1
INTRODUCTION
1.1
Importance of Experiment in Science
The study of the history of the progress of
science reveals that experiment has been the basis of
advancement of science. The main object of
experiment in science is either to discover the natural
law underlying a natural phenomenon or to verify
some theory. Therefore, experiment is an essential
part for the progress and development of science.
The importance of experiment, in physics is
still more, since physics is an experimental science.
Indeed observation and experiment have given birth
to all the theories of physics.
of an object the following procedure should be
followed to avoid errors in measurement :
1.
The scale should be kept along the length of
the object whose length is to be measured and
not in any other manner.
2.
One end of the object which is to be measured
should coincide with a definite mark of the
scale not necessarily with zero-because the
end side of the scale generally gets rounded
and to place the object coinciding exactly
with zero is difficult.
3.
While reading the scale one eye may be kept
closed and the other should be held vertically
above the mark. The reading should not be
taken by keeping the eye slantingly. Fig. 1.1
explains the correct method of viewing.
To understand a scientific theory, it is desirable
for the students to observe the experiments
illustrating that theory. However, if the student himself
does those experiments, takes observations and arrives
at results himself, it would be especially beneficial
for him, because he himself has reached the
conclusion by the experiment.
In this manner the students will learn how to
use the ‘Scientific Method’. This is the main purpose
of including ‘Practical work’ essentially in the
physical curriculum.
In Physics, first of all definite and well defined
object of the experimental work should be known
then, with the help of certain measuring instrument’s’,
arrangement for the experiment & its apparatus is
made. Later on observations are taken in certain well
planned sequence. Important results are obtained by
analysing the observations. In the end result is
ascertained.
Now we shall introduce some general
measuring instruments which are used in most of the
experiments.
1.2
Some Common Measuring Instruments
1.2.1 Meter Scale
A meter scale is used to measure small lengths
or distances in a laboratory. It has markings from 0
to 100 centimeter. Each centimeter is divided into 10
parts. On this scale the value of the smallest parts is
1 millimeter or 0.1 centimeter, which is called the
least count of this meter sale. To measure the length
Fig. 1.1 Length of rod AB=2.4cm & not 1.8 cm
If the eye is placed in E1 and E2 positions,
the object AB will be measured correctly. In case the
eye is placed in E3 or E4 positions the length of AB
will not be measured correctly.
The correct length of AB is 4.4–2.0=2.4cm
but the position E3 shall read 2.2 and the length AB
will be read as 4.0-2.2=1.8cm which is incorrect. This
error is known as the error due to parallax.
4.
If the scale is vertical, the eye should be held
at the correct horizontal level, straight in front
of the mark.
In E1 position, the eye reads correct level of water as
74cm3, but in E 2 position the reading is 76 cm3
which is wrong. This error is also known as error
due to parallax.
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1.2.2 Vernier Callipers
A metre scale has a least count of 1mm only
i.e., with this we cannot measure less than 1mm. To
measure upto 0.1mm correctly we use Vernier
Callipers. (Fig.1.2). It is invented by a French
Mathematician Pierre Vernier.
main scale’s (n-1) divisions i.e. length of main scale’s
(n-1) divisions is equal to that of ‘n’ divisions of the
vernier scale.
∴
(n – 1) S = nV
or
nS – S = nV
or n(S – V) = S
or S – V = S/n
but S – V = Length of one division of main
scale – length of one division of vernier
scale = least count of vernier callipers.
∴
least count of vernier cappllipers = S/n
= Length of one division on the main scale
Fig.1.2
With the help of this instrument one can
measure the diameter of the bob of a simple
pendulum, length of a rod or cylinder, internal or
external diameter of a cylinder, etc. correctly upto 1/
100 th of a centimeter.
In vernier calipers, a main scale is designed
on a metal strip on which divisions are marked in
centimeters. Each centimeter is divided into 10 parts
and each division is of length 1mm. On another strip
a vernier scale is made which slides over the main
scale. With the help of a screw the vernier scale can
be fixed on the main scale at a convenient place. A
pair of external and internal jaws are placed
perpendicular to main scale. The jaw on left end of
main scale is fixed whereas the other jaw is attached
to the moving frame of vernier scale. Second jaw
slides with the movement of vernier scale. Lower pair
of jaws can be used to find the diameter of bobs,
length of rods and external diameters of hollow
cylindrical objects whereas upper jaws help to find
internal diameter of hollow cylinders. With the help
of metallic strip, depth of objects can be determined.
Vernier Constant or Vernier Least count:
The minimum distance or length which can
be measured by a vernier is called least count of the
vernier. The difference of one division of main scale
and one vernier division is called the least count of
the vernier callipers.
Let us assume that the length of one division
of main scale is ‘S’ and that of one vernier scale is
V, Suppose, Vernier’s ‘n’ divisions coincide with the
total no. of divisions on the vernier scale.
For Vernier Callipers S = 1mm = 0.1cm.,
n = 10.
Least count = 0.1/10 = 0.01 cm.
Zero error : when both the external and internal
jaws of the vernier calipers are joined to each other
the zeroes of both the scales should coincide with
each other, if they do not, then vernier is said to have
a Zero error.
Fig.1.3
When both the jaws are joined, then if the
zero of vernier scale lies o the left side of Zero of
main scale then the venier has negative zero error,
(Fig. 1.3). To evaluate this, the number of divisions
of vernier scale which coincide with some division
of main scale-is subtracted from the venier’s total no
of divisions. The number thus obtained is multiplied
with the least count and marked negative. This gives
the magnitude of zero error. Fig 1.3 shows the
negative zero error. In this Fig. 6 th division of the
vernier is shown to coincide with 5th division of the
main scale, and total number of vernier divisions is
3
10. Therefore zero error= – (10-6) × least count = –
4 × 0.01 = – 0.04 cm.
Similarly if the zero of vernier scale lies to the
right side of the zero of main scale then the zero error
is positive (Fig. 1.4). In this fig. the vernier’s 6 th
division is coinciding with 5th division of the main
scale and that vernier zero is on the right side of
main scale zero.
gun metal. Main scale is marked on the reference
line. Main scale has marks in millimeters. On the
hollow cylinder is a cylindrical cap, When the cap is
rotated screw starts moving inside the cylinder.
Circumference of cap is divided into 50 or 100 equal
parts. It is called the circular scale or head scale. Cap
is moved with the help of ratchet R. As the cap is
rotated the screw either moves in or out. Motion of
the screw is such that when its end touches stud A,
then screw either moves I-nor out. Motion of the
screw is such that when its end touches stud A then
screw does not move, only ratchet R rotates.
Least count of the screw gauge :
Reading – zero error (with sign)
When the head scale is moved to complete
one round then the distance covered by the screw on
the main scale is known as pitch or thread distance.
It is the linear distance between two adjacent threads.
It is normally 0.5mm or 1mm. If the pitch of the
screw gauge is ‘p’ and the number of divisions on the
head scale is n, then the least distance that can be
measured by the screw gauge is p/n. This is the is
distance which the screw moves along main scale on
moving the head scale by one division. This is the
least count of the screw gauge.
of Venier Callipers
∴
Fig.1.4
∴
Zero error = + (0 + 6 × 0.01) = 0.06cm.
Now,
Correct reading = Observed
Total observed reading = Main scale reading + No.
of the coinciding vernier scale division × vernier
least count.
1.2.3 Screw Gauge – It is used to find diameter of
very thin wires or thickness of very thin sheets.
Generally thicknesses of the order of 0.001 cm. can
be measured by it. The construction of this instrument
is shown in fig. 1.5.
Fig.1.5
It has a U-shaped metallic frame. On one end
of metallic frame is a small flat rod A which is known
as stud. On the other end, is a hollow cylinder C.
Hollow part carried a cylindrical hub. The hub extends
few millimeters beyond the end of the frame. On the
tubular hug along its axis is drawn a line known as
reference line. A nut is threaded through the hub &
the frame. Through the nut moves a screw made of
Least count of
screw gauge =
Pitch of screw gauge
Total no. of divisions
on the head scale.
If the pitch of the screw gauge is 1mm and no.
of divisions on head scale is 100 then its least count
will be 0.001 cm.
Zero Error
Fig.1.6
In some instruments due to wear and tear or
due to manufacturing defect, the zeroes of the head
scale and linear scale do not coincide. When the
faces A and B touch each other, if the zero of the
Fig.1.7
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circular scale advances beyond the reference line the
zero error is negative and zero correction is positive.
If it is left behind the reference line, the zero error is
positive and the zero correction is negative.
shown in fig. 1.9. Two exactly similar pans of equal
weighs & size are suspended from the two ends of
beam with the help of hooks.
Example – If zero of circular scale advances beyond
reference line by 7 divisions zero correction = (+ 7
× least count) and if it is left behind the reference
line by 7 divisions, zero correction is = (– 7 × least
count).
E1 & E2 are screws with the help of which the
beam is made horizontal. A light pointer p is attached
in the middle of the beam with pointed end vertically
downwards. The pointed end of pointer can move to
and fro over a small graduated scale fixed at the base
of pillar.
Blacklash error.
On account of waer & tear or loose fitting,
some space is left for the play between the screw and
the nut. In such instruments if screw is adjusted by
turning it in one direction and then it is rotated back,
the screw may not move along the axis for an
appreciable rotation of the head. This error is known
as back lash error of the screw. To reduce this error
the screw must always be rotated in the same direction,
for a particular set of observations. If it is necessary
to rotate the screw in opposite direction, first it should
be rotated in the same direction for some distance
and then reading should be taken after moving it in
opposite direction.
Pillar can be raised or lowered with the help
of a lever L fixed to its base. With the help of leveling
screws S1 and S2 the plane of the base of the balance
can be made horizontal. These screws are adjusted in
such a manner that the plumb line PL, hanges exactly
above the mark below. In this situation the base of
the balance is set horizontal. When the beam is raised
it is ensured that the pointer swings equally to both
sides of the zero mark. If it is not so then it is
adjusted by the adjusting screws E1 and E2 provided
at the extreme ends of the beam.
= Main scale reading
The object whose mass is to be measured is
kept on the left pan and the weights are placed one
by one on the right pan till the pointer swings equally
on both sides of the central mark. The object should
neither be wet nor hot.
+ circular scale reading
1.2.5 Stop Watch and Stop clock
– zero error (with sign).
To measure and note the time/time-interval in
a laboratory, a stop watch/stop clock is used.
Total reading of screws gauge
1.2.4 Physical Balance : In some experiments mass
of the object is to be determined. For this purpose a
physical balance is used generally. It is shown in fig.
1.8.
Fig.1.8
It has a beam made of a light but strong metal.
It has a central knife edge or fulcrum of agate turned
downwards. It rests on a flat agate surface fixed to the
top of a vertical support within the central pillar, as
1. The Stop Watch : It is capable of being
stopped and started as desired. It has a long seconds
hand which moves over a circular dial with 60 equal
divisions. Each small division is one second which
is further divided into 5 or 10 parts. There is another
small hand which represents minutes. The least count
of a stop watch is generally 0.2 or 0.1 second. The
operation of the stop watch consists of 3 steps. In the
first step the knob at the top of the stop watch is
pressed just at the beginning and both the hands start
rotating. In the second step the knob is pressed again
at the end of the interval to be measured, and the
hands come to rest. The difference between the two
readings gives the desired time interval. Now knob
is pressed again and both the hands reach zero
position and come to rest. Watch is now again ready
for use.
2. Stop Clock : Stop clock is much bigger in
size than stop watch. It has a different arrangement
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to start and to stop. Its accuracy is less than that of
stop watch. It has a circular dial which is divided
into 60 equal parts. It shows the time in seconds and
minutes the hands of second and minute move on a
dial. It has a third hand also which is called the index
hand. It can be adjusted from outside. Before taking
observations index hand is set over seconds hands.
It does not move. It simply indicates the initial
position of the second’s hand before the beginning
of the event. The clock is started or stopped with the
help of horizontal rod R. The rod comes out of a hole
along both sides of the clock. If the rod is pushed in
one direction the clock starts moving and on pushing
it on opposite direction the clock stops.
1.3 Necessary information regarding calculations :
1.3.1 Calculations by log tables : In the history
of mathematics, development of logarithm is an
important event. Verbal meaning of logarithm is A
rule to shorten arithmetic. Therefore with the help
of logarithm a student can complete calculations
quickly and accurately. Logarithm was invented by
Napier in sixteenth century. Now a days calculations
and computers have replaced logarithm to a large
extent, but even now many universities allow only
log tables for doing calculations in laboratory and in
examinations. Hence we are briefly describing the
method of doing calculations by log tables.
1.
Standard decimal form of a number.
Any number N can be represented in terms of another
number n and a power (m) of 10. For example,
125.6
=
1.256 × 102
0.734
=
7.34 × 10–1
4375.2
=
4.375 × 103
4.8
=
4.8 × 100
0.00 852 =
base must be raised to get the given number. Let the
number be N and the base be b then logarithm of N
will be a number x, provided that N=bx or for
example,
100
23
=
8
4
=
81
Hence log10 100 = 2, log2 8 = 3 and log3 81 = 4
In general, if bX = N then logb N = x. Base of
common logarithms is always 10. Base 10 is generally
not written.
Thus log 125 means log10125. Napier
developed natural logarithm which has a base e. e
is defined with the help of a convergent series.
e = 1+ 1+
1 1 1
+ + +....... up to inf inity
2! 3! 4!
e has a magnitude between 2&3, upto three
places of decimal the value of e is equal to 2.718.
If ey = N then ln N = y. ln N is called Napierian
or natural logarithm of N. It has a base e. It can be
provided that ln N= 2.303 log 10 N l Napierian
logarithm. (ln N) of any number N is obtained by
multiplying, its common logarithm with 2.303. In
ordinary calculations Napierian logarithm is not used,
only common logarithm is used.
3. Fundamental rules of logarithm :
(i)
The logarithm of 1 to any base (a) is o i.e.
loga 1 = 0
(ii)
Logarithm of any number to the same base is
always 1 or loga a = 1 because a1 = a
ln e = 1, here base is implied.
log 10 = 1 here base 10 is implied.
(iii)
In general, any positive number N can be
written as N = n × 10m
2. Definition of logarithm : To define
logarithm of a positive number, a standard number is
chosen as the base of logarithm. Logarithm of a
number to a given base is the number to which the
=
3
8.52 × 10–3
where 1< n<10 and m is a positive or negative
integer.
102
Logarithm of o to any base is - ∞
i.e. Loga 0 = – ∞ since (a)–∞ = 0
(iv)
Logarithm of a negative number (-N) to
positive base a is not defined i.e. loga (-N) =
Not defined, because a positive number raised
to a real power cannot give rise to a negative
number. Or it is impossible to get a real number
x such that aX = –N.
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(v)
The logarithm of the product of two or more
numbers is equal to the sum of their
logarithms.
i.e. loga (mn) = loga m + logan
(vi)
The logarithm of a fraction is equal to the
difference between the logarithms of the
numerator and that of the denominator.
( m) = loga m − loga m
Thus = loga n
(vii)
The logarithm of a number raised to any
power (integral or fraction) is equal to the
product of the index of power and the
logarithm of the number.
Thus : logamn = n logam
Thus we see that in loratithm i) operation of
multiplication is changed to that of addition
ii) Operation of division is changed to that of
subtraction iii) The operation of raising to a
power is changed to the operation of
multiplication. As addition and subtraction are
easier as compared to other operations so
calculations by logarithm are easier.
4. Characteristic of logarithm of a number;
When a number N is represented in standard decimal
form then integral power of 10 is known as
characteristics of logarithm of N to base 10. If N = n
× 10m
then m is known as characteristic of log10N
Characteristic of any positive number to base
10 can be calculated with the help of following rules:
i)
Characteristic of numbers greater than 1 is a
positive integer. Its numerical value is 1 less
than the total number of digits before the
decimal point in the given number.
For example in 9.7, 25.46, 3045.4, 47893
characteristics of logarithm is 0, 1, 3&4
respectively, i.e.
Log 9.7 = 0 ....…. Log 25.461 = 1….
Log 47893 = 4. .........
ii)
For quantities less than 1 (0<N<1)
characteristics is negative and its numerical
value is 1 more than the number of zeroes
after decimal.
For example, characteristics of 0.56, 0.056 &
0.0056 are respectively –1, –2, & –3.
Thus log10 0.56 = (–1) + 0…. Or I…
Log10 0.056 = (–2) + 0 or ……2……
Log 0.0056 = (–3) + 0 or 3… Here I is read
as ‘bar I’, 2 is read ‘bar 2’ Similarly, for
example.
I.34 = (–1) + 0.34 i.e., in I.34 only 1 is negative
& o.34 is positive. Similarly, 3.748 = (–3) +
0.748 = –2.252 = (–2) – (0.252).
5. To find mantissa of a given number with
the help of log Tables :
To find mantissa of a given number, a standard
table has been made which is known as log table.
The value of mantissa does not depend on the position
of decimal point for example each of the numbers:
2546, 2.546, 25.46, 0.2546 will have the same
mantissa.
In case of four figure log tables, we write
down the given number up to four significant digits
For more accuracy the 5 digit table can be used.
But for laboratory calculations a 4 figure table
is sufficient. If for example, the number is 6739.2
then we round it off to 6739.
After this in vertical column we search for 67then in front of this number we move on the
horizontal line where 3 is written in the column. The
number is round to be 8280. Now in difference
columns 1 to 9 we note the mean difference of the
column in which 9 is written (along the line of 67).
Thus we get the number 6. We add this to 8280 and
get 8286. This number is the mantissa of 6739.2 The
characteristic of this number is 3, therefore,
Log 6739.2 = Log 6739=3.8286 (we put
decimal point before mantissa). Similarly, you can
verify that
Log 325.4
=
2.5124
Log 0.6739
=
1.8286
Log 0.0639
=
2.8286
6. Antilogarithm:
Now we shall discuss that if we know log N
then how to find out N?
The number N whose logarithm is x is called
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N = yx10n
the antilogarithm of x hence,
If Log N = x then Anti log x =N
For example, if log N = 2.7345 = x
If we know x how to find out N ?
Mantissa of x = 0.7345
i.
First we consider the mantissa of x. It should
be positive. Now see the anti log table.
a.
First two figures of mantissa should be seen
on the first column of the table. This will
ensure correct row.
b.
The third digit of mantissa is to be seen in
the line of the row obtained in (a) Note
this number.
Now we search for 0.73 in the first column of
log table, then we move horizontally to the
column of 4 and in the same line as 0.73, we
obtain the number 5420. Now in the line of
0.73, we obtain the number 5420. Now in the
line of 0.73 under the difference column 5, we
get the number 6.
c.
ii.
The fourth digit of the mantissa is to be
judged from 1-9 vertical columns of mean
difference and the row decided in (a). The
number obtained is added to (b) This gives
the mantissa of x. Now after the first digit we
put the decimal point Let the number so
obtained be y.
∴
Mantissa of the given number is 5420 + 6
= 5426. Now putting the decimal point after
the first digit we get y = 5.426
The Characteristic of x =2
∴
N = yx102 = 5.426 × 102 = 542.6
Similarly
If log N = 0.8121, then N = 6.487
Now we find out the characteristic of x.
If log N = I.8121 then N = 0.6487
If x has characteristic n then the number
If log N = 3.8121 then N = 0.006487