KAMALJEETH INSTRUMENTS
TORSIONAL PENDULUM:
DETERMINATION OF TORSIONAL
CONSTANT AND YOUNG’S MODULUS
OF A WIRE
Introduction
After the inversion of simple pendulum in 1602 by Galileo, different types of
pendulums came in to existence among them, the Bar pendulum, Torsional pendulum,
Kater’s pendulum, Foucault pendulum and Gridiron pendulum are well known. These
entire pendulums were time sensitive and made to indicate time. In additions these
pendulums were used to study the basic properties of metallic wires, beams and discs.
Determination of moment of inertia, various elastic constants were common. The
torsion pendulum was invented by Robert Leslie in 1793 and the torsion pendulum
clock was first invented and patented by American Aaron Crane in 1841. He made
clocks that would run up to one year on a winding.
Anton Harder of Germany independently invented and patented the torsion clock in
1879-1880. He was inspired by watching a hanging chandelier rotate after a servant had
turned it to light the candles. Although they were successful commercially, torsion
clocks remained poor timekeepers.
A Torsional pendulum, or Torsional oscillator, consists of a circular metallic disk with
sufficient weight suspended with a thick wire. When the mass is twisted about the axis
of the wire, the wire exerts a torque on the mass, tending to rotate it back to its original
position. If twisted and released, the mass will oscillate back and forth, executing
simple harmonic motion. From the study of the period of the oscillation, with its length
from where it is suspended provide information about rigidity modulus of the wire
twisting the mass.
Theory of Torsional Pendulum
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KAMALJEETH INSTRUMENTS
Figure-1: Torsional pendulum and amplitude of oscillation dies as the time pass
Figure-1 shows a Torsional pendulum consists of a circular disc of mass ‘M’, radius ‘R’
suspected from a fixed point through a thick wire (1mm) of radius ‘r’ and length ‘l’. The
disc is give small angular twist ‘θ’ to perform simple harmonic oscillation. The
amplitude will maximum at the beginning and time proceed the amplitude will
decrease whereas the period of oscillation will remain the same as shown in Figure-1
If ‘τ’ is the torque generated by the angular force then we can write
τ = − cθ
…1
Where ‘c’ is couple per unit twist
If ‘I’ is the moment of inertia of the suspended disc, then the torque τ is given by
τ=I
θ
…2
Equating Equations 1 and 2 we get
θ
=−
θ
…3
This is equation represent the simple harmonic motion with angular frequency
ω=
= 2̟f =
̟
Hence the period of oscillation is given be
T = 2̟
…4
The constant c is also known as torsional constant which can be determined by
observing shear taking place in the wire as shown in Figure-2.
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KAMALJEETH INSTRUMENTS
Figure-2: Twisting of the wire and shear produced
Consider a thick cylindrical wire of radius ‘r’ and length ‘l’ whose cross is shown in
Figure-2 fixed at A and other end B is fitted to heavy circular disc. For analysis consider
a circular section of the wire with radius and ‘a’ and ‘a+da’ (the shaded grey segment).
If P is point on this segment at the fixed end of the wire, and Q’ is the corresponding
point at the weight hanging end of the wire. When the wire is sheared or twisted due to
the oscillation, the point Q’ is sheared and shifted to Q by an angle θ. The movement of
the point Q’ to Q also creates an angle φ with respect point B as shown in Figure-2.
From Fioure-2
and θ =
φ=
QQ’ =lφ =aθ
Share φ =
…5
θ
Rigidity modulus n =
n=
…6
φ
Stress required to produce share φ= nφ =
θ
The force giving this stress =Stress x area over which it is acting
F=
θ
2̟rdr =
θ
2̟a dr
…7
This force acts at every point on the section of the cylindrical wire tangential to the
circumference & normal to its radius at the point
The moment of the force about its axis = Force x Perpendicular distance
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KAMALJEETH INSTRUMENTS
Moment of the Force =
…8
θ
2̟a dr x a =
θ
2̟a dr
The moment of the force meeting on the entire cylindrical with of radius ‘r’ is
!"#
τ=
!"& * #
&
$' % (% =
where c =
!"& *
= cθ
known as couple per unit twist or Torsional constant
…9
Substituting Equation-9 in Equation-4, the period of oscillation
T=
+
T2 =
n=
=
̟
-̟
*
+
̟
̟
,
or
or the rigidity modulus of the material of the wire
-̟
...10
*
Hence determining period of the Torsional oscillations for different length, rigidity
modulus n can be determined.
Apparatus used
IR-gated Torsional pendulum consisting of a stand on which the disc allowed to
oscillate, digital microcontroller based counter, circular disc with chuck nuts, iron wire,
brass wire, digital balance (2Kg), digital screw gauge, digital vernier and spirit level.
Figure-3 shows experimental set-up used.
Figure-3: IR-gated Torsional Pendulum
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KAMALJEETH INSTRUMENTS
Experimental procedure
Part-A: Determination of dimensional parameters mass of the disc and wire
Part-B: Determining the period of the Torsional pendulum
Part-A: Determination of dimensional parameters mass of the disc and
wire
1. The mass of the circular disc, chuck nut and pointer fitted to the disc are
determined using a digital balance as shown in Figure-4
M =1.22Kg
The diameter of the disc is measured using a digital screw gauge and radius is
determined
R = 0.786mm = 0.786x10-3m
The values are tabulated in Table-1 and the moment of inertia of the disc about
CG is determined
Figure-4: Determining the mass of the disc
I=
./
=
+.
12'.3-41+'5, 6
=
3.7 31+'58
= 3.768x10@3 Kg m2
2. Now the diameters of both the wires (brass and iron) are determined using
digital screw gauge and the values are tabulated in Table-1
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KAMALJEETH INSTRUMENTS
Table-1: Dimensional parameters of the wire and disc
Mass of the disc
Radius of the disc
Diameter of iron wire
Diameter of brass wire
Diameter of steel wire
1.22Kg
0.125m
0.941mm
0.871mm
0.940mm
Part-B: Determining the period of the Torsional pendulum
3. The Torsional pendulum disc is fitted to the wire (iron) whose rigidity modulus
has to be determined. The other end of the wire is fitted to the stand top using a
chuck nut.
4. The IR-sensor is now fitted to the rear rod of the Torsional pendulum apparatus,
in level with the first red marking on it from the bottom as shown in Figure-5.
5. The sensor cable is connected to counter and switched on. The LCD display of
the counter is shown in Figure-6
6. Now the pendulum is set to oscillate by giving a torque and the pointer fitted to
the circular disc moves inside the sensor cutting IR- connectivity and the counter
states counting the disruption of the connectivity which is nothing but number
oscillations.
Figure-5: IR sensor fitting to the tripod stand
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KAMALJEETH INSTRUMENTS
Figure-6: LCD display showing default oscillation setting
7. The counter stops counting after 10 oscillations as set by the default setting. LCD
display on Figure-7: shows the time taken for 10 oscillations
Time taken for 10 oscillations = 25.655
The length of the shearing wire inside the both chuck nuts are measured using
half meter
Scale as shown in Figure-7
Figure-7: Measuring length of the shearing wire
Length of the wire l = 45.1cm = 0.451m
The readings obtained are tabulated in Table-2
Table-3: Time taken for different length of the Torsional pendulum wire iron
Time for 10 oscillations
Period T (s)
T2 (s2)
Length l
(s)
(m)
0.451
25.655
2.5655
6.581
0.425
24.757
2.4757
6.129
0.400
24.154
2.4154
5.834
0.375
23.399
2.3399
5.475
0.346
22.542
2.2542
5.081
0.318
21.633
2.1633
4.679
0.292
20.652
2.0652
4.265
0.266
19.817
1.9817
3.927
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KAMALJEETH INSTRUMENTS
8. Experiment is repeated another length by positing the sensor at the second red
mark and lifting the disc to oscillate inside the sensor.
9. In each case, the time taken for 10 oscillations are noted for each length and
period is calculated and present in Table-2.
10. A graph is drawn taking l on Y-axis and T2 on X-axis as shown in Figure-8, using
Excel the slope of the straight line is also noted.
0.5
Length (m)
0.4
0.3
0.2
0.1
0
0
2
4
6
8
T2(s2)
Figure-8: Variation of l with T2 for iron wire
Slope = 0.0703 =
11. Rigidity modulus is calculated using equation
n=
-̟
*
=
-A .+BA . - A+'5,
CB.D5 * E
0.0703 = 8.59x1010 =85.9 GPa
12. The experiment is repeated for brass and steel wires, the period and length are
tabulated in Table-4 and Figure-9 for brass and Table-5 and Figure-10 for steel
shows the vitiation of l with T2
Table-4: Time taken for different length of the Torsional pendulum brass wire
Time for 10 oscillations
Period T (s)
T2 (s2)
Length l
(s)
(m)
0.448
56.103
5.6103
31.475
0.419
54.309
5.4309
29.494
0.392
52.523
5.2523
27.586
0.368
50.908
5.0908
25.916
0.340
48.865
4.8865
23.877
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KAMALJEETH INSTRUMENTS
0.315
0.291
0.265
47.136
45.310
43.368
Slope = 0.01437 =
n=
-̟
*
=
4.7136
4.5310
4.3368
22.218
20.529
18.807
F
-A .+BA . - A+'5,
C .7D3A+'5 * E
0.01437 = 2.392x1010 =23.9 GPa
Length l(m)
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
T2(s2)
Figure-8: Variation of l with T2 for brass wire
Table-5: Time taken for different length of the Torsional pendulum Steel wire
Time for 10 oscillations
Period T (s)
T2 (s2)
Length l
(s)
(m)
0.448
28.791
2.8791
8.289
0.419
27.895
2.7895
7.781
0.387
26.821
2.6821
7.193
0.352
25.523
2.5523
6.514
0.318
24.334
2.4334
5.921
0.280
22.840
2.2840
5.216
0.272
22.482
2.2482
5.054
Slope = 0.054 = F
n=
-̟
*
=
-A .+BA . - A+'5,
CB.-3DA+'5 * E
0.054 = 6.62x1010 =66.2 GPa
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Length l(m)
KAMALJEETH INSTRUMENTS
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
T2(s2)
Results
The rigidity or the shear modulus of iron and brass determined are listed in Table-5
Rigidity or
Shear
modulus
(GPa)
Iron (MS)
Expt.
Std.
85.9
78-76
Brass
Expt.
Std
23.9
37-40
Steel
Expt.
66.2
Std
78-76
Discussion
The value of rigidity modulus obtained is in order of standard value. Unless the exact
material content is known one cannot conclude the accuracy of the n value. The 10%
higher value of rigidity modulus obtained for iron wire indicates that the sample
material is not pure iron it may be reprocessed iron wire. Similarly brass is an alloy of
copper and zinc, the zinc is added in various proportions from 5 to 40% accordingly the
brass is graded as brass 95/5 or 60/40. Since we do not know the exact content the
accuracy of rigidity modus cannot be ascertained.
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