Measurement of Time Period of A Simple Pendulum using an Electronic Circuit

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Measurement of Time Period of A Simple Pendulum using an Electronic Circuit

Bhuvnesh, Phurailatpam Hemantakumar

Department of Physics, Hindu College, University of Delhi

Abstract:- This project was taken up in the hope of building an electronic circuit which enables us to measure the time period of a simple pendulum accurately, taking into account the parallax, human reflex and random errors.

Measuring the time period of a simple pendulum by counting the number of oscillations and noting down the time using a stop watch is one of the simplest experiments one can perform to find the value of g, i.e. acceleration due to gravity. This experiment has restricted accuracy due the above mentioned errors. But the problem can be overcome to certain extent by employing an electronic circuit which reads the pendulums

movement, as well as count the time interval between oscillations. In this project an attempt is made to detect the pendulum using a laser detector with which a circuit consisting of timers and counters is employed to measure the number of oscillations and time of the journey simultaneously. The required result is allowed to display using 7447 IC and SSD (seven segment display).

The concepts involved in designing this project is well familiar with and read by any college student pursuing Bsc. Physics Hons.

LASER DETECTOR

DESCRIPTION OF THE COMPONENTS

LDR (light depending resistor) is an electrical component which changes its resistance according to how much light intensity falls on it.

A laser from a source is allow to fall on it continuously which keep the resistance of the LDR low. As the oscillating pendulum cuts the laser, the resistance of LDR goes high. This change in the LDR resistance is read with a circuit using two transistor. The transistor on the right is used as a switch and the output is derived from its collector terminal.

555 TIMER (AS MONOSTABLE MULTIVIBRATOR AND ASTABLE MULTIVIBRATOR)

R 4 8

+Vcc

Monostable multivibrator:

It has a stable and a quasistable state. A pulse at the trigger switches the

7 3

555 TIMER

5

output to quasistable state and stay for predetermined length of time. Then it switches back to the stable state and wait for the next pulse.

It is used to get a digital output wave with sharp edges.

6

C 1

2

TRIGGER

O.01uF

+Vcc

Figure 1: Monostable Multivibrator

Astable multivibrator:

R1 4 8

7 3

Neither the digital state is stable. Therefore the output switches back and forth between the two unstable state and it is periodic, rectangular

R2 555 TIMER

5

6

O.01uF

waveform.

This is used for timing the journey of the oscillating pendulum.

2 1

C

Figure 2: Astable Multivibrator

+5

A a

b

c

c

B 7447 d

e

C f

g

D

ICs (74160 and 74373):

+5

  1. Seven segment display (common anode):

  2. Digital output from counter is received by 7447 IC and is

  3. converted to numerical form needed by SSD. The SSD

  4. display the numerical output corresponding to the digital

  5. output given by the counter.

    f

    g

    Figure 3: Seven Segment Display with 7447 IC

    74160 is a decade counter which can make digital count from 0000 to 1001, and repeats itself after each cycle. Every count is triggered through the clock pin.

    74373 is an IC with 20 pins. It is internally D-flip flops which can be control with the enable pin provided. It also acts as a buffer to derive SSD display.

    LASER DETECTOR 555 TIMER

    MONOSTABLE

    555 TIMER ASTABLE

    +Vcc

    CARRY OUTPUT

    DECADE COUNTER 74160

    DECADE COUNTER 74160

    DECADE COUNTER 74160

    DECADE COUNTER 74160

    CLOCK

    CARRY CLOCK CARRY CLOCK CARRY

    DECADE COUNTER 74160

    DECADE COUNTER 74160

    CLOCK

    DECADE COUNTER 74160

    CLOCK

    DECADE COUNTER 74160

    OUTPUT OUTPUT OUTPUT OUTPUT

    74373 D-flip flop LATCH

    74373 D-flip flop LATCH

    7447 IC 7447 IC 7447 IC 7447 IC

    SSD SSD SSD SSD

    Figure 4: Schematic diagram of the circuit used

    Figure 4: Schematic diagram of the circuit used

    EXPERIMENT

    The pendulum is allowed to oscillate between the laser source and the detector. When at rest the laser, the bob of the pendulum and the LDR are made collinear. As the pendulum oscillates it cuts the laser which makes the detector to send a pulse and trigger the 555 timer (monostable). The timer outputs time period is set to be higher than the time the detector is obstructed while crossing the laser and lower than the time it takes to return to the mean position, i.e. when the timer is triggered again. The timer is then connected to a decade counter (74160 IC), which increase its count as the laser is cut, i.e. for every half oscillation. The carry output of 74160 IC goes to each enable pin of 74373 ICs which later will help in latching the output of the series counters.

    The 555 timer (astable) is made to oscillate with a known frequency, by adjusting the value of capacitor and resistor used (87.5878Hz, for this experiment). It is then interface with a series of decade counters. These counters start counting as soon as there is an output from the 555 timer (astable) and the process continues. But the experiment dictates the requirement of time interval in certain number of oscillations. In order to achieve this 74373 ICs are employed to latch the counters output.

    Each 74373 IC is control through enable pin by the carry output from the counter connected to 555 timer

    (monostable). This counter counts from 0000 to 1001 and then starts from 0000 with a high carry output. As long as it is high, it enables the 74373 ICs and the output of the series counters is made available to be displayed by SSDs. When the former counter changes 0000 to 1000, its carry output goes low, thus disenabling the 74373 ICs. As a consequence the output display in SSD is latch till 74373 ICs are enable again.

    Numbers displayed on SSDs are noted after every five oscillations for a particular pendulum length. Such ten readings are taken for nine different pendulum lengths and graph is plotted for each set, between the SSDs readings and number of oscillation. A line is drawn that fits the data points and the slope of this line will give the number of count made by the astable 555 timer per oscillation. The required time period of the pendulum can be obtained by multiplying the value of the slope with the least count of the astable 555 timer.

    Comparison between the experimental results and theoretical values are made by plotting a graph between time period (T) and length of the pendulum (l). Further comparison can be achieved by plotting graph between l and T2.

    The formula T=2/ is used to find the value of g.

    OBSERVATIONS

    Least count of the astable 555 timer = 0.011417 s

    Theoretical value of g= 981cm/s2 g =acceleration due to gravity T= 2/ l = (g/42)T2 l = length of the pendulum

    T = time period of the pendulum

    Following are the graphs and tables to find the time period of the given length:

    1. Pendulum length= 100 cm

      Graph 1 Table 1

      NO. OF OSCILLATIONS

      SSDs READINGS

      9568

      10

      10357

      15

      11227

      20

      12095

      25

      12965

      30

      13839

      35

      14715

      40

      15588

      45

      16464

      50

      17339

      NO. OF OSCILLATIONS

      SSDs READINGS

      5

      9568

      10

      10357

      15

      11227

      20

      12095

      25

      12965

      30

      13839

      35

      14715

      40

      15588

      45

      16464

      50

      17339

      Slope=173.608 Time period= 1.982 s g=1000.49 m/s2

    2. Pendulum length= 90 cm

      Graph 2

      Table 2

      NO. OF

      SSDs

      OSCILLATIONS

      READINGS

      5

      3544

      10

      4375

      15

      5206

      20

      6040

      25

      6874

      30

      7706

      35

      8540

      40

      9375

      45

      10209

      50

      11045

      Table 2

      NO. OF

      SSDs

      OSCILLATIONS

      READINGS

      5

      3544

      10

      4375

      15

      5206

      20

      6040

      25

      6874

      30

      7706

      35

      8540

      40

      9375

      45

      10209

      50

      11045

      Slope= 166.696

      Time period= 1.903 s

      g=981.12

      cm/s2

    3. Pendulum length = 80 cm

      Graph 3

      Table 3

      NO. OF

      SSDs

      OSCILLATIONS

      READINGS

      5

      5840

      10

      6615

      15

      7390

      20

      8166

      25

      8943

      30

      9722

      35

      10503

      40

      11283

      45

      12065

      50

      12847

      Slope=155.719

      Time period= 1.778 g=999.04 cm/s2

    4. Pendulum length= 70 cm

      Graph 4 Table 4

      Slope=

      NO. OF

      OSCILLATIONS SSDs READINGS

      5 5941

      10 6675

      15 7410

      20 8145

      25 8882

      30 9623

      35 10296

      40 11035

      45 11771

      50 12504

      145.525

      Time period= 1.661 s g=1001.65 cm/s2

    5. Pendulum length= 60 cm

      Graph 5 Table 5

      NO. OF OSCILLATIONS

      SSDs READINGS

      5

      1955

      10

      2623

      15

      3292

      20

      3961

      25

      4629

      30

      5298

      35

      5966

      40

      6635

      45

      7303

      50

      7971

      NO. OF OSCILLATIONS

      SSDs READINGS

      5

      1955

      10

      2623

      15

      3292

      20

      3961

      25

      4629

      30

      5298

      35

      5966

      40

      6635

      45

      7303

      50

      7971

    6. Pendulum length= 50 cm

    Graph 6 Table 6

    NO. OF OSCILLATIONS

    SSDs READINGS

    5

    4310

    10

    4943

    15

    5577

    20

    6211

    25

    6841

    30

    7468

    35

    8094

    40

    8720

    45

    9346

    50

    9971

    NO. OF OSCILLATIONS

    SSDs READINGS

    5

    4310

    10

    4943

    15

    5577

    20

    6211

    25

    6841

    30

    7468

    35

    8094

    40

    8720

    45

    9346

    50

    9971

    Slope= 125.771

    Time period=

    1.436 s

    Slope=133.701 Time period=1.526 s g=1017.18 cm/s2

    g=957.24 cm/s2

    7. Pendulum length= 40 cm

    Graph 7

    Table 7

    NO. OF OSCILLATIONS

    SSDs READINGS

    5

    5443

    10

    6003

    15

    6563

    20

    7123

    25

    7682

    30

    8242

    35

    8803

    40

    9311

    45

    9870

    50

    10431

    Slope= 110.668

    Time period= 1.263 s

    g=989.94 cm/s2

    8. Pendulum length= 30 cm

    Graph 8

    Table 8

    Slope= 97.0436

    Time period=

    1.108 s

    g=964.72

    cm/s2

    NO. OF

    5

    9364

    10

    9848

    15

    10334

    20

    10819

    25

    11305

    30

    11790

    35

    12275

    40

    12761

    45

    13245

    50

    13730

    5

    9364

    10

    9848

    15

    10334

    20

    10819

    25

    11305

    30

    11790

    35

    12275

    40

    12761

    45

    13245

    50

    13730

    OSCILLATIONS SSDs READINGS

    9. Pendulum length= 20 cm

    Graph 9 Table 9

    Slope=

    NO. OF OSCILLATIONS

    SSDs READINGS

    5

    5848

    10

    6242

    15

    6635

    20

    7028

    25

    7422

    30

    7815

    35

    8208

    40

    8602

    45

    8996

    50

    9390

    Slope=

    NO. OF OSCILLATIONS

    SSDs READINGS

    5

    5848

    10

    6242

    15

    6635

    20

    7028

    25

    7422

    30

    7815

    35

    8208

    40

    8602

    45

    8996

    50

    9390

    78.6958

    Time period= 0.896 s

    g=983.49 cm/s

    COMPARISON BETWEEN THE EXPERIMENTAL RESULTS AND THEORETICAL VALUE

    Table 10: Time period of the pendulum in a particular length

    X-AXIS

    Y-AXIS

    l (cm)

    T (s)

    20

    0.898

    30

    1.108

    40

    1.263

    50

    1.436

    60

    1.526

    70

    1.661

    80

    1.778

    90

    1.903

    100

    1.982

    Graph 10: Time Period VS Pendulum Length

    Table 11: Relation between (time period)2 and length of the pendulum

    X-AXIS

    Y-AXIS

    T2 (s2)

    l (cm)

    0.8064

    20

    1.2277

    30

    1.5952

    40

    2.0621

    50

    2.3287

    60

    2.7589

    70

    3.1613

    80

    3.6214

    90

    3.9283

    100

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    Graph 11: Pendulum Length VS (Time Period)2

    RESULT

    It can be seen from the graph that the experimental data and the experimental curve are fairly close enough to the theoretical curve which are drawn with the assumption that g is 981cm/s2.

    PRECAUTIONS

    1. Least count of the astable 555 timer should be found accurately using a CRO, or a multimeter.

    2. The counting done by the decade counter which is connected to monostable 555 timer is monitored with caution using LEDs at its output terminals, so that is doesnt skip its count.

    3. Light condition of the room should not change as it may interfere with the desire detector output.

    CONCLUSION

    This project provides a platform where students learned to integrate various topics studied in digital electronics and classical physics. It also give exposure to troubleshooting, datasheets, design parameters and experimentation.

    Besides this project, the method involved can be made to use in various other fields, like measuring rpm of a wheel etc.

    ACKNOWLEDGEMENT

    Special thanks to Maam Adarsh Singh for supervising the project.

    REFERENCES

    1. Digital principles and applications By Donald P. Leach & Albert Paul Malvino, (Glencoe, 1995).

    2. Microprocessor Architecture, Programming, and Applications with the 8085 By Ramesh S. Gaonkar, (Prentice Hall, 2002).

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