For example, if the average IQ is 100, and the standard deviation is 15, then the average category consists of people scoring between 85-115 (68.3% of the population). Those scoring above 115 ( +1 ) comprise 2.3% of the population notice that this is half of 100% - 95.4% = 4.6% because we are only considering the part of the population with above-average intelligence. CALCULATION OF MEAN AND STANDARD DEVIATION USING INDIVIDUAL LENGTH MEASUREMENTS
x i ( x i x) ( x i x) 2 15.68 0.15 0.0225 15.42-0.11 0.0121 15.03-0.50 0.2500 15.66 0.13 0.0169 15.17-0.36 0.1296 15.89 0.36 0.1296 15.35-0.18 0.0324 15.81 0.28 0.0784 15.62 0.09 0.0081 15.39-0.14 0.0196 15.21-0.32 0.1024 15.78 0.25 0.0625 15.46-0.07 0.0049 15.12-0.41 0.1681 15.93 0.40 0.1600 15.23-0.30 0.0900 15.62 0.09 0.0081 15.88 0.35 0.1225 15.95 0.42 0.1764 15.37-0.16 0.0256 15.51-0.02 0.0004 326.08 m (-0.05 m) 1.6201 m 2 From the above table we can make the following calculations for N = 21 measurements x= N x i i=1 N =326.08 21 =15.53m N i=1 (x σ= i x) 2 ( N 1) = 1.6201 =0.29 m 20 N i=1 ( xi x) σ= 2 [ N ( N 1)] = 1.6201 [(21)(20)] =0.062m Hence x±σ = 15.53m ± 0.062 m. This says the average is 15.53m, which average has an uncertainty of 0.062m. But, the uncertainty or spread in individual measurements is σ = 0.29m. Remember, when you calculate a non-zero value for σ or σ random uncertainty is present; the values of tell you how large the magnitude. Increasing the number of individual measurements reduces the statistical uncertainty (random uncertainties); this improves the "precision". On the other hand, more measurements do not diminish systematic uncertainty in the mean because these are always in the same direction; the "accuracy" of the experiment is limited by systematic uncertainties. Often we must compare different measurements. Consider two measurements A±σ A B±σ B
If you want to compare, say, their difference, the expected uncertainty σ is given by σ 2 =σ A 2 +σ B 2 The reason we sum the squares is that we must sum the squares of the variances ( x i x) 2, in order to get the standard deviation. (Look again at the above equations for the standard deviation.) We should expect that A B < σ, if the two data sets are statistically the same. If they diverge greatly from the expected standard deviation, A B > σ, they are then statistically different. Often, a value of 2σ is used as a simplified reference for being significantly different. Procedure (Reaction Time) What's this all about: here you will measure the time it takes for you to respond to a visual (or aural) cue. You will do this repeatedly, find your mean, minimum and maximum reaction times, plot the distribution of reaction times, and note the spread (standard deviation) of the data, as well as the standard deviation of the mean, which is a measure of how reliable the mean value is. For reaction time data, each partner should determine and compare her/his reaction time distributions and their means for the Space bar key on the keyboard. 1. Open Stopwatch a program in the Lab Apps folder on the Mac's desktop. Click on Reaction time; Click Start. Each time you hear the ding, hit the Return (Enter) key to stop the timer. The time in seconds between the signal and your response will be recorded. 2. Practice; select Stopwatch to stop, then Clear record. Start Reaction Time, click Start and collect data for five minutes. When you are finished with your last button press, Click on 'Stop watch' in the program so that it will no longer run the timer. 3. Select reaction times you have just generated by clicking and dragging with the mouse, until you have gotten all the points. Once they have been highlighted, copy them to the Mac clipboard (first part of copy and paste ) either by using Command-C (or right-clicking and choosing Copy). 4. Disconnect the power cable to your LabPro interface (the green translucent flat box on your lab table into which are plugged multiple cables). Open Logger Pro, also in the Lab Apps folder. You should see a gray-outlined data table on the left, and an empty graph on
the right. You should see two data columns: X and Y. Double-click on the X heading, which brings up a windows called Column Options. Type in Trial for Name and check the Generate Values box; click Done. Double-click on the Y heading and change the name to indicate which lab partner did the trial (for example, Lisa ), then click Done. Click on the first cell of this column and paste (Ctrl-V or right-click Paste) the data you copied from Step 3. 5. Data is automatically plotted in the graph window (Lisa vs. Trial). If you have any outliers (points which are obviously bad, such as those resulting from not being ready to click the button), remove them from the data column by highlighting the cell and pressing the Delete key. Go to Analyze-->Statistics to determine the Minimum, Maximum, Mean and Standard Deviation. Note that this Standard Deviation is and not, but that you can get the latter from the former with minimal calculation by carefully examining the mathematical definitions for both. Record these values in the hand-in sheet. 6. Create a histogram, which is a plot of 'Frequency of Value' vs. 'Value'. Click on the existing graph window (so that the new histogram window you are about to create will have the same width) and select Insert-->Additional graphs-->histogram. Double-click on the middle of the new blank graph and check the quantity (for example, Lisa) you wish to plot. Ideally you should see a bell curve shape if your histogram looks too blocky, decrease the Bin Size (the resolution of the histogram) by double clicking on the histogram itself, then selecting the Bin and Frequency Options tab and entering a lower value in the Bin Size box. If you see bins with zeros in them, enter a higher value of Bin Size. 7. To check that your reaction time data's distribution indeed resembles a Gaussian, or bell curve, you will need to pick a function that fits the data this is called Curve Fitting. Go to Analyze-->Curve Fit-->select Gaussian in General Equation and click on Try Fit. If for some reason this yields a straight line as the fit curve, you may have to help the fitting program as follows: a) Increase bin size if hist-data is too widely scattered. b) Manual fit, to provide reasonable starting parameters for the auto fit. Input estimated A (maximum height), B (center of peak) and C (width), D (vertical offset). Adjust these parameters for visual fit. Then switch to automatic and Try It. If the D parameter remains, the auto fit can sometimes do really stupid things, such as finding a large positive A and a large negative D, with the difference being around your manual fit starting value for A, or giving a horizontal line fit.. If so, try getting rid of the D parameter, then c) Define function - choose Gaussian, then remove D parameter (vertical offset), manual fit, switch to auto and Try It. 8. The other lab partner should repeat Steps 1-7. If you are working alone, repeat using your non-dominant hand, which will almost certainly yield longer reaction times. Print
out only one curve-fitted point plot from either partner, making sure it is properly labeled (Name, Lab Partner's name, Section, Date). 9. Plug the power cable back in to the Lab Pro interface. Partner 1 Times x x MIN x MAX σ Partner 2 Times x x MIN x MAX σ Calculate the difference in the reaction times of you and your lab partner. Estimate the uncertainty in the difference. (HINT: You can determine σ from σ if you can find a simple relationship between their mathematical definitions). Questions Were your and your lab partner's reactions times statistically similar or different? Explain. What criteria did you use?