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
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.
Verniers, Verniers, Everywhere…
… 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.
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 many oscillations should we choose to measure the time
interval to determine the periodic time of a simple pendulum?
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.
We can deal with this problem by raising the least
count of the measuring instrument higher by one order
to 0.1 s !
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.
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.
If exact magnitude of the systematic error can be
calculated using appropriate formula these errors from the
data can be eliminated. The corrected data could then have
only random errors.
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.
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.
LCR Series Resonance
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
1) Slope
2) Intercept
1 2
s  ut  gt
2
1 2
s  ut  gt
2
s/t
s 1
 gt  u
t 2
1
g
2
t
1 2
s  ut  gt
2
s/t2
s u 1
  g
2
t
t 2
1
g
2
1/t
1 2
s  ut  gt
2
2s 2
t    u
gt g
t
2
g
2

g
s/t
A few Graphs…
A few “Bad” Graphs…
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)?
Extrapolating line where
there are no data points.
Neglected Point???
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)
log (y) = n log (x)
Slope = n
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.
For example, for a spring-mass system,
Ms 
g

Static Case: Extension, e   M 
k
2 
Ms
M
3
T

2

Dynamic Case:
k
After linearizing,
g
 g  Ms
Static Case : e   M   
k
k 2
Graph of e against M can be plotted.
2


4

2
M
Dynam icCase : T  
 k 
 4 2  M s

 
 k  3
Graph of T2 against M can be plotted.
Below, assuming that the uncertainties in the independent
variable are negligible, uncertainty bars are shown to represent
the uncertainties in the dependent variable only.
The lines crossing all the error bars and with maximum and
minimum slopes will then be giving the extent of uncertainty of
the best fit line.
Let S be the slope of the best fit line and let S1 and S2 be the
slopes of two lines with the maximum and minimum slopes
respectively.
S S1 ~ S2 /2
Error in slope,

S
S
Error in slope,
S S1 ~ S2 /2

S
S
If there is large scatter and the distances of various points
from the best fit are more than the length of the error bar, …
…then showing the error bars becomes useless for
evaluating uncertainty in the slope because random errors (or
varying systematic errors) in data are large.
Estimation of Uncertainty due to Random Errors
Standard Error in the Result
AN EXAMPLE
To illustrate the use of uncertainty analysis, let us
consider how to estimate the standard error in the
value of Young’s modulus determined by Searle’s
method.
MgL
Y
 r 2l
 
l


Y
 L   r  
M
 
  2   
l
Y
 L   r 
 M
2
2


2
 
Δ l
ΔY
ΔL
Δr

 
 
M
 
  2
 
l
Y
 L   r 
 M
MgL
Y
2
r l
2
2
Y   Y   Y   Y 
2
 Y 


 Y 
2
r
2
L
2
l
M

 Y   Y   Y 

 
 

 Y r  Y  L  Y  l M 
2
2
2


2
MgL
Y
2
r l
 
Δ l
ΔY
ΔL
Δr

 
 
M
 
  2
 
l
Y
 L   r 
 M
2
2
 Y
  MgL   2

Y r    r   
 3  r 
 r
  l  r

 Y   MgL   2
 1

 
 3  r   
 Y  r  l  r
 Y
2
 MgL   2
r 
 r l  

 3  r  
   2 
r 
 MgL 
 l  r


2
MgL
Y
2
r l
 
Δ l
ΔY
ΔL
Δr

 
 
M
 
  2
 
l
Y
 L   r 
 M
2
2
 Y
  Mg

Y L    L    2  L
 L
  lr

r l   L 
 Y   Mg

   2  L 

 
MgL  L 
 Y  L  lr
2


2
MgL
Y
2
r l
 
Δ l
ΔY
ΔL
Δr

 
 
M
 
  2
 
l
Y
 L   r 
 M
2
2
 MgL   l M  
 Y

Y  l M    l   l M    2  l 2 
  M 
  lr   M  
2 l 
l

  l M  
gL   M   r M
 Y 


   2  l 2  
   l
 Y   l M   r   M   gL    M  


2
Flat Spiral Spring
Modulus of rigidity
16 2 R 3 N

r4
T2 
 
M 
16 2 R 3 N

 Slope
4
r
 R   r   Slope 

 3
   4   

 R   r   Slope 

2
2
2
Flat Spiral Spring
Young’s Modulus
32 NR  I 
Y
 2
4
r
T 
32 2 NR
slope
Y
4
r
2
or
 
  
 I 2
Y
 R   r  
T
 
 4  
Y
 R   r   I 2
 T
2
2
2
Inferences and conclusions
The final result should always be stated with its
uncertainty.
[What in earlier textbooks was described as ‘estimation of
error’ is now called evaluation of uncertainty. Error is the
difference between the true value and measured value and
it can never be known. What we can know is the
uncertainty.]
The uncertainty is expressed only in one or two significant
digits rounding the number.
Inferences and conclusions
The final result is rounded to have the last digit at the same
decimal place as the last digit in the uncertainty.
In any measurement the uncertainty in the measured
quantity is at least equal to the least count of the measuring
instrument.