Your completed report should include the following sections:
Your abstract must contain all four of the above elements: theory, activity, findings, and agreements. You will have to examine your laboratory activity carefully and decide which data and calculations are important enough to be included in your abstract.
An example of an abstract describing one of the experiments you will do is given below:
We tested the validity of Newton's First and Second Laws of Motion. The acceleration of a glider on an inclined air track by the earth's gravity was measured as a function of glider mass. The motion was measured with an ultrasonic ranging device. We observed that the acceleration was 9.8 ± 0.05 m/s2, independent of object mass, and consistent with the currently accepted value for the gravitational acceleration at sea level. We studied the relationship between force, mass, and acceleration by allowing a gravitationally accelerated object to exert a force on the glider. Our results show a linear relationship between force, mass, and acceleration, and are consistent with Newton's Second Law of Motion.
All of your data and calculations must appear in your laboratory notebook. Computer generated graphs should be taped into the notebook. (It's a good idea to make two copies of important plots, one for your report and one for your notebook.) Your teaching assistant will periodically check your notebooks as you are taking data and may, in some cases, ask to see your notebook in order to corroborate the data in your report.
Your laboratory report will in general contain only a representative fraction of your data and graphs. These should be printed from your computer screen or copied from your laboratory notebook. Note: your laboratory notebook should remain intact---do not tear your data sheets out of your notebook! Clearly mark on any plot in your report which regions of data you actually used in making any calculations or drawing any conclusions. Mathematical calculations should be concise: briefly and succinctly show the derivation of formulas you use, or state the source of the formulas, and then show tables or plots that summarize the application of the formulas to your data.
An example of part of a data analysis for the inclined plane experiment is shown below. The highlighted areas indicate which parts of the graphs were used for least-squares fitting. Note that the areas chosen avoid noisy regions of the data and that we do NOT need to show every possible plot we have considered. If you repeat a measurement several times to assess the uncertainty in a measurement, show one representative plot but state uncertainties and indicate how you obtained them. The numbers below are those obtained by one experimenter; your measurements for the same experiment may differ. Finally, it is sometimes useful to include simple sketches of the experimental setup even though none have been included in the sample below.
We measured the acceleration due to gravity of a glider on an inclined airtrack. The mass of the glider (including the reflecting flag) was 0.218 ±\ 0.001 kg. The separation of the supports under the glider is 1.0 ± 0.001 meters. The inclination angle was fixed by placing one or more spacers (each of height 0.008 meters) under one of the airtrack supports. We used the ultrasonic ranger to determine position (and velocity) of the glider as a function of time; the acceleration is the slope of the velocity vs time graphs. Four measurements (referred to as trials in the table below) were made for each angle of inclination. The uncertainty in the measurement of track elevation and track length was 0.001 meters (Note: The table below is typed, but you may handwrite yours if you wish).
| Trial | Height (meters) | Acceleration (m/s2) |
|---|---|---|
| 1 2 3 4 |
0.008 |
0.077 0.080 0.076 0.080 |
| 5 6 7 8 |
0.016 |
0.158 0.152 0.160 0.161 |
Representative graphs (trial 5) of distance and velocity as a function of time are shown below.
From the data shown, we find that the average accelerations for the 0.008 meter and 0.016 meter heights are 0.0785 ± 0.002 m/s2 and 0.158 ± 0.003 m/s2, respectively. The uncertainty is estimated from the spread of the data points. We compute g from the acceleration data using the relation a = g sin(theta), the acceleration of an object on a frictionless incline at angle theta to the horizontal. The angle of inclination, theta, for height, h, and 1 meter separation of the airtrack supports is theta ~ tan(theta) = h/1 = h. (The small angle approximation is valid here.) Our uncertainty on the measurement of theta is derived by summing the percentage errors in our length measurements. The percentage error for h and for the nominal one meter length used for angle determinations were 10% and 0.1%, respectively. Hence, the percentage error in theta is about 10%. Our values for g for each trial are:
| Trial | Inclination angle (radians) | value of g (m/s2) |
|---|---|---|
| 1 2 3 4 |
0.016> |
9.625 10.00 9.500 10.00 |
| 5 6 7 8 |
0.016 |
9.875 9.500 10.00 10.06 |
Instead of propagating errors mathematically, we averaged our results and estimated the error from the spread in values around the average. For the data shown in the table, we find
We next added mass to the glider in increments of 100 grams and again made several measurements of acceleration for each of our two heights. The analysis of these trials follows exactly as described above for the unweighted glider. In each case, the uncertainty on the glider mass is 0.001 kg and the uncertainty on the determination of g is 0.2 m/s2, or about 2% fractional error. Our results for all measurements are summarized below:
| Trial | Glider mass (kg) | value of g (m/s2) |
|---|---|---|
| 1 2 3 |
218 318 418 |
9.82 9.80 9.79 |
We discuss the significance of these results in the next section.
Your analysis should also describe difficulties and shortcomings that you encountered during your experiment. You must suggest experimental redesigns to eliminate and overcome these troubles. Now note that there is a subtle point here. Some uncertainties come about because of limitations in our measuring apparatus. A meter stick can only give you length measurements to about a millimeter or so, no matter how skillfully you use it. We can only repeat measurements so many times in order to determine the best average value. We lump these kinds of uncertainties into the term {\it statistical} error. Other kinds of uncertainties come about because we are not able to completely eliminate external influences. For example, we use an airtrack for the glider because we want to eliminate friction, but we can't eliminate air resistance. The same flag which reflects ultrasonic pulses so well also makes a good parachute for our glider. We need to think about how these kinds of external influences, often referred to as systematic errors or uncertainties, might fool us or limit the accuracy of our experiment.
You should identify and give real or estimated numerical sizes for the most important sources of uncertainty in the activity. It is important that you provide evidence for these sources of uncertainty and that you not simply guess wildly at possible sources. Use numbers and observations to justify changes to the activity. Discuss how the apparatus could be redesigned to reduce uncertainties or to streamline experimental technique. Also discuss how the measurement methods themselves can be changed to reduce or eliminate both statistical and systematic uncertainties. A small diagram might be appropriate when describing the exact situation.
We are interested primarily in shortcomings in measurement techniques and not in your personal errors (i.e. mistakes). However, if a particular method encourages personal errors, you should describe this as well.
Your analysis section must contain the following three elements: major sources of uncertainty and true or estimated numerical sizes for them, suggested changes to the apparatus to reduce or eliminate uncertainties, and suggested changes to the measurement techniques to reduce or eliminate uncertainties. You must identify the most important portions of your activity and analyze these, as you cannot hope to discuss everything in the space provided. The important thing is to use numbers wherever you can to justify courses of action.
For example, the analysis of an experiment to measure g with an inclined airtrack might go as follows:
The best estimate we have found for the value of g in Philadelphia is 9.802 ± 0.001 m/s (reference Physics by Eugene Hecht, pg. 73). According to Newtonian theory, this value should be the same for all objects regardless of mass. Our result for three different glider masses shows that g is independent of mass and are consistent within their 2% uncertainty. The average value of g from all three masses is 9.80 ± 0.01. The uncertainty on g is obtained by forming the quadratic sum of the uncertainty on each measurement of g (0.02 m/s2). Our results are consistent with the value given in Hecht. There are several sources of uncertainty in the determination of the gravitational acceleration: the measurement of lengths with a meter stick; 2) the intrinsic measurement accuracy of acceleration with the MacMotion program; and 3) the influence of air resistance on the glider. The measurement of the length between the airtrack supports is appropriate for the meter stick since the length to be measured is comparable to stick length and the fractional error is small. However, using the same stick to measure the small height of the blocks used for inclination leads to a much larger fractional error. These lengths would be better measured using calipers or a micrometer. The uncertainty in the height accounts for essentially all of the uncertainty on the determination of the inclination angle. Since the angle stays fixed for several trial measurements of the acceleration, this error does not play a role in determining the spread of acceleration values.
The uncertainty in the determination of acceleration accounts for essentially all of the spread in our values of g. This spread is presumably partially due to actual small differences between the trials and partially to limitations in the instruments and software. Fitting various parts of a section of good data and repeating this for several data plots indicate that we have 2% uncertainty in our acceleration measurements, which leads to the 2% spread in our determination of g for a fixed incline angle. Some method for directly determining velocity, e.g. with direct timing measurements over short intervals might reduce this uncertainty.
Finally, air resistance needs to be reduced. Air resistance gives us lower values of acceleration than we would otherwise measure, hence it should tend to push our measured values of g lower than the true value. It is difficult to measure the extent of this effect. We see little evidence for it in our measurements, presumably because air resistance increases as the speed of the glider increases and all our velocities for small inclination angles are small. This effect might be minimized by switching to an ultrasonic reflection flag which is streamlined.
Your conclusions require these two elements: examples where the theory and methods are useful or relevant, and modifications to the laboratory techniques and apparatus that describe its application. Again, due to space restrictions, you must discriminate and discuss only key ideas. Your TA's will probably give higher grades for more original ideas expressed in the conclusion. An example is given below for the inclined airtrack experiment.
The determination of the independence of gravitational acceleration on mass or content of a falling body represents a major achievement in scientific thought. It allows the use of gravitationally accelerated objects to be used in testing concepts of inertia and the relationship between force, mass, and acceleration. In addition to verifying Galileo's original observation, our experimental technique can be used to measure g to about 1% in any local environment. We have suggested methods by which even more accurate values of gravitational acceleration may be determined.
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