Instruments, Accessories
and Related Topics
The development of science depends upon the close observation of nature’s events and their explanation.
Thus, practical work is of utmost importance for the development of science. The laboratory courses
are designed to train the student in the use of laboratory apparatus and techniques used in physics
measurements and to make him competent with the art of drawing the valid and worthwhile
conclusions from the observed data.
Before performing an experiment, a student must know that what physical quantity he has
to measure by that experiment or which law or which formula he has to verify. Apart from it, he
must know what apparatus he has to use to perform that experiment and how he has to use that
apparatus.
ACCORDING TO LORD KELVIN
When you can measure what you are speaking about and express it in numbers, you know
something about it but when you cannot measure it in numbers, your knowledge is meagre and
unsatisfactory.
In experimental Physics, we always measure quantities and try to have some idea about it. Mere
measurements do not satisfy us, as we make errors in our measurements. We ought to know the
sources of errors and the accuracy of our measurements. Moreover, if we know the sources of
errors we can take care of these, provided it is within our control.
A sensitive and good apparatus would not give necessarily good results. The observer himself
should have a good experimental skill. It is often found that a skilled observer or one with a good
experimental hand can do a job better with the same apparatus than an average observer. However,
with care and practice one can develop a good skill for performing an experiment.
The reliance of the results obtained in an experiment depends on a number of factors, such as
the skill of the observer, the apparatus used, the constancy of the condition under which he is
working etc.
The errors one commits in performing an experiment can be broadly classified under the
following heads:
(i) Constant error
(ii) Systematic error
(iii) Random error
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(i) Constant error: When a number of observations are made and error occurs by a constant
amount, the error is said to be a constant error. For example, when we are measuring the length
with a scale of faulty graduation, say 100 cm is read as 99 cm, the length measured will always
be smaller than the actual length. This can be detected by comparing the scale used in the measurement
with a standard scale.
(ii) Systematic error: While performing an experiment it may happen that certain errors occur
according to some rule, such errors are known as systematic errors. These errors can be eliminated
if the sources of error are detected and the rule governing them is known. The following methods
are adopted to eliminate systematic errors:
(a) In some cases, the errors are determined in the beginning of the experiment and corrected
while taking observations. For example, in making measurements for the radius of the wire
with a screw gauge, the zero error is noted and subtracted to (or added with) the
observations accordingly. Another such example is the bench error in optical bench experiment.
(b) In some experiments, the experiment is performed with full knowledge that certain errors
are occurring. Those errors are eliminated by making additional observations. For example,
in some of the experiments on heat, it is known that certain amount of heat radiates out
and thus makes some error. At the end of the experiment, radiation correction is made to
eliminate the error.
(c) In some cases, the error is eliminated by performing the experiment under changed conditions.
For example, in the Carey-Foster bridge experiment, the contact resistance at the end of
the wire is eliminated by interchanging the position of the two resistances. In the resonance
air column experiment, the end effect is eliminated by taking observation resonance for
fundamental and for the first overtone.
(iii) Random error: This may be defined as the error due to a cause of which the law of action
is unknown. This is caused by the following:
(a) Small changes in the conditions of the experiment.
(b) Due to observer’s incorrect judgement about the observations.
The source is accidental and cannot be traced as in systematic or constant error. As will be seen
these cannot be usually eliminated, but still some knowledge of these might help in performing the
experiment and dealing with them.
For example, the null point in an electrical experiment might change due to the heating effect
of the current causing random error.
Random errors may be both positive and negative, i.e., it may be on both sides of the correct
value. Usually, the error is not large and the value is governed by the law of probability. Since it is
possible that an error may be on both sides of the correct value, hence taking an arithmetic mean
of a large number of observations will minimise this type of error. The result will be close to the
correct value.
IMPORTANT OBSERVATIONS, REJECTION OF OBSERVATIONS ETC.
In any experiment a number of quantities are measured to obtain a result. Sometimes in the same
experiment it is not necessary to measure some quantities more accurately than the others. It is
because of the fact that some quantities occur in the first power and some in higher powers. The
one occurring in the highest power should be measured with utmost accuracy. For example, take
the case of determining the modulus of rigidity by torsion table. In this experiment, the power of
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T is 2 and that of r is 4. This shows the need of more accurate measurements of r and, to some
extent, of T as compared to other quantities occurring in the first power.
While taking a large number of observations, it is seen quite often that one or two of them
deviate considerably from the rest. These observations can be rejected provided that it is known that
some errors are there in the deviated observations. In case the fault is detected while performing the
experiment, it is better to repeat the same.
MEAN ERROR, ROOT-MEAN SQUARE AND PROBABLE ERROR
Different measures are adopted for estimating the magnitude of errors, the choice depends upon the
prevailing conditions. Some of the measures are as follows:
(i) Mean error or mean deviation
(ii) Root-mean square deviation
(iii) Probable error
(i) Mean error: This is the arithmetic mean of all the errors neglecting their sign. Let x1, x2,
x3,  xn be the values of n different observations of the same quantity × (true value). The arithmetic
mean (say x ) of all these (x1, x2, x3,  xn) is given by
x  x  x3    xn
x 1 2
n
For a large number of observations and for a normal of distribution x should be equal to X.
But for actual measurement x may not be equal to X. Deviations of individual observations may be
obtained as follows:
x1 – x = d1
x2 – x = d2
x3 – x = d3
––––––
––––––
xn – x = dn
d1  d 2  d3    d n
and if d is the average of all these deviations, i.e., d =
we may call it the mean
n
error and write X = x ± d.
(ii) Root-mean square deviation: Another important quantity is the square root of the average
of the squares of deviations called root-mean square deviation. This is also called the standard
deviation D. This is obtained in the following manner:
D =
d12  d 22  d 32    d n2
n
X =x  D
(iii) Probable error: Probable error denotes the limits on either side of the most probable value
of a quantity. The probable error P can be shown to be approximately equal to 0.6745 D.
P = 0.6745 D
X = x P
The results obtained in determining the focal length ‘f ’ of a convex lens are presented in below
table, column I. The deviation, mean deviation ‘d’, standard deviation ‘D’ and the probable error ‘P’
as calculated using the methods mentioned above are presented in the subsequent columns.
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Table 1.1
Focal length
f (cm)
316
312
319
313
314
316
Deviation (cm)
+
–
+
–
–
+
Mean deviation
d (cm)
Standard deviation
D (cm)
Probable error
P
± 02
± 023
± ~ 016
01
03
04
02
01
01
We shall describe the procedure to make observations with a given apparatus.
(i) Note the least count of the measuring instruments: Each instrument has a limit to
accuracy of measurement which depends on the least count of the instrument. As an example, a
metre rule can measure upto 01 cm or 1mm, a vernier calliper can measure upto 01 cm or
0.1 mm etc. One should note the least count of the instrument to be used before recording the
observations. Such details must be mentioned at the top of the observation table.
The least count of an instrument is that least measurement which can be done by that instrument.
The method of finding the least count of some instruments is described below:
(a) Metre rule: A metre rule can measure accurately upto one-tenth part of one cm. Hence, its
least count is 0.1 cm.
(b) Vernier callipers: A vernier callipers is provided with an auxillary scale in addition to the main
scale. This scale is called the vernier scale. The vernier scale can slide along the main scale. The
vernier scale is so graduated that the length of total number of divisions on it is smaller by the length
of one division on main scale. The least count of vernier callipers is equal to the difference in the
value of 1 main scale division and 1 vernier scale division. This is calculated by using the following
formula:
Value of 1 main scale division ( x)
Total number of divisions on vernier scale ( n)
Generally, the value of 1 main scale division on vernier callipers is 0.1 cm and there are 10
divisions on the vernier scale, i.e., x = 0.1 cm and n = 10.
0  1 cm
 Least count of vernier callipers =
= 001 cm.
10
In the travelling microscope generally used in laboratory has 10 division in 1 cm on its main
scale and the length of 20 divisions on its vernier scale is equal to the length of 19 divisions on its
main scale, i.e., x = 01 cm and n = 20.
0  1 cm
 Least count of travelling microscope =
= 0005 cm.
20
But if the travelling microscope has 20 divisions in 1 cm on main scale and the length of 25 divisions
on its vernier scale is equal to the length of 24 divisions on its main scale, then x = 005 cm
and n = 25. Then
0  05 cm
Least count of travelling microscope =
= 0002 cm.
25
(c) Screw gauge and spherometer: They work on the principle of screw. They have a linear scale
called the main scale and a circular scale. The circular scale can be rotated by a head screw.
Least count of vernier callipers =
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On turning the screw, the circular scale advances linearly on the main scale. The distance moved
by the tip of screw when it is given one complete rotation, is called the pitch of the screw. Dividing
the pitch of screw by the total number of divisions on the circular scale, we get the distance which
the screw advances on rotating the screw by 1 division on its circular scale. This distance is called
the least count of the instrument. Thus,
Pitch x
Least count =
Total number of divisions 'n' on the circular scale
Generally, in a screw gauge or spherometer the screw advances by 1 division on main scale
when the screw is given one rotation. If there are total 50 divisions on its circular scale and the value
0  1 cm
= 0002 cm. But, if there are 100
of 1 division on main scale is 01 cm, then least count =
50
0  1 cm
divisions on circular scale, then least count =
= 0001 cm.
100
(d) Physical balance: A physical balance is used to determine the mass of an object. Its least
count is the mass of smallest weight (which is generally 1 milligramme) available in the weight box
provided with the physical balance.
(e) Stop watch: A stop watch is used to measure time. The second’s arm completes one
revolution in 1 minute (= 60 sec). Therefore,
60 sec
Least count of stop watch =
Total number of divisions on its circular scale
Generally, there are 120 divisions on the circular scale of a stop watch (i.e., each second is
divided in two parts). Hence,
60 sec
= 05 sec.
120
(ii) Find the zero error of the measuring instrument: There may be some zero error in the
measuring instrument. If there is some zero error in the measuring instrument, it should be determined
and mentioned at the top of the observation table with its sign and in calculation it must be subtracted
from the observed readings with sign to obtain the true reading. The method of finding the zero error
of some instruments is described below:
(a) Vernier callipers: On bringing both the jaws together, if the zero mark of vernier scale is on
the right hand side of the zero mark of the main scale, the zero error is said to be positive. To find
this error, we note that division of the vernier scale which coincides with any division on the main
scale. The number of this vernier division multiplied by the least count of the vernier, gives the zero
error. This error is subtracted from the observed reading to find the actual reading. For example,
in Fig. 1.1 5th division of vernier scale coincides with a main scale division. Therefore, the zero error
= + 5 × least count of vernier = + 5 × 001 cm = + 005 cm.
Least count of stop watch =
0
Main scale
Vernier scale
0
Fig. 1.1
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But if on bringing both the jaws together, the zero mark of vernier scale is to the left of the
zero of the main scale, the error is said to be negative. To find this error, we note that division of
the vernier scale which coincides with any division of main scale. The number of this vernier division
is subtracted from the total number of divisions on the vernier scale and then the difference is
multiplied by the least count. Since the zero error is always subtracted from the observed reading
with its sign, so the negative zero error gets added to the observed reading. In Fig. 1.2 the sixth
division of the vernier scale coincides with a certain division of the main scale. The total number
of divisions on vernier are 10, therefore the zero error = – (10 – 6) × least count = – 4 × 001 cm
= – 004 cm.
1
0
Main scale
Vernier scale
0
Fig. 1.2
(b) Screw gauge: If on bringing the flat end of the screw in contact with the stud, the zero mark
on the circular scale is below the base line on the main scale, the error is said to be positive. To
find it, we find the division of the circular scale which coincides with the base line. This number
when multiplied with the least count of the screw gauge gives the positive zero error. In Fig. 1.3,
the 6th division of circular scale coincides with the base line. If the least count of the screw gauge
is 0001 cm, then zero error = +6 × 0001 cm = + 0006 cm.
15
10
0
5
0
95
Fig. 1.3
The true reading is obtained by subtracting it from the observed reading.
But if on bringing the flat end of the screw in contact with the stud, the zero mark on the
circular scale is above the base line of the main scale, the error is said to be negative. To find it,
we look for the division of the circular scale coinciding with the base line. This number is subtracted
from the total number of divisions on the circular scale and is then multiplied with the least count
of the screw gauge. This gives the negative zero error. In Fig. 1.4, 91th division of circular scale
coincides with the base line and the total number of divisions on the circular scale are 100.

Zero error = – (100 – 91) × 0001 cm
= – 0009 cm.
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0
95
0
90
C
85
80
Fig. 1.4
(c) Spherometer: While using a spherometer, generally we take the difference of two observations.
Hence there is no need to find its zero error.
(iii) Never estimate the values less than the least count of the measuring instrument:
In Fig. 1.5 (i), OA will be expressed as 1.2 cm and in Fig., 1.5 (ii), OB will be expressed as 13
cm. The reason is that the least count of scale is 0.1 cm and in Fig. 1.5 (i), the end A is nearer
to 12 mark than 13 mark, while in Fig., 1.5 (ii), the end B is nearer to 13 mark than 12 mark.
O
O
A
0
B
0
1
1
(i)
( ii)
Fig. 1.5
Similarly in Fig. 1.6 (a), the least count of thermometer is 01°C and its reading is 705°C, while
in Fig. 1.6 (b), the least count of thermometer is 1°C and its reading is 71°C.
110
74
10 0
73
90
72
80
71
70.5ºC
70
70
60
69
50
71ºC
40
2
30
1
20
0
10
0
(a)
(b)
Fig. 1.6: In (a) the reading of thermometer is 70.5ºC and in (b) 71ºC
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In Fig. 1.7 (a), the least count of protractor is 1°, so the angle AOB will be read as 69° and
not 6875°.
B
100
110
120
13
60
0
90 80
100
7
0
0
1
60
20 1
1
5
0
0
13
80
70
0
20
10
0
170 18
60
0 1
15
0 0
20 1
30
0
0
14
40
30
40
18017016
0 15
50
01
40
9
A
Fig. 1.7 (a): AOB is 69º, and not 68.75º
In Fig. 1.7 (b), since the least count of ammeter in 01 amp., so the ammeter reading will be
expressed as 07 amp., and not 074 amp.
+
–
1
0
2
3
Ammeter
Fig. 1.7 (b): Ammeter reading is 0.7 A and not 0.74 A
(iv) Avoid error due to parallax: This is the most common error in all measurements involving
scale reading. Figure 1.8 illustrates the error due to parallax in measurement of length of a rod PQ
with a metre scale. If observations are taken from the positions A and B (i.e., the line of sight at
both ends being normal to the scale), the reading corresponding to the end P is 10 cm and that of
end Q is 34 cm, so the length of the rod is 34 – 10 = 24 cm whereas, if readings are taken by
keeping the eye in positions C and D, the end P appears to be at 12 cm and the end Q at 30 cm,
so the length comes out to be 30 – 12 = 18 cm, which is smaller than the actual length and so
it is incorrect.
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2
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P
Q
Fig. 1.8 The length of rod PQ is 2.4 cm and not 1.8 cm
(v) Record the observations upto the proper significant digits: Do not add extra zeroes on
the right side of the decimal in any observation. This misleads the observation. For example, if you
measure the diameter of a wire with a screw gauge and it comes out to be 123 mm, it should not
be written as 1230 mm, because in this measurement the Fig 1.8 is doubtful. The actual diameter
can vary from 1226 mm to 1234 mm. But if there is no digit before the decimal, put a zero before
the decimal to make the decimal prominent. For example, if the current measured is
.6 A, write it as 06 A.
(vi) Record observations as far as possible in a tabular form: This helps in comparing the
different sets of observation. At the top of the column, write the quantity being measured and then
its unit immediately below.
(vii) Drawing of graphs: The plotting of graphs originated from descartes analytical geometry
and the method is widely used in physics, observations plotted in graphs have several advantages
over the tabulated form. They are as follows:
(i) It gives us an immediate feeling about the observed values.
(ii) It shows immediately whether a particular observation is widely different from others, as
the point will lie outside the resultant smooth curve.
(iii) Any intermediate or extrapolated values not available in the observations can be obtained
(within the limit of extrapolation) without any difficulty.
(iv) The significant feature in the data are obviously seen, i.e., maxima or minima or regular
variation in the curve etc.
(v) It smoothens out the observations.
It may be mentioned here that these may also be deduced from tabulated observations but that
would need some careful scrutiny.
Observations are conveniently plotted in graphs and calculations are made. The desired results
may also be obtained from the slope and intercept of the curve.
(a) Dependent and independent variables: When two variables are represented by a formula
and a graph is to be plotted, the independent variable is plotted along the X-axis and the dependent
variable along the Y-axis. While performing an experiment, the value of one of the variables is
arbitrarily fixed and the values of the other quantities are observed. Obviously, the former is an
independent variable and the latter is a dependent variable.
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(b) Choice of scale: The scale should be properly chosen. In case the intercepts on the axes
are not required for getting the results, it is not necessary to take the origin as (0, 0). In such a case,
the scale should be so chosen that the major portion of the graph paper is used up. This has got
its own limitation. The scale is usually chosen such that one small division of the scale is greater
or equal to the least count of an apparatus used in the measurement. At the most one can choose
a scale such that two small divisions are equal to the least count of the apparatus.
The graph should be properly labelled, i.e., the variables plotted and the scale chosen along both
axes should be indicated and the axes should be properly shown.
The size of the circles or crosses (O, ×) used for showing the observations have not got some
meaning. The least count of the apparatus used is known and this may be used in showing the points
so that one immediately gets the idea of accuracy claimed in the observations.
Each pair of data are plotted and the points are marked. The points are then joined to have a
smooth curve. In few cases, the points may be far away from the smooth curve and may be ignored
while joining the points.
Least square fit: For finding the values from graph for a straight line with the observed data,
one of the methods is the method of least square fit.
Suppose x and y are the two quantities, such that y = f (x) and the equation relating the two
is y = mx + c where m and c are two constants known as the slope and the intercept respectively.
For the evaluation of m and c, two sets of data will be sufficient. Hence, if more sets of data are
available these are not of any use. Thus it is essential to evolve a method for utilising the extra data
available. Let us examine how the experimental data may be fitted to a straight line using all the
observations.
In case x and y are all exact values, they would lie on the straight line. If they are not exact,
y .
then instead of points we get a small ellipse where axes are in the ratio of
Because of the errors
x
in the measurements, the centres of these ellipses will not lie on the straight line, but can be expected
to be distributed equally on the either side of the line. When we try to draw a straight line, we may
not always judge a certain line to be the best one. It may vary from person in judging which straight
line is the best one.
Let y1, y2, y3, yn , be the measured values and let ŷ be the most probable value, i.e.,
yˆ 
y1  y2  y3    yn
n
n

y
 nn
1
and the arithmetic average of the deviation will be zero. However, the squares of the deviation are
all positive, and therefore the sum of these squares does not vanish.
We assume that x is measured exactly, while all errors are concentrated to the other variable
ŷ . Graphically, this fact can be represented by vertical lines as shown in the Fig. 1.9. The vertical
deviation d1 of the point (x1, y1) from the straight line
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