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Doing Experiments in Physics. Shirish R. Pathare. Homi Bhabha Centre for Science Education Tata Institute of Fundamental Research V. N. Purav Marg , Mankhurd . Mumbai – 400 088. shirish@hbcse.tifr.res.in. Experimental Work in Physics.
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Doing Experiments in Physics Shirish R. Pathare HomiBhabha Centre for Science Education Tata Institute of Fundamental Research V. N. PuravMarg, Mankhurd. Mumbai – 400 088. shirish@hbcse.tifr.res.in
Experimental Work in Physics We can think of experimental work in physics as consisting of the following steps: Conceptualization Data collection Data processing and analysis Inferences and conclusions
Conceptualization of Experiment In analyzing a phenomenon one first tries to identify the physical quantities that are involved in the phenomenon. Think of ways to keep all except one variable constant and observe the phenomenon by varying it in a systematic way.
To collect meaningful and reliable data, it is necessary to set the experimental apparatus in the right way. This becomes possible only when the mental picture of the experimental setup and the procedures is clear.
… so Conceptualization of the experiment consists of thinking about the objective of the experiment, the principles of physics involved, the procedures to be followed in the measurements and the arrangement of the apparatus.
It is also essential to plan about the range of independent variable over which the data is to be collected and the appropriate precisions (or least counts) of the measuring instruments. To make proper choice of measuring instruments with appropriate precisions, sensitivity analysis using the formula of propagation of uncertainty in the result is helpful.
Each quantity measured in the experiment has some uncertainty in the measured value. So, the best one can do is to see that contributions to the uncertainty in the result from them are of the same order. e.g. In the case of simple pendulum, the uncertainty in the measurement of length, l and the uncertainty in the measurement of time t should be of same order.
As a primary goal the choice of precision of measuring instrument must be such that the uncertainty in the measured value is less than 1%.
How is the uncertainty in measurement related to the precision or resolution of the measuring instrument?
Every measuring instrument has some scale with markings or digital display. • The readings of the instrument are multiples of the smallest unit (least count) of the scale. • Any fraction of the least count cannot be included in the reading. • Thus the measurement is uncertain to the extent of that unmeasured part and so is dependent on the resolution of the measuring instrument.
How many oscillations should we choose to measure the time interval to determine the periodic time of a simple pendulum? The answer to this question depends upon the resolution of the watch as well as some other considerations. To keep the uncertainty due to the limit of resolution of the measuring device less than 1% … … the choice of either the measured quantity or the measuring instrument must be such that the reading is at least 100 units of least count.
Least count of the stopwatch used = 0.01 s Hence to limit the uncertainty below 1% in the reading due to this least count of 0.01 s, the minimum interval could be … 1 s !!! So one oscillation should be enough. Then why waste so much of time by choosing an interval with more oscillations…??? See what values you get for the period of the pendulum by repeating the measurement. Is it possible to achieve repeatability in the readings using the watch with least count of 0.01 s ???
No. The measurements vary randomly. We cannot know the true value of the measured quantity because of the random errors present. This is done by either truncating the reading to ignore the last digit or rounding the previous digit in the reading of the watch. By this we have increased our uncertainty to such an extent that the random errors are now smaller than it.
With this least count the minimum time interval to limit the uncertainty below 1% would be 10 s. Knowing the approximate period determine the number of oscillations in an interval of 10 s, and then choose some round number N above that for your measurements.
Random errors change the readings when repeated and thus their presence can easily be detected. But suppose the watch is running slow. Then the time measured may show same readings when repeated. The error cannot be detected from repeating the readings because it remains constant and does not change randomly. Such errors are called systematic errors. The most important thinking exercise is about searching possible sources of systematic errors in the data and devise methods to apply corrections for those errors. Sound theoretical knowledge about all things involved in the experiment is essential for this search.
Systematic errors generally produce changes in the readings in the same direction. They may be due to incorrect setting of the apparatus, wrong procedures followed in taking readings, faulty instruments and such other factors.
Some common systematic errors Zero error in micrometer screw and vernier callipers readings. The ‘backlash’ error. When the readings on a scale of microscope are taken by rotating the screw first in one direction and then in the reverse direction … … the reading is less than the actual distance through which the screw is moved. To avoid this error all the readings must be taken while rotating the screw in the same direction.
The ‘bench error’. When distances measured on the scale of an optical bench do not correspond to the actual distances between the optical devices, addition or subtraction of the difference is necessary to obtain correct values.
To detect systematic errors due to inaccurate apparatus or due to unsuspected effects of the apparatus in use, one way is to replace the apparatus by another of similar type, another is to make all such possible changes in the arrangement of the apparatus which are not expected to bring about any changes in the readings of the instruments and then observe if any changes are there. (e.g. unwanted magnetic or thermal or optical disturbances) These steps will show if the calibrations of the instruments are inaccurate or if the instruments in some way produce some disturbances.
The deformation of a body under external forces is a slow process and is not instantaneous. Similarly, when a substance is heated by keeping it in a constant temperature bath its temperature does not attain the temperature of the bath instantaneously.
Often the experimenter has to take readings without waiting for the steady state. For this reason, the reading of extension of a wire for a particular load while loading will be less than corresponding reading while unloading. Similarly, the reading of resistance of a thermistor while heating will be more than that while cooling at the same temperature.
This type of error depends on the direction of change of the quantity being measured and is called ‘error due to hysteresis’. The mean values of the two readings approximately eliminate this type of error.
Data Collection The data should be collected first by varying the independent variable uniformly over the entire range of the study and then additional readings may be taken where the study is to be more specific. The range of the study should be as wide as is permitted by the experimental set up. If the relation between the independent variable and the dependent variable is expected to be linear, then six to eight readings are adequate. Always state the absolute uncertainty in the readings while taking the readings.
More readings should not be taken if they do not provide any additional information, because that will be only waste of time and energy. When the relation is non linear, after covering the entire range uniformly, more readings (by varying the independent variable) are needed to locate the points of maxima and minima, points of inflexion etc. Additional necessary readings are to be taken wherever needed after examining the observations taken over the entire range.
Data Processing and Data Analysis The data recorded from observation is to be processed to be able to draw inferences. Often secondary data is created using the primary data and then it is used for analysis. While calculating this secondary data the uncertainties in them are also to be estimated.
Graph Plotting One efficient tool of analysis is graph plotting. The independent variable (or secondary data obtained from it) is plotted on the x-axis and the corresponding data representing dependent variable on the y-axis. Often the primary data is combined in such a way that the relation plotted on the graph is linear. The linearizing of the relation is found to be useful to get the results from slopes and intercepts of graphs using all available data.
Example: • Linearize the following relation to find out the acceleration • due to gravity from • Slope • Intercept
s/t t
s/t2 1/t
t s/t
Always prefer slopes… When the relationship between two variables is that of direct proportionality, constant systematic error in the measurement of one of the variables would appear as an intercept on the graph but the error would have no effect on the slope. t T2 s/t l
Many times, with experimental data, straight-line graphs, which are expected to pass through the origin are found to give some intercepts showing the presence of constant systematic error in the data. T2 T2 l l
When the data contains some constant systematic error, all the points on the graph are shifted by the constant amount but the slope of the the best fit line would be free of that systematic error. The intercepts, however, will be shifted due to that error. T2 T2 l l
If calculations are made from the data, the error remains in the calculated result. Hence, whenever a linear relationship is expected, the slope should be used in the formula instead of the mean of the ratios of the two quantities. Calculation using the mean of the ratios includes the systematic error and should not be used. By calculation By graph
If calculations are made from the data, the error remains in the calculated result. Hence, whenever a linear relationship is expected, the slope should be used in the formula instead of the mean of the ratios of the two quantities. × Calculation using the mean of the ratios includes the systematic error and should not be used. By calculation By graph
The scales along the x and y axes should be chosen in such a way that the graph occupies maximum part of the graph paper. The quantities plotted should be written along with their units beside the axes. ????
Just three points for a straight line !!! × Assuming (0, 0) to be an obvious data point … When should one set origin as (0, 0)?
Something wrong with the collected data… Online plotting helps …
The smallest unit on graph is a simple multiple of the unit of data. Taking scale like 3 cm of graph paper equal to 1 unit of data makes life difficult.
Graphs for handling power relations Consider a relation, y = xn To determine n, one can take logarithm of both sides. log(y) log(y) = log (xn) Slope = n log(y) = nlog (x) log(x)
Calculating Uncertainty from Graph When the graph is supposed to be a straight line, it is essential to see which is the best fit line. If all the points appear to lie on a straight line then it is simple to identify the line by simply drawing the line passing through all the points. If we believe that there is only one line representing the data, then there will be no uncertainty in the slope of the line and it will mean that there is no error in data plotted. …obviously, this can’t be correct.
We know that the data have uncertainties. Then the position of a point on graph representing a value from data also must have the same uncertainty. The uncertainty will form a rectangular region around the marked point. Generally, considering the uncertainty in independent variable negligible, error bars are shown for the dependent variable when all the points on the graph appear to lie on a straight line. This is essential to determine the error in the slope.
