1
Simple Pendulum
Kamaljeeth Instrumentation & Service Unit
Experiment-365
F
DIGITAL SIMPLE PENDULUM
Jeethendra Kumar P K, Ajeya PadmaJeeth and Santhosh K
KamalJeeth Instrumentation & Service Unit, No-610, Tata Nagar, Bengaluru-560092. INDIA.
Email: [email protected]
Abstract
Using an IR-LED pair and a micro-controller, “stop clock cum counter” is designed to
count oscillations of a simple pendulum and measure time taken for oscillations. The
instrument is tested using a simple pendulum. The values of g and the length of the
second’s pendulum are determined and compared with the corresponding standard
values.
Introduction
Galileo Galilei discovered the simple pendulum in??1602. For over 300 years, since its
discovery, until the development of the quartz clock in the 1930s, the pendulum was considered
as the standard all over the world for timekeeping. In addition to clock pendulums, free swinging
seconds pendulums were widely used as precision time keepers for scientific experiments in the
17th and 18th centuries.
Pendulum was once considered as the standard for defining meter, which is the fundamental unit
of length in the International System of Units (SI). A second’s pendulum has the length as
0.9937 m at 45° north latitude. Hence meter was defined as the length of the pendulum which
has period of two seconds at 45° north latitude. However, the situation changed after 1930,
because of variation of its length with latitude. In 1983 the meter was officially defined in terms
of the length of the second and the speed of light as "the length of the path travelled by light in
vacuum during a time interval of 1 ⁄ 299,792,458 of a second"[1].
Pendulums are used to regulate pendulum clocks, scientific instruments such as accelerometers
and seismometers. Historically they were used as gravimeters in geophysical surveys to measure
the acceleration due to gravity.
Simple pendulum
A simple pendulum consists of a weight suspended from a pivot with a weightless thread. It can
swing freely on either side of its equilibrium position. When a pendulum is displaced sideways
from its resting equilibrium position, it is subjected to a restoring force due to the earth’s gravity
that moves it back toward the equilibrium position. When released from the displaced position
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Simple Pendulum
Kamaljeeth Instrumentation & Service Unit
on either side of the equilibrium position, the restoring force combined with the pendulum's mass
causes it to oscillate, swinging it back and forth. The time taken for completing one complete
oscillation, which means starting from the extreme left point of the swing, then reaching the
extreme right point of the swing, and coming back to the initial position from where it started, is
called the time period of the pendulum.
A pendulum swings with a specific period which depends mainly on its length. The acceleration
due to gravity, g, is given by [2]
g = 4π2
…1
From Equation-1, it is seen that the period of a pendulum does not depend on its weight
depends only on its length and the value of acceleration due to gravity at its location.
but
Second’s pendulum
A simple pendulum with the time period of two seconds is known as the second’s pendulum. It
ticks twice per swing and hence four times in one complete oscillation. The length of the
second’s pendulum is, therefore, given by
T =2 s,
L=
=
.
= 0.993
This is the length of the second’s pendulum for the value of g = 9.8m/s2. It may vary a little from
place to place, depending on the value of g at the location.
Digital simple pendulum
Determination of ‘g’ using a simple pendulum is one of the most fundamental experiments in
physics lab practical classes. Using a split cork, thread, metal bob and retort stand simple
pendulum as a mechanical stop clock or digital stop clock has been routinely employed in
physics labs. Using this arrangement the value of ‘g’ was estimated in the range 8-12m/s2, as per
the laboratory records available with us. However, it appears that due care was not taken, by the
concerned teachers and/or manufacturer of instruments, to ensure that the instruments used
were accurate and were properly used. Further, care was not taken to perform the experiments,
particularly
accuracy of the results obtained. In many cases, the retort stand used for holding
the pendulum also swings along with the pendulum because of the faulty base. The length
measurement was also not done accurately and in some cases the stop clock used also was not
accurate.
In view of the importance of correctly estimating the value of ‘g’, the simple pendulum used in
the class room has been digitized by us, employing a digital microcontroller based clock and a
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Simple Pendulum
Kamaljeeth Instrumentation & Service Unit
counter. Infra-red light emitting diodes (IR LEDs) are used as sensors. One can set the number of
oscillations of the pendulum by pressing the ‘set-oscillation’ switch. A ‘clear’ switch is provided
to clear the number of oscillations and the time taken from the display. This modification
provides more accurate value of g compared to that obtained from a mechanical stop clock or a
digital stop clock.
The heart of this instrument is the microcontroller 89V51RD and the associated software. The IR
sensor emits IR radiation and the receiver placed on the opposite side of the sensor receives the
radiation which resets the clock. As the pendulum bob intercepts the IR radiation at the
equilibrium position, the microcontroller receives a signal, which initiates the counter and the
timer placed inside it. After the pre-set number of oscillations has occurred, the counter sends a
signal to the microcontroller to stop the clock and display its reading on the display. The entire
process is controlled by the software loaded in the microcontroller.
Figure-1 shows the digital simple pendulum. The stand has been redesigned by employing a
heavy iron base and 18mm rod with a guiding slot for fixing the pivot clamp. The length of the
pendulum can be changed by moving the pivot clamp up or down. The sensor holder can also be
moved up and down and fixed at any desired position.
Apparatus used
Digital vernier calipers, brass pendulum bob tied to a thread and fixed to an adjustable vertical
stand with heavy base, sensor based clock and counter, and measuring tape.
Experimental procedure
The experiment consists of three parts, namely
Part-A: Determination of g
Part-B: Determination of the length of the second’s pendulum
Part-C: The Damping test
Part-A: Determination of ‘g’
1. Using a digital vernier calipers, the diameter of the brass pendulum bob is noted and its
radius is calculated
Diameter of the pendulum bob = 2.40cm
Radius of the bob (r) = 1.2cm =12x10-3m
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Simple Pendulum
Kamaljeeth Instrumentation & Service Unit
Figure-1: Simple pendulum fixed on the stand and the counter and clock system
2. The clamp of the pendulum stand holding the thread is fixed at the top at its highest
position and the sensor holder carrying the IR-sensor pair is fixed about 40-45 cm below
the top clamp and exact length of the thread from the bottom edge of the clamp to the top
of the pendulum is noted using measuring tape.
Length of the thread from top of the clamp to the bob (l) = 48.3cm
Hence length of the pendulum
L = 48.3+1.2= 49.5cm = 0.495m
3. The pendulum sensor is now connected to the counter/clock system. The pendulum
sensor is aligned exactly below the pendulum bob so that when the pendulum crosses
over the sensor holder, it intercepts the IR radiation.
4. Holding the pendulum by hand, the counter is reset by pressing the clear button and the
LCD display shows
Time = 000.000 s
OSC. 000/010
As shown in Figure-2, by default the number of oscillations is set to 10. If the number
oscillations are to be increased, one can do it by pressing the “set oscillations” button.
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Simple Pendulum
Kamaljeeth Instrumentation & Service Unit
Figure-2: Setting the number of oscillations
5. The bob is pulled to one side up to the level of the pendulum base and released. The
pendulum starts oscillating. When the bob passes over the sensor, the counter starts
counting the number of oscillations. The time taken for 10 oscillations is displayed on the
LCD display at the end of 10 oscillations, as shown in Figure-3.
Figure-3: Displaying the time after completion of 10 oscillations
Time taken for 10 oscillations = 14.133s, hence
Period T = 1.4133s
g = 4π2
= 4π2
.
.
= 9.783 m/s2
6. The experiment is repeated for different lengths of the pendulum and the corresponding
value of g is determined from which the average value of g at Bengaluru is calculated.
The readings obtained are tabulated in Table-1.
Table-1
Length of the
Length of the
Time for 10
thread l, (cm)
pendulum, L = l+r (m) oscillations (s)
48.3
0.495
14.133
45.0
0.462
13.606
44.1
0.453
13.504
42.4
0.346
13.293
41.0
0.422
13.049
39.3
0.405
12.794
37.4
0.386
12.491
36.0
0.372
12.263
33.9
0.351
11.913
31.9
0.331
11.573
29.5
0.307
11.152
Average value of g at Bengaluru ( m/s2)
Period
T(s)
1.4133
1.3606
1.3504
1.3293
1.3049
1.2794
1.2491
1.2263
1.1913
1.1573
1.1152
Part-B: Determination of the length of the second’s pendulum
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g (m/s2)
9.783
9.850
9.80
9.74
9.78
9.78
9.77
9.77
9.76
9.76
9.75
9.776
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Simple Pendulum
Kamaljeeth Instrumentation & Service Unit
7. The clamp holding the sensor is raised to about 30cm and the exact length of the
pendulum is noted as shown in Figure-4, using measuring tape.
Figure-4: Measuring the length of the pendulum using measuring tape
8. Length of the pendulum L = l+r = 29.5+1.2 = 30.7 cm = 0.307m
9. The ‘Clear’ button is pressed which clears all the previous readings. The “Set
Oscillation” button is pressed to set the number of oscillations as 20 and the pendulum is
made to oscillate and its time period is calculated.
Time taken for 20 oscillations = 22.365s, hence Period T = 1.1183s
The value of g is calculated as
g = 9.70m/s2
The readings obtained are recorded in Table-2.
Length of the
thread l (cm)
29.5
31.2
33.0
36.3
37.4
38.9
40.4
42.0
43.5
Length of the
pendulum L = l+r (m)
0.307
0.324
0.342
0.375
0.386
0.401
0.416
0.432
0.447
Table-2
Time for 20
oscillations (s)
22.365
22.973
23.573
24.569
25.020
25.448
25.892
26.349
26.856
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Period
T(s)
1.1183
1.1487
1.1706
1.2284
1.2510
1.2724
1.2946
1.3174
1.3428
T2
g (m/s2)
1.250
1.319
1.370
1.508
1.565
1.619
1.675
1.735
1.803
9.70
9.70
9.72
9.81
9.74
9.78
9.80
9.82
9.78
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Simple Pendulum
Kamaljeeth Instrumentation & Service Unit
45.0
47.2
48.1
50.4
51.0
53.2
0.462
27.301
1.3650
0.484
27.951
1.3975
0.493
28.174
1.4087
0.516
28.793
1.4396
0.522
28.995
1.4497
0.544
29.624
1.4812
Average value of g at Bengaluru ( m/s2)
1.863
1.953
1.984
2.072
2.101
2.194
9.79
9.80
9.80
9.82
9.80
9.79
9.776
10. Experiment is repeated for different lengths (about 30-50cms) of the pendulum. In each
case the sensor holder is placed in line with the pendulum axis and period of oscillation is
determined. The readings obtained are tabulated in Table-2.
11. A graph is drawn taking L along the X-axis and T2 along the Y-axis, as shown in Figure5. The straight line graph is extrapolated up to L=1m. From the graph, length of the
period corresponding to the value of T2=4 is noted. This gives the length of the second’s
pendulum.
4.5
4
3.5
T2
3
2.5
2
1.5
1
0.5
0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Pendulum length L (m)
Figure-5: Plot of L versus T2
12. From the graph, the length of pendulum for T2 = 4 sec is found to be exactly 1m. To
confirm this, the length of the pendulum is set as 1m and its period is determined.
13. The period, T, obtained is exactly 2 s
Part-C: The damping test
In this part of the experiment, the counter/clock system is tested for damping of the pendulum
motion due to friction with air and if its period is reduced. This is done in two ways. In the first
case, the pendulum is made to oscillate with a fixed length and the time taken for 10 oscillations
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Simple Pendulum
Kamaljeeth Instrumentation & Service Unit
is recorded continuously, noting down the time taken and resetting the clock after each 10 sec up
to a maximum of three minutes. The period is calculated in each case and the value of g is
calculated.
In the second test, the period of the pendulum is calculated by making it to oscillate with
different number of oscillations, starting from 10 and reaching up to 100 in steps. In each case
the period is calculated and the value of g is determined. The average value of g is found to be
the same as obtained in the Part- A and Part-B. The readings obtained are tabulated in Table-3
and Table-4 respectively.
Length of the pendulum, L =0.544m
No of
oscillations
Time taken for
10 oscillations (s)
10
14.788
20
14.786
30
14.784
40
14.779
50
14.779
60
14.775
70
14.775
80
14.769
Average period, T =1.47739 s
Table-3
Period
No of
T(s)
oscillations
1.4788
1.4786
1.4784
1.4779
1.4779
1.4775
1.4775
1.4769
g = 9.83m/s2
Time taken for 10
oscillations (s)
Period
T(s)
14.772
14.768
14.766
14.763
14.760
14.759
14.761
14.760
1.4772
1.4768
1.4766
1.4763
1.4760
1.4759
1.4761
1.4760
Time
taken(s)
80.007
93.644
106.886
120.102
133.629
Period
T(s)
1.3334
1.3377
1.3360
1.3344
1.3362
90
100
110
120
130
140
150
160
Length of the pendulum = 0.44m
No of
Time taken(s)
oscillations
10
13.327
20
26.768
30
40.129
40
53.482
50
66.746
Average period, T =1.3358 s
Table-4
Period
No of
T(s)
oscillations
1.3327
60
1.3384
70
1.3376
80
1.3370
90
1.3349
100
g = 9.73m/s2
Average value of g from Table-3 and Table-4, = 9.78m/s2
Results
Acceleration due to gravity, g, at Benagluru = 9.776m/s2
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Simple Pendulum
Kamaljeeth Instrumentation & Service Unit
Length of the second’s pendulum =1m
Conclusions
1. Automation of simple pendulum has provided accurate value of g, 9.78 m/s2, at
Bengaluru which agrees well with the recorded value (9.7764) [3].
2. Easy operation of the pendulum makes it convenient for laboratory practical.
3. In the damping test, a slight decrease in the period with time shows the effect of friction
of air in damping the movement of pendulum. However, the average period remained
almost the same.
Reference
[1]
Seventeenth General Conference on Weights and Measures (1983). Resolution-1.
International Bureau of Weights and Measure.
[2]
Ranganayaki Rao and M Y Vishwanath Sastry, A laboratory manual in Physics, I PU, Page13.
[3]
http://www.physicsclassroom.com/class/circles/u6l3e.cfm
Vol-12, No-2, June-2012