Science is the organization of apparently unrelated facts into a useful whole. Science goes beyond verifying what we understand. We use science to challenge what we "know" and develop methods to prove or disprove our hypothesis.Physics is that branch of science that serves as the foundation for all other scientific and engineering disciplines. Laboratories for physics are the classrooms where the relationships of physics, and science in general, are illustrated in a manner that encourages and even depends upon your participation. It is the goal of the physics laboratories to introduce you to the relationships of physics. Lab exercises will provide the hands-on experience necessary to appreciate the methods of science and scientific discovery. In order to understand physics, and science in general, you must be involved in the laboratory.
The physics laboratory is designed to do more than simply illustrate proven relationships. Its ultimate goal is to provide you with the tools and experience necessary for you to pursue your own discovery. In these labs you will explore some of the fundamental concepts of physics. More importantly, you will be introduced to some of the tools of science and experience the methods of scientific discovery.
This manual contains lab exercises based on class discussion. Each exercise requires a certain level of familiarity with the physical concepts to be studied. Study the exercises before coming to the lab and review any additional material as may be necessary. Each exercise is designed and written to minimize the amount of outside reference material required. However, a careful reading of the material covered in lecture will compliment the laboratory experience. Each exercise contains the following: .
- Overview of the physical concept to be explored
- What you should achieve and learn having successfully completed the lab
- A list of required equipment
- A step by step procedure complete with all necessary data tables, illustrations, and room for calculations
- Analysis designed to reinforce and review what you learned
The magnitude or greatness of a value, properly recorded to indicate the appropriate significance of the measurement, is not yet a true physical quantity. In order for your data to have true significance it must also express what the quantity is physically. This is done by assigning each value measured an appropriate unit. In these labs you will be working within the metric system, or System International (SI). In this system the three basic units of length, mass, and time are the meter, kilogram, and second, respectively. Additionally we will be using the ampere for current, volt for potential difference, and the Kelvin or degree Celsius for temperature.
The mks (meter, kilogram, second) convention is simply a matter of convenience when measuring and reporting physical quantities. An alternate convention used in some labs are the cgs (centimeter, gram, second) units. The decision to use either the mks or cgs convention is actually determined by the relative magnitude of the quantities measured. At times it is simpler to work with centimeters and grams than it is to use the meter and kilogram. However, once a decision is made to use either the cgs or mks convention, it is crucial that you remain within that convention. The data tables for each exercise are annotated with the recommended unit of that particular entry. Any additional discussion of the choice of units for a particular lab will be detailed in the procedure section of that lab exercise.
At times it may be necessary to record a measurement using kilograms as your choice of units but the value itself is on the order of a hundredth of a gram. You write in your data table .00017 kilograms. Although the representation of this number is entirely correct, it does get rather cumbersome carrying around all of those zeros. The solution to this is to represent the value in scientific, or powers-of-ten, notation. To express your value in this manner, insert a decimal point after the first non-zero figure then multiply by a power of ten to locate the true decimal point. In locating the true decimal point, count the number of places that the decimal point was moved in order to represent it in scientific notation. In our example, we count four places to the right. Our value of .00017 kilograms would then be expressed as 1.7 x 10-4 kilograms. A value of 17000 kilograms would become 1.7 x 104 kilograms. If you move your decimal point to the right your exponent is a negative number, to the left the exponent is a positive number. Another advantage to scientific notation is the ease in which significant figures are determined for your data. When using scientific notation, you only write those values that are significant. Our value of .00017 kilograms only has two significant figures, therefore we express it as 1.7 x 10-4 kilograms. A more detailed discussion on significant figures will be given in a subsequent section.
Using the appropriate units and recording your data, is simply a matter of noting the value you read from the instrument. Read the instrument to the resolution of the tool. Keep in mind that your instrument needs to have sufficient sensitivity to give meaningful results. You need to be careful and consistent to maximize the precision of your measurements. You must use the proper tool, calibrated to be as accurate as possible. It's that simple. If you apply these concepts every time you take a measurement and write a value with its appropriate units in the data tables, every lab exercise can be successfully completed.
After obtaining all data, you are ready to start evaluating and determining whether or not it agrees with the expected results. You will also be trying to determine how well your data are in agreement with what you expected. Determining how well your data are in agreement to some known standard and assigning a mathematical value to that agreement is called quantitative analysis. However you need to do a qualitative analysis first.
Qualitative Analysis
is asking yourself if the data you just measured or the value you just calculated makes sense. For example, try asking yourself "is this about what I expected to get?" If you are measuring the velocity of a small child riding a bike and you get a value of 29 meters per second (65 mph), something is probably wrong. If, however, that child is in a car traveling down a highway, 29 meters per second may be reasonable. If you are not sure that the measured or calculated values are reasonable, ask your instructor. Qualitative analysis is a good way to catch mistakes in the experimental setup and data collection to avoid repeating the entire experiment.
Quantitative Analysis
is where you statistically determine how accurate and precise the data are. The extent to which you analyze the experimental results will normally be stated in the procedure section of your lab. This may be as simple as finding a percent difference from some known or standard value. It could involve comparing your experimental error to the error you might reasonably expect from your setup and available equipment. Regardless of the extent to which you analyze your data, it all begins with proper use of significant figures.
In scientific work, most numbers are measured quantities and are not exact. All measured quantities are limited in significant figures (SF) by the resolution of the instrument used to make the measurement. The measurement must be recorded to show the resolution of the instrument - no more, no less. Calculations based on the measured quantities can have no more (or no less) precision than the measurements themselves. The results of your calculations must be recorded to the proper number of significant figures. To do otherwise is misleading and improper.
It is important to be aware of some of the sources of experimental error and some basic statistical methods of determining the value or values that best represent the system in question. Sources of error that affect the outcome and results of your lab normally result from one of two general categories: random error that you have little control over but which can be addressed using statistics, and systematic error that you either can control, minimize or measure.
Random Error
is the error that occurs at the limit of the resolution of your instruments. It is random error that necessitates the proper use of significant digits. Every instrument has a limit as to how well it can record a value. When listing a value to three significant digits in data tables, you are making a statement that implies you are unsure of what the next figure should be. Is my last reported value close to being rounded up or down? On average you are just as likely to be at one end of the scale as the other. The way we address this issue is by taking an average of several values. The Standard Deviation of your values will determine the ÒspreadÓ or variance of your data.
Systematic Error
is the error we create. Systematic error includes such things as consistently reading a scale too high or too low, having an instrument not calibrated or set properly, trying to use an instrument outside of its range of sensitivity, improper experimental setup, or poor assumptions as to whether the setup will illustrate the concept that you are trying to explore. Systematic error can be minimized by the proper use of equipment. Each time you use a new piece of equipment, make sure it is properly calibrated. The best and easiest way to calibrate an instrument is to make sure it reads 'zero' when it should. If zeroing the instrument is not appropriate, you might try checking the instrument against some known standard. To minimize a personal bias to read an instrument too high or too low, let different people read the measurement. Vary your method. When measuring the length of an object, do so at several different locations. While systematic error may still be present, you can minimize it so that it is smaller than your random error.
The distinction needs to be made at this point between "error" and performing the experiment improperly. When we talk about error, we are referring to the errors that affect our data at the limit of our ability to use the equipment or at the limit of the resolution of the equipment. You might be tempted to say my error was that, "I did something wrong." Some call this "personal" error. Granted, personal error will certainly affect your final results. However, it should be corrected as soon as it is discovered and that portion of the experiment repeated.
You also need to know how to compare an experimental value to either another experimental value or to some known or accepted value. When comparing an experimental result to another experimental result, calculate a percent difference. The percent difference is the absolute value of the difference between the values that you are comparing divided by the average of the two values,
percent difference = | (value1 - vlaue2) | /average x 100%.
When comparing your experimental results to some known or accepted value, we will calculate a percent error. The percent error is the absolute value of the difference between the two values divided by the accepted value,
percent error = [(exp value - accepted value) ] /accepted value x 100%.
The difference in these two methods is that with the percent error we know or at least have a much better understanding of the accuracy of the accepted value. We know the value well enough so we can say it is essentially exact in comparison to our experimental value. Therefore, we can calculate an absolute deviation. When calculating a percent difference, we have no reason to believe one value is any better or worse than the other. Therefore, we will find the deviation from the average of the two values which should better represent the actual value.
In physics one of the best methods for visualizing and interpreting data is with the use of graphs. Graphs allow you to transfer data from a table onto a scaled, clearly labeled grid. The primary purpose of graphing data is to understand better the functional relationship between variables. Once your data is properly graphed, it is simple to predict an approximate value for one variable given you know the other, or to verify extreme relationships between your variables.
Most physical relationships in nature can be graphically analyzed using one of three different types of plotted curves. Fortunately, all three of these can be graphed as a linear relationship with only minor mathematical manipulation. They are:
Before you start graphing your data, it is important to properly scale and label your graph paper. Plot the independent variable along the x-axis and the dependent variable along the y-axis. For example, our relationship y = mx + b has the form such that y is our dependent variable and x is our independent variable. Is the distinction between independent and dependent variables just a matter of how our equation is expressed? Could we have rearranged the equation where x was dependent on y? Yes, but the distinction between dependent and independent variables is actually a consequence of what you can actually measure in the laboratory. For the relationship y = mx + b, we were evidently able to vary x and measure y. Therefore, y is dependent on x. Once you have decided which variable belongs to which axis, you must choose a scale.
There are two important steps in scaling your graph, first, use a scale that allows your plotted curve to fill as much of the graph paper as possible. Secondly, use a scale that is convenient and easy to read such as one that increases as a factor of either 1, 2, or 5 of your units of measure.
Example:Your data ranges from 1.103 meters to 2.047 meters. The graph paper is divided into twenty, evenly spaced grids. If you subtract your two extreme data points, your values encompass a range of .944 meters. This value evenly distributed among all twenty divisions of your graph paper would give .0472 meters per grid, a difficult number with which to work. Make the graphing simpler by rounding up the .0472 to the next multiple of 1, 2, or 5. In this case you would choose .05. Label your graph axis starting with 1.10 meters and continue in increments of .05. Your next value would be 1.15, 1.20, and so on. Once you have plotted all your points, be sure to label all axes with the appropriate units, give your graph an appropriate title, and include any additional scaling information as required.
In the linear relationship y = mx + b,
The slope of your graph is the ratio of the change in the y values to the change in the x values. The slope of your graph is determined from the "line of best fit" that you draw on your graph paper. The line of best fit is a straight line you think best depicts the average of your values.
The power relationship is difficult to graph and difficult to work with in the form y = ax^m
However, if you take the Log of both sides of the equation you obtain;
Log(y) = Log(a^xm)
rearranging gives
Log(y) = m Log(x) + Log(a).
The third of our relationships, the exponential relationships, can be handled by taking the natural log of the equation y = be^mx;
ln(y) = ln(be^mx)
rearranging gives
ln(y) = ln(e^mx) + ln(b)
ln(y) = mx + ln(b)
In this form, a plot of ln(y) versus x would give a linear graph with slope m and y intercept of ln(b). A graph of this type is a semi-log graph.
Becoming proficient at graphing your data and properly interpreting the results is a matter of practice. The only way to truly understand and appreciate the usefulness of graphical analysis is to work your way through some labs. The most important point of any graphing exercise is to be neat. When graphing, clearly label your axis and name your graph. When labeling your axis, use the appropriate units as well as magnitudes. The "line of best fit" is really just an approximation of what you think the average values represent. The slope that is determined from your "line of best fit" does have physical significance and must be calculated with the units.