IB1 Physics HL/SL
Lab Experiment 1 – Simple Pendulum
Introduction:
IB1 Physics HL/SL
Lab Experiment 1 – Simple Pendulum
Introduction:
A simple pendulum is an ideal pendulum consisting of a point mass attached to a massless
and inextensible string which can oscillate around the vertical axis without friction.
The forces acting on a simple pendulum are the weight of the pendulum bob mg⃗
and the
tension in the string T⃗⃗.
For small angles of oscillations (θ < 10o), a real pendulum acts as an ideal pendulum since the
resistive forces (friction and air resistance) would be negligible.
The period of oscillation of a simple pendulum is defined as the time taken for the pendulum
bob to make one complete oscillation.
The period of a simple pendulum is given by the relation:
T = 2π√L/g............ (1)
where L is the length of the string (in meters)
g is the gravitational acceleration (in m.s-2)
Aim:
The aim of this investigation is to study the period T of a simple pendulum for different
lengths L and use the relation of the period to determine the value of the gravitational
acceleration g at the location of the experiment both graphically and analytically.
1
Material:
• Stand
• String
• Mass
• Ruler
• Vernier software with sensor.
• Laptop
Procedure:
1) Attach the mass m to the end of the string.
2) Choosing a certain length for the string, attach the other end of the string to the stand.
3) Make sure that the mass can easily oscillate between the arms of the sensor. 4) On
your laptop, open the “Logger Pro” software.
5) Go to “File”, then choose “Open”.
6) From the list, choose “Physics with Vernier” and then open “Pendulum Periods”. 7)
Displace the mass (and hence the pendulum) by approximately 10oand release it from
rest. Press the “Collect” button and allow the software to record the periods of the
oscillations of the pendulum.
8) The software will measure the successive periods of the pendulum. Choose the first
period recorded and use it as your data for the first trial.
9) Repeat steps [7-8] 9 more times to end up with a total of 10 trials per length. Each
member of the pair uses 5 of the 10 values obtained.
10)Repeat steps [7-9] 7 more times to end up with a total of 8 different lengths.
Variables:
Independent: Length of Pendulum (since it is the one we manipulatively change to observe
the changes in the corresponding T)
Dependent: Period of one Complete Cycle (Since it depends on the Independent Variable l)
Controlled Variables:
·
The massless string used (same height and material)
·
No air resistance (Although this is only an assumption and is a main cause of
deviation from the literature value; unfavorable lab conditions).
·
No initial push on the ball. Only and consistently let it go.
·
Constant 20º angle from which the ball is released.
·
Same equipment and devices are used. No switching from one instrument to another
(for example, we use the Lab Pro device to give readings of the period throughout the whole
experiment. Never in the experiment do we start suddenly taking manual readings.)
·
The endpoints of the strings on the horizontal rod are 10 cm apart always.
Data Collection:
Record the collected data in a clear table. The table should have a clear design, with title,
uncertainties and correct significant figures.
Use the table design below as a reference.
Table 1: Raw data for the time taken for 10 oscillations
Length
of the
string
L/m
ΔL = ±0.001
Period of Oscillation T/s
ΔT = ±0.000001s
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
0.51
1.416401
1.421725
1.416812
1.417306
1.414532
0.45
1.326772
1.3266052
1.3544048
1.336474
1.3354068
0.40
1.2672556
1.2695792
1.2664008
1.266536
1.2661044
0.35
1.1744028
1.1756272
1.1740542
1.1745588
1.1755052
0.30
1.080898
1.0802364
1.082782
1.0822136
1.083996
0.25
0.9968
0.9943072
0.9962084
0.9950048
0.9943892
0.20
0.885238
0.8895116
0.8866
0.8914256
0.8838956
2
Since I used a ruler and my lengths came out to 2 decimal places the most accurate
uncertainty is 0.1 cm or 0.001m, for the periods I used 0.0000001s to maintain as much
accuracy as was given to me by the software.
II- Processing Data:
In order to continue with the experimentation, an average of the 5 trials per every length must
be taken and is determined by first getting the summation of the values and then finding the
quotient of the sum and the number of trials.
After finding the average period for each length, we must calculate the uncertainty of each
period found by dividing the difference between the greatest and the smallest value of the
period by two.
As an application, take the first value. When l= 0.53 ± 0.001 m,
Notice that the T average must abide by the number of decimal places of all the periods (since
they all have 6 decimal places).
After doing the same for all values of l, we tabulate the values of Taverage with their average
uncertainties for each of the lengths.
Table 1.2: Raw Qualitative Measurements of Lengths and Processed Qualitative
Measurements of Their Corresponding Average Periods
Raw Qualitative Measurements of Lengths and Processed Qualitative Measurements of Their
Corresponding Average Periods
Length l in m
(±0.001 m)
Average Period (Taverage)/s
The Corresponding Uncertainties of Average
Periods ΔT/s
0.510
1.417
± 0.003
0.450
1.336
± 0.003
0.400
1.267
± 0.003
0.350
1.175
± 0.001
0.300
1.082
± 0.003
0.250
0.995
± 0.002
0.200
0.887
± 0.002
A noteworthy comment is that there is no common uncertainty for all values. This is a clear
misconception. The is obtained as shown above.
Manipulating data:
The equation that links Period to Length can be found in the formula:
Square both sides of the equation to get rid of the radical. The equation becomes: . After that,
make g the subject. The equation becomes:
Testing out our experimental values:
As an example,
For l = 0.930± 0.005m,
g=4π2l/T2=4π2(0.510/1.4172)=10.0274432393m.s(−2)≅10.027m.s(−2)
For l =.450 ± 0.005m, g = 8.84774403464 m.s-2 ≅ 8.85 m.s-2
For l= .400±0.005 m, g= 7.86466136412 m.s-2 ≅ 7.86 m.s-2
For l = 0.350± 0.005m, g= 6.88157869361 m.s-2 ≅ 6.88 m.s-2
For l =0.300± 0.005m, g= 5.89849602309 m.s-2 ≅ 5.90 m.s-2
For l = 0.250± 0.005m, g= 4.91541335258 m.s-2≅ 4.92 m.s-2
Note that the uncertainties for the individual values of g are not calculated, as they are not
needed. As we move further along this lab report, we need to find the average of g. The
independent uncertainties would not be taken into account. Rather, it will be calculated as
half the difference between the highest and the lowest value of g. (For experimental
purposes, however, to calculate the uncertainties independently, you must take into account
the uncertainties in the length and the period. Pie is a real number and thus has no
uncertainty.
To find the average acceleration,
To find average uncertainty
Therefore, average g found= 9.6 ± 0.2 m.s-2
Graphing:
Evaluation:
A) Identifying weaknesses:
I- Possible Causes of Random Error:
·
·
·
·
·
There may have been atmospheric contributors to the small imprecision of values.
Variations in the air breeze vary the period of one complete cycle and thus may
cause random errors.
The Photogate device may not have been properly aligned to allow the ball to pass
fully and well above it allowing for improper readings of the period.
The pendulum may not have been correctly placed. That is, both strings of the
pendulum may not have been of equal lengths.
There is significant uncertainty in the measurement of the pendulum. Readings
may have been wrong due to improper use of the ruler that must be placed
perpendicular to the midpoint of the endpoints of the two strings. The length
measured must be from that point till the highest point on the ball. Improper
readings may result in random errors.
Taking only 5 trials for each height. If further trials were taken and then averaged,
·
the random errors would have been significantly decreased.
Perhaps more sensitive instruments should be made available, that allow for total
isolation of the pendulum from external factors. Using the instrument that we did
brought forth some random errors.
II- Causes of Systematic Errors:
·
·
·
·
The angle at which the ball was swung from was never kept constant (which
makes it a systematic error) I.e. at those points, the angle may have been greater
than 20 degrees.
Our stand was not of great quality. The ruler that is stuck on it, has a great off-set
error as measurements of the pendulum start at 3.5 cm and not at zero.
Perhaps there is a fault in the calibration of the instrument used to determine T.
(This, however, is unlikely since measurements and data are collected digitally
rather than manually).
Improper set-up that could cause all values to shift from the true values.
B) Methods of Improvement:
I- Reducing Random errors:
·
·
·
·
·
Allow for the best possible set-up of the pendulum where it is as close as possible
to isolation from any externalities.
Allow for perfect and proper alignment of the Photogate device. Place it just under
the pendulum and tilt it diagonally to allow the ball to pass completely through it
for better readings.
Perfect adjustment of the pendulum where both massless strings are of equal
lengths.
Correct use of the ruler to measure the length of the pendulum or using a stand that
already includes a stuck-on ruler with proper measurements and no offset error
(which could produce systematic error). You can also use a set-square to allow for
the perpendicularity of the set-up.
Taking more than 5 measurements of the period for each length to allow for an
averaged value that is closer to the real literature value.
II- Reducing Systematic Errors:
·
·
·
·
Using the attached protractor properly to measure a constant angle to use
throughout the whole experiment.
Making sure there is no offset error in the Universal Lab Interface device by
registering the period when the pendulum is not swinging. If there is an offset
error, the Logger Pro program will give a period. If not, no reading will be
recorded. In case of an offset error, ask the teacher for guidance on how to
adjust the device.
Increasing the range of lengths studied to further the reliability of the results,
which would make them closer to the true value. (Note that this does not mean
that you should increase the number of trials or measurements taken. This
would be part of reducing random errors. Multiple readings do not decrease the
presence of systematic errors).
Using a specialized ruler to determine the verticality of the set-up so as not to
cause shifts in the experimental values.