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Instruction Manual
Physics Laboratory
MP-101
Applied Physics Department
Sardar Vallabhbhai National Institute of Technology,
Surat395007. India
2
Contents
S.N Practical Name Page
A Note to Students 3
Error Analysis 6
1 Volume of an Object by Vernier Calipers 11
2 Diameter of a wire using Screw Gauge 15
3 Radius of curvature by Spherometer 18
4 „g‟ by Simple Pendulum 22
5 Parallel axis Theorem by Bar Pendulum 24
6 Elastic Constant by Torsional Pendulum 29
7 Young‟s Modulus by Cantilever 32
8 Modulus of Rigidity by Barton‟s apparatus 35
9 Frequency of Tuning Fork by Meld‟s Method 38
10 Prism angle by Spectrometer 42
3
A Note to Students
Introduction:
The objective of Lab Experiments along with the theory classes is to understand the basic
concepts clearly. The experiments are designed to illustrate important phenomena indifferent
areas of Physics and to expose you to different measuring instruments and techniques. The
importance of labs can hardly be overemphasized as many eminent scientists have made
important discoveries in homemade laboratories. In view of this, you are advised to conduct the
experiments with interest and an aptitude of learning.
This manual will provide the basic theoretical backgrounds and detail procedures of
various experiments that you will perform in the Physics laboratory. Before that, here are some
specific instructions for you to follow while carrying out the experiments. It also outlines the
approach that will be undertaken in conducting the lab. Please read carefully the followings.
Specific Instructions:
1. You are expected to complete one experiment in each class. Come to the laboratory with
certain initial preparation. The initial preparation will involve a prior study of the basic
theory of the experiment, the procedure to perform the experiment so as to have a rough
idea of what to do. In addition, it will also involve a partial preparation of the lab
report (journal) in advance as mentioned later in this section.
2. You must bring with you the following materials to the laboratory: This instruction
manual, journal (lab report)and graph sheets if necessary, pen, pencil, measuring scale,
calculator and any other stationary items required.
3. The format of a lab report (journal) shall be as follows:
(a) The first sheet (page) will contain your name, branch name and roll number, date and title
of the experiment.
(b) Each experiment should contain the following in order. Experiment Number, aim of
experiment, apparatus needed, a brief theory with working formulae, observation tables
with units, figures or diagrams whenever necessary. Write procedure in brief.
4
(c) Experimental observations: Data from experimental observations should be recorded in
proper tabular format with well documented headings for the columns. The data tables
should be preceded by the least counts of the instruments used to take the data and
numerical value of any constant, if any, used in the table.
(d) Graphs: (whenever applicable) Always label your graph properly. Be very clear to write
the proper units, scale, experiment number, etc.
(e) Relevant calculations: Calculations should be done neatly and carefully in proper unit.
Error analyses are must to present your result.
(f) Final results along with error estimates.
(g) Remarks if any.
4. You must record your data directly in your lab record (Journal).Switch off any power
supply etc. used and put back the components of the apparatus in their proper places.
Complete the rest of the relevant calculations, error analysis, graphs (if necessary),
results, and conclusion and obtain your grade of the performed experiment before
leaving the laboratory from concerned Faculty (Instructor).
5. Last but not the least - please handle the instruments with care and maintain utmost
discipline and decorum of the laboratory.
6. Be honest in recording your data. Never cook up the readings to get desired/ expected
results. You never know that you might be heading towards an important discovery.
Graphs
A graph is simply a diagram illustrating the relationship between two quantities, one of
which varies as the other is changed. The quantity that is changed is called “independent
variable”, the other is called the “dependent variable”. The following general points should be
noted:
1. Scale must not be too small – loss of accuracy, scale should not be too large –
exaggeration of accidental errors. Scales on each axis are chosen usually the same unless
one variable changes much more rapidly than the other, in which case it is plotted on a
smaller scale.
2. The independent variable is placed / plotted horizontally and dependent variable placed /
plotted vertically.
3. The origin need not represent the zero values of variables – unless definite reference to
the origin is required.
5
4. Graph should be titled. It should have captions containing - a – standard name of variable
– b – its symbol, if such a thing exists, and – c – standard abbreviation for the unit of
measure.
5. Numerals representing scale values should be placed outside the axis. Values less than
unity should be written as 0.47, not .47. Use of too many ciphers should be avoided. Thus
if scale numbers are 10,000; 20,000; 30,000 etc. They should be written as 1.0, 2.0, 3.0
with the caption – say pressure in 104 N/m
2. Similarly scale numbers 0.0001; 0.0002;
0.0003 etc. Should be written as 1.0, 2.0, 3.0 with the caption – say pressure in 10-4
N/m2
6. All letterings should be easily readable from the bottom of the graph.
Example:
0 1 2 3 4 5
20
40
60
80
100
120
140
160
180
200
Tem
per
atu
re (
deg
. C
)
Distance (cm)
Sample 1
Sample 2
Sample 3
6
AN INTRODUCTION TO ERROR ANALYSIS
Suppose the length of an object is measured with a meter scale and the result is given as
11.3 cm. Does it mean that the length is exactly 11.3 cm? The chances are that the length is
slightly more, or slightly less, than the recorded value but as the least count of the scale is 1 mm
(it cannot read fraction of a mm) the observer rounds off the result to the nearer full mm. Thus,
any length greater than 11.25 cm and less than 11.35 cm we can only conclude that the actual
length is anywhere between 11.25 and 11.35 cm. The maximum uncertainty (on either side) or
the maximum possible error, δ1, is 0.05 cm which is half the less count of the scale.
Let the object under consideration be a glass plate. To obtain the volume of the plate, suppose we
measure the width „b‟ with slide callipers and the thickness „t‟ with a screw gauge – whose least
counts are respectively 0.1 mm and 0.01 mm. Let the result obtained, after averaging over many
measurements, be
b = 2.75 cm
t = 2.52 mm = 0.252 cm
and 1 = 11.3 cm
as measured by a meter scale with one end at zero exactly! We note that the coincidences noted
in the vernier scale on the head scale of the screw gauge might not have been exact and represent
only the nearest exact reading. Hence these measurements also include the corresponding
uncertainties each equal to half the least count. So we have
1 = 11.3 ± 0.05cm
B= 2.75 ± 0.005 cm
t = 0.252 ± 0.0005 cm
Note that ± 0.05 cm, ± 0.005 cm, ± 0.0005 cm are actually instrumental errors. Personal errors –
like reading 11.3 as 11.2 or 11.4 are not taken into account. To avoid personal errors average
values of many readings has to be used. The volume calculated from the recorded values of 1, b
and t is
V = (11.3 2.75 0.252) = 7.8309 cm3
Take care to avoid writing cm as mm, mm as cm etc. This is also personal error but a careless
one at that.
However, since each observation is subject to an uncertainty, there should be an uncertainty in
the result V too. How can the cumulative effect of the individual uncertanities on the final result
be estimated?
Let the maximum error in V due to δ1, δb, and δt be δV. Then,
(V ± δV) = (1±δ1)(b ± δb)(t ± δt)
7
V + δV will corresponds to maximum positive values of δ1, δb, δt,
(V + δV) = (1+ δ1)(b +δb)(t + δt)
Or
V(1 + δV/V) =1bt (1+δ1/1)(1 +δb/b)(1 + δt/t)
Cancelling V = 1bt from both sides and using the approximation
(1 + x) (1 + y) (1 + z) = 1 + x + y + z as x<<1, y<<1, z<<1,
We obtain
δV/V = δ1/1 + δb/b + δt/t
The relative error in the product of a number of quantities is the sum of the relative errors of the
individual quantities.
δ 1 /1 = 0.05 / 11.3 =0.0044
δb / b = 0.005 / 2.75 = 0.0018
δt / t = 0.0005/ 0.252 = 0.002
δV / V = 0.0082
From the value V= 7.8309 , we have
δV = 7.8309 0.0082 = 0.064213 cm3
(rounded off to one significant digit).
The result of the measurements is therefore
V = 7.8309 ± 0.06 cm3
An important point to be noted is that writing the volume as 7.8309 cm3 would convey the idea
that the result is measured accurate to 0.0001 cm3. We know from the calculated error that this is
not the case and error is in the second decimal place itself. We are not certain that the second
decimal is 3 but it may be 3 + 6. The volume may be anywhere in the range 7.77 to 7.89 cm3. As
the second decimal place is subject to such an uncertainty, it is meaningless to specify the
subsequent digits. This result should therefore be recorded only up to the second decimal place.
[The error could be much larger if the least counts themselves are taken into account].
Thus, V = (7.83 ± 0.06) cm3
It is the calculation of the maximum error in the result, based on the least counts of the different
instruments used that can indicate the number of significant digits to which the final result is
accurate. Suppose we now measure the mass of a plate correct to a milligram and the result is
8
m = (18.34 ± 0.005)gm
The density „d‟ can be calculated from m and V.
d = m/ v = 18.34/ 7.83 = 2.3423 gm cm-3
To estimate the uncertainty in d , we write
(d + δd) = m + δm/ - δν
As the maximum density will correspond to the greatest mass and least volume.
d(1 + δd/d) =
1 + δd/d = -1
As and are very much less than 1,
δd/d = +
The relative error in the quotient of two quantities is (also equal to the sum of the individual
relative errors).
= 0.0085 2.3423 = 0.02 gm/ cm-3
Therefore d= (2.3423 ± 0.02) or (2. 34 ± 0.02) gm / cm-3
[The error in measurements may be many times the least count if the instrument is not properly
designed. Least count may often signify readability/ resolution and not the accuracy. Repeated
measurements falling outside the least counts are indicative of this]
Other situations:
1. Suppose x is the difference of two quantities a and b, whose measurements have
maximum possible errors as δa and δb. What is δx?
X =a – b
(x ± δx) = (a ± δa) – (b ± δb)
The maximum value of the difference x corresponds to maximum a and minimum b
(x + δx) = (a + δa) – (b - δb)
= (a - b) + (δa + δb)
9
Cancelling x = a – b,
δx =δa + δb
In a sum or difference of two quantities, the uncertainty in the result is the sum of the actual
uncertainties in the quantities – (Not the relative uncertainties).
2. If p =
First δ(1 + m) = δ1 + δm
Y2 can be dealt with as a product of y and y.
Questions:
1. Suppose x = (a+b)/ (c-d). To minimize the uncertainty in x, which of the four quantities
must be measured to greatest accuracy, if all four quantities a, b, c, d are of the same
order of magnitude?
2. The period of a simple pendulum is measured with a stop watch of accuracy 0.1 second.
In one trial 4 oscillations are found to take 6.4 secs, in another 50 oscillations take 81
secs. In this relative uncertainty depends only on the least count of the instrument – in
this case the stop watch? How can the relative uncertainty in the period be minimized?
3. The refractive index of a glass slab may be determined using a vernier microscope as
follows. The microscope is focused on a marking on an object placed on a platform and
the reading, a, on the vertical scale is noted. The glass slab is placed over the object. The
object appears raised. The microscope is raised to get the image is focus and the position
on the scale, b, is again noted. The last reading, c, is found raising the microscope to
focus on a tiny marking on the top surface of the slab. The least count of the vernier scale
is 0.01mm. The readings a, b, c are respectively 6.128 cm, 6.497 cm, and 6.128 cm.
Calculate the refractive index and the percentage error in the result. Express the result to
the accuracy possible in the experiment, along with the range of error.
10
Note:
In the above case cited we have used our judgment i.e. the ability to estimate the reading to ONE
HALF the least count of the instrument. If we take that the actual error is ONE least count on
either side of the measured quantity all the errors calculated in the above cases would be
doubled.
References
1. Practical physics – by G.L.Squires, Cambridge University Press,4th edition, 2001.
2. A text book of Practical Physics by M.N. Srinivasan, S. Balasubramanian and R.
Ranganathan, Sultan Chand and Sons, First edition , 1990.
11
EXPERIMENT-1
VOLUME OF AN OBJECT BY VERNIER CALIPERS
Aim: To measure the length and diameter of the given solid cylinder using vernier calipers and
to calculate the volume of the solid cylinder.
Apparatus: Vernier calipers and solid cylinder.
Theory:
The vernier calipers consists of main scale (MS) graduated in cm along the lower edge of the
frame and inches along the upper edge of the frame. To one end of the frame, two fixed jaws are
provided. There are two movable jaws, which slide over the main scale, and it can be fixed at
any point using a screw S. The movable jaw attachment carries a vernier scale (VS).
Procedure
i) To find the Least Count
Least Count is the minimum measurement that can be made with the given instrument. The value
of 1 main scale division (MSD) and number of divisions (n) on the vernier scale are determined.
The least count of the instrument is calculated using the formula
Least Count (LC) =
ii) To find the Zero Error (Z.E) and Zero Correction(Z.C)
The instrument is checked to find initial error. If there is any initial error, suitable correction is to
be done. When the two jaws are in contact, the graduations are made such that the zero of the
MS coincides with the zero of the VS. In this case, the instrument has no error. If the zero of the
VS is on the right side of the zero of the MS, it has positive error. Then the zero correction is
negative. If the zero of the VS is on the left side of the zero of the MS, it has negative error. Then
zero correction is positive. In each case, note the division of the vernier scale(VS) which is
coinciding with any one of the main scale divisions. This is vernier scale coincidence (VSC).
Applying this in the given formula, the value of zero error is determined. Hence zero correction
(ZC) is also determined.
12
iii) To find the length and diameter of the solid cylinder
The given cylinder is gently placed in between the two lower jaws of the vernier calipers such
that the length of the cylinder is parallel to the scale. The completed main scale reading (MSR) is
taken by noting the position of zero of the vernier scale (VS). The vernier scale coincidence
(VSC) is then noted. VSC is that particular division which coincides with any one of the main
scale divisions, making a straight line. The MSR and VSC are noted in the tabular column I for
different settings of the cylinder. Then the Vernier Scale Reading (VSR), Observed Reading
(OR) and Correct Reading (CR) are calculated using the formula given. The average value of CR
is determined, which gives the length of the given cylinder. Similarly the readings for diameter
of the solid cylinder are taken by placing the cylinder suitably in between the lower fixed jaws
and the readings are noted in the tabular column II.
No error Positive error Negative error
No correction Negative correction Positive correction
VSC = 3 div VSC = 7 div
ZE = + (VSC) x LC ZE = -(10-VSC) x LC
= + 3 x 0.01 cm = - (10-7) x 0.01cm
= + 0.03 cm = - 0.03 cm
ZC = - 0.03 cm ZC = + 0.03 cm
Zero error and zero correction of the given vernier callipers
Formula:
Volume of the solid cylinder = πr2l
13
Where
l = length of the cylinder
r = radius of the cylinder
Length / diameter of the solid cylinder = Observed reading +Zero correction
= {MSR + (VSC x LC)} + ZC
Where MSR = Main Scale Reading
VSC = Vernier Scale Coincidence
LC = Least Count
ZC = Zero Correction
Vernier Scale Reading (VSR) = Vernier Scale coincidence (VSC) x Least Count (LC)
Observed Reading (OR) = Main Scale Reading (MSR) + Vernier Scale Reading (VSR)
Correct Reading (CR) = Observed Reading (OR) + Zero Correction (ZC)
Observations
(i) Least count:
Value of 10 MSD = cm
Value of 1 MSD = cm
No. of division on the vernier scale (n) =
(ii) The three possible initial error and initial correction:
Least Count (LC) = =…………=……….cm
Observation Table:I
To find the length of the solid cylinder
LC = ………cm ZE = ………cm ZC =……… cm
Sr.
No
MSR VSC VSR = VSC x LC OR = MSR + VSR Length
CR = OR + ZC
Unit cm div cm cm cm
1
2
3
4
5
14
The average length of the solid cylinder (L)=……… cm
= ………..x 10-2
m
Observation Table: II
To find the diameter of the solid cylinder
LC =…….. cm ZE =……… cm ZC =………. cm
Sr.
No
MSR VSC VSR = VSC x LC OR = MSR + VSR Diameter
CR = OR + ZC
Unit cm div cm cm cm
1
2
3
4
5
The average diameter of the solid cylinder (D) =……… cm
=………x 10-2
m
Calculation
1. The average radius of the solid cylinder „r‟ =………. x 10-2
m
2. Volume of the solid cylinder = π r2l = …………..m
3
Result
1. The length of the solid cylinder l = -------x 10-2
m
2. The diameter of the solid cylinder d = x 10-2
m
3. The volume of the solid cylinder v = x 10-6
m3
15
EXPERIMENT-2
DIAMETER OF A WIRE USING SCREW GAUGE
Aim : i. To measure the diameter of a wire by using micrometer screw gauge.
ii. To measure the thickness of a metal glass plate by using micrometer screw gauge.
Apparatus: Micrometer screw gauge, wire, metal plate or glass plate etc.
16
Procedure:
1. Find out the distance (X) through which screw travels on the main scale in five rotations
of the circular scale. Hence distance travelled by the circular scale in one rotation is
called the pitch of the screw.
2. Note down the number of divisions on the circular scale. Hence knowing pitch, L.C.
(Least count) of the screw can be determined (See diagram - 1).
3. Find out zero error of micrometer screw. This can be found out as follows: When the two
jaws of micrometer screw gauge are brought in contact with each other without applying
unduer pressure, the zero on the circular scale should coincide with the reference line on
main scale. If this happens then there is no zero error (See diagram - 2). If this is not
observed then the micrometer screw has zero error.
There are two possibilities
Zero of circular scale is above the reference line:
When zero of the circular scale is above the reference line as shown in diagram - 3, then
zero error is negative. Count the divisions from zero of circular scale coinciding with
reference line (m). Negative sign is for negative error. The value of zero error can be
found out by multiplying it by least count. [(m) L.C.]
Zero of circular scale is below the reference line:
When zero of the circular scale is below the reference line as shown in diagram - 4, then
zero error is positive. The division of circular scale coinciding with reference line is
noted (+m). Positive sign is for positive error. The value of zero error can be found out by
multiplying it by least count. [+(m) L.C.]
Note that any micrometer screw gauge apparatus can have one and only one
possibility out of the three i.e. no zero error, negative zero error and positive zero
error. Also zero error is related to the apparatus.
4. Hold the wire, between the two jaws of micrometer screw gauge, whose diameter is to be
measured without applying undue pressure. Note down the reading of the main scale
(M.S.R.) or pitch scale (P.S.R.). This is „a‟.
5. Note down the circular scale reading coinciding with reference line. This is C.S.D. (n).
Multiply this by L.C. to obtain circular scale reading (C.S.R.).
17
6. The total reading (T) can be found out by adding main scale reading and circular scale
reading i.e. T = a + b.
7. To obtain correct reading, zero error which is found out in procedure (3) is subtracted
from total reading. Thus correct reading of the diameter is obtained.
8. Five readings of the diameter of the wire are taken by using procedure (4) to (7). Hence
mean corrected diameter of the wire can be measured.
9. To find the thickness of the metal plate (or glass plate) instead of the wire, glass or metal
plate is held between the jaws of the micrometer screw gauge. The procedure (4) to (8) is
repeated, resulting in mean thickness of the plate (metal or glass).
Observations:
(i) Distance (X) travelled by the circular scale in five rotations = .......... cm
(ii) Pitch of the screw = = ...........
(iii) Number of divisions on the circular scale = ...........
(iv) L.C. = Pitch No. of divisions on circular scale = ............
(v) Zero error, Z = ±(m) (L.C.) = ............
Observation table:
Object Obs. M.S.R.
a (cm)
C.S.D.
n (div)
C.S.R.
b=n x
(L.C.) cm
Total
Reading
cm
Correct
Reading=
T-Z (cm)
Wire 1
2
3
4
5
Plate 1
2
3
4
5
Result:
1. Diameter of wire = .......... cm
2. Thickness of the plate (metal or glass) = .......... cm
18
EXPERIMENT-3
RADIUS OF CURVATURE BY SPHEROMETER
Aim – To find the radius of curvature of a convex lens using spherometer.
Apparatus – A spherometer, a plane glass slab, a convex lens of appropriate diameter (about
8cm to 10 cm), a vernier calipers and wooden blocks for supporting the lens
Theory – It is a device to measure the thickness of a thin plate and the radius of curvature of any
spherical surface (concave /convex mirror or a lens). It carries a small vertical scale usually
divided into millimetres. The body of the instrument is supported on three legs whose lower tips
form an equilateral triangle and lie in one plane. A screw which carries a circular scale (having
100 or 50 divisions) at its top is so supported that the tip of this screw is at circum- centre of the
triangle formed by the tips of the legs. The distance through which the screw advances along the
vertical scale in one full rotation is called pitch of the spherometer. It is usually 1 mm or 0.5 mm.
If the pitch is 1 mm and the circular scale has 100 divisions, then this means that when the
circular scale is rotated by 100 divisions, the screw moves through distance 1mm. Therefore
when rotated through one division, it moves through 0.01 mm. This is the least count of the
instrument.
.
19
Zero Error- If the instrument is correct, then the zero of the circular scale should coincide with
the zero of the vertical scale when the tip of the screw is in the plane of the tips of the legs. This
is seldom so and hence the instrument has initial error called zero error. The correction whether
positive or negative depends upon the final reading whether taken by moving the screw
downward or upward. It is not quite necessary to find the zero error if the number of
divisions through which the circular scale is rotated is measured correctly and carefully.
To determine the radius of curvature of a spherical surface-
The formula for calculating R can be easily derived.
Fig 1 Fig 2
From the geometry of the circle (Fig 1), we have
AD2= ED. DF
or r2 = h(2R-h)
or 2 R h = r2 + h
2
20
or R = r2/2h + h/2
If l is the length of each side of the equilateral triangle ABC formed by joining the tips of
the three outer legs, then as shown in Fig 2,
I/2 = r cos = r
Substituting r = , we have
Procedure –
1. Find the least count of the spherometer.
2. Raise the screw by tuning its head so that it may be above the plane containing the tips of
the three legs. Place the spherometer on the page of your notebook on which experiment
is to be done. Press the spherometer to get the impression of the tips of three legs. Mark
the position of each of the three points by drawing a small circle around each point.
Measure the length of each side of the triangle formed by joining these points and let the
mean value. Let the mean length be l cm.
3. Set the convex lens firmly on a horizontal surface (table).Place the spherometer on the
surface of convex lens after raising its central leg through four or five complete rotations
of the circular scale. Turn the screw so that the central leg while moving downwards
touches the surface of the lens. Note the circular scale reading against the edge of the
vertical scale. Repeat this step three times
4. Place the spherometer on the surface of the plane glass slab or a table. Turn the screw in
the same directions as in the previous step till tip of the central leg just touches the plane
surface. Count the number of complete rotations and the additional number of circular
divisions moved.
5. Repeat the step three times by moving the central leg every time in the upward
direction through a sufficient large distance as compared to the value of ‘h’.
Observations:
(i) Distance moved by the screw in 4 complete rotations =
(ii) Pitch of the screw =
(iii)Total number of divisions on circular scale (n) =
(iv) Least count of the spherometer = Pitch/ n =
(v) Mean distance between the legs l =
21
Observation Table:
Sr.No. No. Complete rotation
(m)
No. of additional
circular division
moved (X)
h= (m × pitch) +
(x×L.C.)
Mean
h
1 2 3 4 5
Calculation:
Formula-
Radius of curvature of the lens
Results: Radius of curvature of the lens R=............. cm
22
Experiment-4
„g‟ BY SIMPLE PENDULUM
Aim: Determine the gravitational acceleration „g‟ using the simple pendulum‟s period
Apparatus: Meter stick, simple pendulum, timer.
Theory:
A pendulum consists of a "bob" attached to a string that is fastened such that the
pendulum assembly can swing or oscillate in a plane. For an ideal pendulum, all the mass is
considered to be concentrated at a point in the center of the bob.
Some of the parameters of a simple pendulum include the length (L), the mass of the bob
(m), the angular displacement θ through which the pendulum swings, and the period T of the
pendulum, which is the time it takes the pendulum to swing through one complete oscillation.
Fig.1.
When the angular displacement is minimal (θ < 100) the period of a pendulum can be
determined with the following equation.
Notice that the period of a pendulum is independent of the mass of the bob. Also, for small
displacement angles, the period is independent of θ. Thus, when the period T and length L of a
pendulum can be accurately ascertained, it is possible to accurately determine the acceleration of
gravity.
23
where L/T2 represents the slope of data collected and graphed.
Observation Table
Length
of string
L (cm)
Time taken for 20 oscillation
t (sec)
Period of
oscillation
T=t/20 (Sec)
T2
(Sec2)
L/T2
(cm/sec2)
t1 t2 t =(t1+t2)/2
90
80
70
60
50
40
From The Calculation
From the Graph(LT2)
=-------------- cm/sec2
( =------------ cm/sec2
Result:
1. Gravitational acceleration „g‟=........................ cm/sec2 (from the calculation)
2. Gravitational acceleration „g‟=........................ cm/sec2 (from the calculation)
24
EXPERIMENT-5
ACCELERATION DUE TO GRAVITY USING BAR PENDULUM
(Parallel axis Theorem by Bar Pendulum)
Aim: To determine acceleration due to gravity using bar pendulum.
Apparatus: Bar pendulum, wedge, stop watch , meter rod and graph paper
Theory:
The purpose of this experiment is to use the angular oscillations of a rigid body in the form of a
long bar for determining the acceleration due to gravity. The particular form of the body is
chosen for the sake of simplicity in performing the experiment. The bar is hung from a knife-
edge through one of the many holes along the length, as shown in Fig. 1. It is free to oscillate
about the knife-edge as the axis. Any displacement θ, from the vertical position of the
equilibrium would give rise to an oscillatory motion just as in the case of a simple pendulum.
The difference is that since it is a rotating rigid body we have to consider the torque of the
gravitational force which give rise to the angular acceleration. Hence the period of oscllation is
given by
(1)
where I is the moment of inertia of the bar about the axis of rotation through O, M is its mass and
d the distance between O and the centre of mass of the bar C (Fig. 1).
Fig. 1. Experimental set-up of bar pendulum.
25
The parallel axis theorem gives
(2)
I0 being the moment of inertia of the bar about the parallel axis through its centre of mass; so that
equation (1) becomes
(3)
Note that T becomes very large as d → 0 as well as d → ∞. You can check that T as a function
of d undergoes through a minimum. The plot of T as a function of d has a shape shown in Fig. 2.
Fig. 2. Plot between time period T, and distance d.
From equation (3),
For a given T,it is a quadratic equation in d. So there are two values d1 and d2, the sum of which
is
or,
26
(4)
Procedure:
1. Find the centre of mass of the bar by balancing it on a knife-edge and put a mark there.
This is necessary to be able to measure the distance of the various points of suspension
from the centre of mass, i.e., ds.
2. The bar has a number of holes in it. A Knife-edge can be passed through a hole and
secured to the bar by means of two nuts at the end of the edge. The bar is then suspended,
with edge resting on the glass plates of a bracket fixed to the wall. Check that it is the
edge, not rough top, of the knife-edge that is resting and that the bar is free of the bracket.
3. Decide on how many oscillations to time by using error estimates in all the
measurements. Set the bar oscillating with a small amplitude and time the oscillations
Decide on where do you start the count. Take three readings for each hole, i.e., the same
value of d. Start at one end of the bar and systematically take readings for determining T
for each available value of d till you reach closest to the centre of mass.
4. Plot a graph of T against d as in Fig.(2). Draw a smooth curve through the points. Draw a
line parallel to the d-axis cutting the curve at points A and B. Then QA = d1 and QB =
d2. Also T = CQ. Get g from equation (4). Do it for at least two values of T.
Observation
(i) Weight of bar M=………….gm
(ii) Length of bar L=………….. cm
(iii) Moment of inertia of a bar w.r.t. the axix passing through center of gravity
(C.G.), Icg=Ml2/12=…………. gm.cm
2
27
Observation Table:I
Sr. No. d (cm) No of
Oscillation
Time of Oscillation (sec) Mean
Time
t (sec)
Time
period
(sec) 1 2 3
1
2
3
4
5
Calculation:
= …………… cm/sec2
Error Calculation:
Observation Table: II
Sr.
No.
Distance
from C.G.
to
suspension
point
d (cm)
d2
cm2
Time for 20
oscillations
sec
Aveg.
t
sec
Time
period
T=t/20
sec
I=Mg/4Π2
[T2d]
gm.cm2
Md2
gm.cm2
I=Icg+Md2
gm.cm2
t1 t2
Calculation
I=Mg/4Π2 × [T
2d]
(compare with I=Icg+Md2)
From Graph (Id2)
Slope=AB/BC=Weight of Bar=.....................gm
28
Intercept=Icg=..................... gm.cm2
Results:
1. Gravitational acceleration „g‟=........................ cm/sec2 (from observation Table I)
2. The Parallel axis theorem in verified and obtained results from the observation table 2.
(i) Weight of Bar M=............... (From Graph)
Weight of Bar M=............... (From Calculation)
(ii) Moment of Inertia Icg=..................(From Graph)
Moment of Inertia Icg=.................. (From Calculation)
29
EXPERIMENT-6
TORSIONAL PEDULUM RIGIDITY MODULUS
Aim: To determine the rigidity Modulus of the given wire by dynamical method.
Apparatus: Torsional pendulum, stop watch, screw guage, vernier calipers, scale.
Theory:
A heavy cylindrical disc suspended from one end of a fine
wire whose upper end is fixed constitutes a Torsional
pendulum. The disc is turned in its old plane to twist the
wire, so that on being released, it executes torsional
vibrations about the wire as axis.
Let ϴ be the angle through which the wire is twisted.
Then the restoring couple set up in it is equal to
(Πna4θ)/2l=cθ
Where (Πna4)/2l=c (1)
is the twisting couple per unit (radian) twist of the wire.
This produces an angular acceleration (dw/dt) in the disc
therefore if “I” is the moment of inertia of the disc about
the wire we have
Idw/dt=-C. θ
dw/dt= -(C/I). θ
i.e the angular acceleration (dw/dt) of the angular displacement(θ) and therefore its motion is
simple harmonic hence time period is given by
(2)
30
From 1 and 2
In case of a circular disc whose geometric axes coincide with the axis of rotation. The moment of
inertia “I” is given by
where M is the mass of disc and “R” is the radius of the disc.
Procedure:
1. The radius of the suspension wire is measured using a screw gauge.
2. The length of the suspension wire is adjusted to suitable values like 40 cm, 60 cm, 80 cm,
100 cm. etc.
3. The disc is set in oscillation. Find the time for 20 oscillations twice and determine the
mean period of oscillation ' T '.
4. Calculate moment of inertia of the disc using the expression, I = (1/2)MR2
.
5. Determine the rigidity modulus from the given mathematical expression.
Observation:
(i) Radius of the wires ( ) =................ cm
(ii) Radius of the disc (R)=.................. cm
(iii) Mass of the disc (M) =.................... gm
Observation Table:
Length of
string
L (cm)
Time taken for 20
oscillation
t (sec)
Period of
oscillation
T=t/20 (sec)
T2
(sec2)
L/T2
(cm.sec-2
)
Mean
L/T2
(cm.sec-2
)
t1 t2 t
=(t1+t2)/2
40
60
80
100
120
31
Calculation:
From Calculation:
=-------------------- dynes/cm2
From Graph:
Plot a curve for l Vs T2 and calculate the slope.
=------------------- dynes/cm2
Result: The rigidity modulus of the given wire using dynamical method
η= ------------------dynes/cm2 (From Calculation)
η = ------------------dynes/cm2 (From Graph)
32
EXPERIMENT-7
YOUNG’S MODULUS BY CANTILEVER METHOD
Aim: To determine Young‟s modulus by cantilever method
Apparatus: Meter rule, half-meter rule, G-clamp, retort stand and clamp, Thread,
50 g slotted mass hanger, set of 50 g slotted masses, wooden block,
pair of vernier calipers, micrometer screw gauge, scissors
Theory:
A cantilever is a beam supported on only one end. The beam carries the load to the
support where it is resisted by moment and shear stress. Cantilever construction allows for
overhanging structures without external bracing. In most applications, objects are assumed to be
rigid for the purpose of simplification. When supposedly rigid materials are subject to great
forces, there is a permanent deformation. When subject to a particular stress, or force per unit
area, materials will respond with a particular strain, or deformation. If the stress is small
enough, the material will return to its original shape after the stress is removed, exhibiting its
elasticity. If the stress is greater, the material may be incapable of returning to its original shape,
causing it to be permanently deformed. At some even greater value of stress, the material will
break.
There are different types of stress: tension or tensile stress, compression or compressive stress,
shear stress, and hydraulic stress. The quantity for all types of stress, however, can be defined as
follows:
where F is the force applied and A is the cross-sectional area of the material. The standard units
of stress are [N/m2]. The quantity strain can be defined as follows:
where L is the original length of the material, and ∆L is the change in length that results after the
stress is applied.
33
The Young‟s modulus E of wood of the meter rule is given by
where M = mass of the slotted mass, d = deflection of the end of the ruler.
s = gradient of graph d against M
Fig. 1.
Procedure:
1. The average thickness of the meter rule is measured by using a vernier caliper and the
reading is recorded.
2. By using a micrometer screw gauge, the average width of the meter rule is measured and the
reading is recorded.
3. The apparatus is set up as shown in Fig. 1.
4. A 10 g of slotted mass is hung and the length of the deflection is observed.
34
5. The reading obtained is tabulated in a table.
6. Step 4 and 5 is repeated by using a slotted mass of 20 g, 30 g, 40 g, 50 g and 60 g.
7. A graph of deflection of the end of the ruler, d against mass, M is plotted.
8. The Young‟s modulus, E of wood of the meter rule is calculated.
Observation:
(i) Reading of the thickness and width of the meter rule
(1) t1= ……. cm (2) t2= ……. cm (3) t3 = ………cm (4) t= (t1+t2+t3)/3=………… cm
(2) b1= ……. cm (2) b2= ……. cm (3) b3 = ………cm (4) b= (b1+b2+b3)/3=………… cm
Observation Table:
Sr.No. Mass (g) Distance (d) cm Average
d=
(d1+d2)/2
Loading
(d1)
Unloading
(d2)
1 0
2 10
3 20
4 30
5 40
6 50
7 60
8 70
9 80
10 90
11 100
Calculation:
From Calculation:
=……………N/m2
From Graph:(dM)
=………….. N/m2
s = gradient of graph d against M
Results:
35
EXPERIMENT-8
MODULUS OF RIGIDITY BY BARTON’S APPARATUS
Aim: To determine the modulus of rigidity of the material of a given rod by static method.
Apparatus: Torsion apparatus , Slotted weights with hanger , Thread , Meter scale and Screw
gauge.
Theory:
Working Formula:
C= 2ΠR
Where
η is the modulus of the rigidity of the materials of a given rod.
M is the mass suspended
g is the acceleration due to gravity (g =980 cm/sec2)
L is the length of the rod between the two pointers
36
C is the circumferences of the pulley
R is the radius of the pulley.
r is the radius of the rod.
ϴ1 is the angle of twist produced at pointer No.1
ϴ2 is the angle of twist produced at pointer No.2
Observations:
(i) Least count of screw gauge LC = ____________
(ii) Circumference of the pulley C = ____________ cm.
(iii)Radius of the pulley R = ____________________cm.
(iv) Length of the rod between the two pointers = L=
Observation Table:
Sr.No. Mass
Suspended
M (g)
Reading on Scale(ϴ) ϴ= ϴ1+
ϴ2
M/ ϴ Average
M/ ϴ ϴ1
loading
ϴ2
unloading 1 0
2 500
3 1000
4 1500
5 2000
6 2500
Calculation:
Actual Value, η= 4.55 × 1011
dynes/cm2
Percentage of error =
From Graph (M )
Slope=M/ ϴ
×Slope
37
Results:
= ---------------------( calculation) dynes/cm2 Percentage error =………… dynes/cm
2
= ---------------------( graph) dynes/cm2 Percentage error =………… dynes/cm
2
38
EXPERIMENT-9
FREQUENCY OF TUNING FORK BY MELD’S METHOD
Aim: To determine the frequency of AC mains by Melde‟s experiment
Apparatus: Electrically maintained tuning fork, A stand with clamp and pulley, A light weight
pan, A weight box, Analytical Balance, A battery with eliminator and connecting
wires etc.
Theory:
STANDING WAVES IN STRINGS AND NORMAL MODES OF VIBRATION:
When a string under tension is set into vibrations, transverse harmonic waves propagate
along its length. When the length of string is fixed, reflected waves will also exist. The incident
and reflected waves will superimpose to produce transverse stationary waves in the string.
The string will vibrate in such a way that the clamped points of the string are nodes and
the point of plucking is the antinode.
Fig. 1. The Envelope of a standing waves
A string can be set into vibrations by means of an electrically maintained tuning fork, thereby
producing stationary waves due to reflection of waves at the pulley. The loops are formed from
the end of the pulley where it touches the pulley to the position where it is fixed to the prong of
tuning fork.
Fig. 2
39
(i) For the transverse arrangement, the frequency is given by
where L is the length of string in fundamental modes of vibrations, T is the tension applied to
the string and m is the mass per unit length of string. If p loops are formed in the length L of the
string, then
(ii) For the longitudinal arrangement, when p loops are formed, the frequency is given by
Procedure:
Constant length
1. Set experiments apparatus in longitudinal position and put some weight in pan and adjust
length of string in such a way that you find loops.
2. Rotate apertures by 900 to set a B position for same value of weight in pan and same
length repeat same process. The loops formed is twice at that in (1)
3. Note readings and repeat process.
Constant Mass
4. Set experiments apparatus in transverse position and put some weight in pan and adjust
length of string in such a way that you find loops.
5. Change length of string for same value of weight in pan as the no. of produced loops
should increases or decreases.
40
Observations:
(i) Mass of the pan, m1 =……… gm
(ii) Mass of unit length of string, m =……… gm/cm
Part A: Constant length
Sr. No. No of
loops (p)
M=m1+m2
(g)
T=Mg
p2T Average
p2T
Calculation
f=............... Hz
Part B: Constant Mass
Sr. No. Mass
Suspended
on string
M=m1+m2
(g)
T=Mg
No of loops
(p)
Length of
one loop
L=l/p
Average
41
Calculation:
f=............... Hz
Results:
1. Frequency of string = f=…………..Hz
2. Frequency of tunning fork= N= 2f= …………Hz
42
EXPERIMENT-10
PRISM ANGLE BY SPECTROMETER
Aim: To determine the angle of the given prism
Apparatus: A Sodium lamp, Spectrometer, a spectrometer prism, a spirit level, magnifying
glass, a table lamp etc.
Theory:
When a beam of light strikes on the surface of transparent material (Glass, water, quartz
crystal etc.), a portion of the light is transmitted and the other portion is reflected. When a beam
of light strikes on a plane surface, the angle of reflection will be the same as angle of incidence.
If the angle between two reflected ray is measured as θ, then the angle of the prism is
A= θ/2
Procedure:
1. Cross wires of the telescope are adjusted properly and then the telescope is focused on a
distant object. A clear and sharp image is got in telescope with the proper adjustment of
the telescope screws. This will make the telescope, set for parallel rays.
2. The telescope is now turned to a position where the axis of telescope and collimator are
in the same line. The slit of collimator is adjustment so that a fine and sharp image of the
43
sodium lamp that illuminates the slit is obtained in the telescope. Now slit is at focus of
telescope wire and is adjusted for parallel rays.
3. Determination of angle of prism. To find the angle of prism, the prism is paced at centre
of prism table with its base perpendicular to the axis of collimator and the edge towards
collimator. Parallel rays from collimator will be incident on refracting surface AB and
AC are reflected by reach face.
4. The telescope is now turned to receive the light reflected from face AB and its position is
so adjusted that the image of slit is at the centre of vertical cross-wires of telescope. The
position of both venires of telescope is recorded.
5. The telescope is then turned and it receives the light reflected from face AC. Then it is
adjusted till the image of both slit of collimator coincides with the centre of vertical cross
wires of telescope. The position of both the venires is again recorded.
6. The mean difference of these two reading gives the angle 2A, and half of this angle of
prism A.
Observations:
(i) Vernier constant of spectrometer = …………
44
Observations Table:
Sr.No. Venier V1 Position of Telescope
Venier V2 Position of Telescope
Face AB Face AC Difference
Face AB Face AC Difference
1
2
3
Result:
1. Mean value of = ……………………………
2. Angle of prism A = /2 = …………………….
