M1 - Theory and Procedure

Experiment M1 Purdue University - Physics 22000 M1-1 © Purdue University 2001-22 Lab 1: Measurements and Experimental Errors (M1) The goal of an experiment is usually to measure something, such as the mass of an object, the position of an object or the velocity of a projectile. When the measurement is finished you have new knowledge. However, real measurements cannot be done without experimental error . It could be very small, but it is always present. To be able to compare the results of different measurements with each other, or to compare them with a theoretical model (like the ones we will be discussing in class), it is essential that we have a way to estimate the uncertainty, i.e., the experimental error , associated with our measurements. In this lab, you will collect, analyze experimental data and calculate the experimental error. This lab will also introduce you to the computerized, ultrasonic motion sensors that you will use in many of Physics 22000 experiments. You will use them to measure position of various objects. Objectives:  Become familiar with the operation and use of the motion sensors and the associated software.  Examine motion by creating distance (position) vs. time graphs.  Discover the relationship between position vs. time and velocity vs. time graphs.  Learn about how to calculate experimental results using the average value and the sample standard deviation. Equipment: You have the following materials at your disposal.  A computer equipped with the PASCO motion sensor hardware and software. Descriptions of Motion • Position : the location of an object with respect to the origin (the motion sensor is the origin ( x = 0) from which distances are measured). • Average Velocity : the displacement (change in position) of an object during an interval divided by the time that elapsed during that interval.

M1-2 Experiment M1 Purdue University - Physics 22000 Velocity is a vector. It has speed (how fast something is moving - the length of the velocity vector) and direction. Average velocity for a constant speed is equal to the slope of the position as a function of time graph for that time interval. Therefore, for a faster velocity, the slope of the position vs. time graph is more inclined (has a larger slope) than that for a slower velocity. • Instantaneous velocity : indicates how fast an object is moving and the direction of the motion at each instant of time. Remember: instantaneous velocity does not always equal average velocity. For example, suppose a person takes a road trip in the car and travels the distance of 60 miles in one hour. The average velocity for that trip is 60 miles per hour. However, at a given instant in time, the instantaneous velocity could be 70 miles per hour, 50 miles per hour, -40 miles per hour or any other value, so long as the average of all the instantaneous velocities is 60 miles per hour. For one-dimensional motion, the sign (+ or -) of the velocity indicates the direction of the motion. +v = moving away from the origin (the motion sensor). -v = moving towards the origin (the motion sensor). The motion detector is the origin ( x = 0) from which all distances are measured. Procedure: Activity 1: Introduction to Ultrasonic Motion Sensors The ultrasonic motion sensors work by sending out ultrasonic pulses and measuring the time between moments when a pulse is emitted and when its reflection returns to the sensor. The ultrasonic is identical to ordinary sound, except that its frequency is too large to be heard (we use 50,000 Hz in this lab). The motion sensor acts as a transmitter, when it sends out pulses, and as a receiver, when it listens for echoes. The software computes the delay ∆t between sending the pulse and receiving the echo. Then, using the speed of sound in air ( v sound = 343 m/s at 20ºC = 68ºF; the speed of ultrasound is the same as the speed of audible sound waves), the software determines the distance x = ½*∆t*v sound from the detector to the object that reflected the pulse. The factor ½ is there because the ultrasound pulse must travel the distance twice – from the sensor to the object and back. Usually, the distance is plotted as a function of time. Some animals, e.g., bats, use the same method of ultrasonic pulses to measure distances. Each time the motion sensor sends out an ultrasonic pulse it also makes an audible sound. This clicking sound is what you hear when the motion sensor is emitting pulses.

M1-4 Experiment M1 Purdue University - Physics 22000 When you are ready to start collecting data, click on the button located below the graph. In most activities, the computer will stop collecting data automatically after the preset time (25 sec in this case). It usually takes a few attempts to get a reasonably good match. When you are happy with the result, print the graph by selecting “ Print” from the “ File” menu. Attach this printout to your lab report. Do not save any changes! Activity 2: Prediction of Velocity from the Position vs. Time Graph. Open the document for Activity 2. You should see two graphs: the position versus time (top) and velocity versus time (bottom) graphs. Click on the button to start data recording. Keep looking at the computer monitor and move to match the position vs. time graph. It is the same pattern as in Prelab Question 1.

Experiment M1 Purdue University - Physics 22000 M1-5 Using the position versus time data, computer will create the velocity versus time graph. Examine that graph and if you have a good match print it. If you do not match the provided pattern, simply try it again. Print a copy of your best run. Next, select Print from File menu. Label the graph: "Velocity vs. Time from a Position vs. Time Match". Activity 3: Average Value and Standard Deviation Experimental errors. In most areas of science and technology, you must be familiar with taking measurements and calculating results based on those measurements. You must also be able to understand the accuracy of your measurements and determine the uncertainties in your results. In many experiments you ever perform (including this one), you will make multiple measurements of a quantity, and will then want to combine them to get a final value. The key questions are: (a) How should we combine multiple measurements to obtain a single “best'” value? (b) How should we estimate the uncertainty (or “experimental error”) in this best value?

Experiment M1 Purdue University - Physics 22000 M1-7 s x = x i - x ave ( ) 2 i = 1 N å N - 1 = x 1 - x ave ( ) 2 + x 2 - x ave ( ) 2 + x 3 - x ave ( ) 2 + x 4 - x ave ( ) 2 + x 5 - x ave ( ) 2 4 (2) Statistics predicts that for large N values, the chances are about 68% that x average will differ from the exact answer by an amount equal to s x or less (this assumes that we are dealing with random errors , as we will explain in a moment). It is customary to write our result as 𝑥 = 𝑥 ௔௩௘௥௔௚௘ ± 𝑠 ௫ ( units ) In our example above, we have: 𝑥 ௔௩௘௥௔௚௘ = 1.010 m and 𝑠 ௫ = 0.011 m Therefore, the result should be written as: x = 1.010  0.011 m. There is one important, though obvious, point we should make about the use of s x from Eq. (2) as an estimate of the experimental uncertainty (experimental error) in x . We have assumed implicitly that there is no “bias” in our measurements. In other words, the probability that a value of x i will be larger than the perfect value is the same as the probability that it will be smaller than the perfect value. Therefore, the average – Eq. (1) will provide a good estimate of x perfect . Errors of this kind are called random errors since their sign (positive or negative) is random. There is also a second kind of error known as systematic errors . Systematic errors are instrumental or methodological errors causing consistent deviation of results in one direction from the true value. One very typical example of measurement with large systematic error is a scale that was not properly zeroed. That scale would show values that are systematically smaller (or systematically larger) value than the true weight of measured objects. Another example is if the temperature in the room (and consequently the speed of sound) would be significantly lower than the usual 20˚C (= 68˚F). Computer needs the exact value of sound speed to properly calculate distance: x = ∆t*v sound . It would then provide values of x i , which always tend to be larger than x perfect , and the average of these x i values could differ from x perfect by much more than 𝑠 ௫ . The only way to cope with systematic errors is to carefully analyze the way your experiment works, to understand how it might be biased and by adjusting equipment. The graphs below illustrate both random and systematic errors.

M1-8 Experiment M1 Purdue University - Physics 22000 Very close agreement between trials, but systematically off the target. Widespread, random variation of results. Small random error, but large systematic error. Large random error and small systematic error. The best situation, i.e., small random error and small systematic error. It is possible to reduce the systematic error by meticulously designing the experiment and adjusting experimental apparatus (for example, zeroing scales). The random error can be reduced by repeating measurements or by getting better equipment. In case of multiple measurements, we use the average value to reduce the random error. Standard deviation provides a good approximation of the random error. However, no matter how good is our apparatus and no matter how many times we repeat measurements the random error cannot be completely eliminated . A similar statement is also true for systematic error. Even the most careful calibration would not eliminate the systematic error. Calibration procedure would reduce or minimize the systematic error, but it cannot eliminate the systematic error. The total experimental error depends on the larger od the two types of errors - random or systematic. Open “M1 Activity 3” by clicking on it. The application program: “ Capstone ” by PASCO Scientific will configure for Activity 3 . In this activity, you are going to make multiple distance measurements for an object that have a fixed position using ultrasonic sensor. Since the sensor measures the distance several times per second, we should get many data points in relatively short time. Then you will export data to Excel spreadsheet and calculate the mean (average) value and the standard deviation. It is possible to do using a calculator instead of Excel, but that would be very time consuming. The ultrasonic motion detector makes 10 measurements per second, so after 60 sec of measurements, you should have 600 data points. Entering those manually into a calculator would be very tedious.