Research Question
How accurately can the gravitational acceleration constant be determined by measuring the oscillation
period against different lengths of the pendulum?
Rationale
My intrigue to determine the gravitational acceleration constant g (9.81 m𝑠−2) was fueled by a curiosity
to comprehend its foundational principles, which has been an integral part of my physics education.
Looking back at myself as a physics student, I realized that I have always subconsciously been using 9.81
m𝑠−2 as the gravitational acceleration constant, without questioning its validity and origin -just because
it was stated in the data booklet. Therefore, I wanted to take this chance to determine the gravitational
acceleration constant on my own, through a hands-on experiment involving the simple harmonic motion
of a pendulum. I specifically chose to make connections to simple harmonic motion, as its ubiquity
intrigued me. Upon further research, I found out that there are more applications and occurrences of it in
real life than I thought there to be, ranging from swings and guitar strings which can all be modeled
through a simple harmonic motion graph. In the simple harmonic motion of a pendulum, it was intriguing
how the oscillation period would remain constant as long as the length stays the same, regardless of the
suspension point of the pendulum bob -unlike how I expected the oscillation period to be shorter for a
higher suspension point as it meant more potential energy.
Theoretical Background
A simple harmonic motion (SHM) is a type of motion where the restoring force is directly proportional
and opposite in direction to the displacement of the object from equilibrium, given by the equation
a = -𝜔2 𝑠
where a is the instantaneous acceleration
of the mass (𝑠𝑠−2 ), 𝜔 is the angular frequency,
and x is the displacement from the equilibrium
position. There are different types of simple
harmonic motion, but this IA will focus on the
simple harmonic motion of a pendulum, which
can be modeled in Figure 1 (Johnwell Physics).
When the pendulum bob is released at a fixed
point of suspension O, it will oscillate back and
forth from point A to point C. Ideally, in a closed
system, the pendulum would oscillate an infinite
amount of times as the energy will not be lost to
its surroundings. However, in real life, energy is
lost to the surroundings due to different sources
(ex. air resistance), which will eventually bring
the oscillation to an end. Position B is referred to
as the equilibrium position, which is the position
the pendulum is at when it is at rest. During
oscillation, B is also the position where the
pendulum bob’s instantaneous velocity is at its maximum and the acceleration is zero.
1
The velocity and acceleration of the pendulum
motion would look like the following graphs in
Figure 2 (BYJUS). The graphs show the
variation of displacement, velocity, and
acceleration of the pendulum over time. The
displacement-time graph is a reflection of the
acceleration-time graph over the x-axis, which is
explained by the definition of a simple harmonic
motion -the restoring force is directly
proportional in the opposite direction to the
displacement of the pendulum from equilibrium.
The velocity-time graph and acceleration-time
graph are also closely related, as acceleration is
the rate of change of velocity -hence the
derivative of the velocity-time graph.
The equation for a simple harmonic motion of a pendulum will play a key role in this IA, which can be
expressed as:
𝑠
T = 2π√𝑠
Where T is the oscillation period of the pendulum in seconds, L is the length of the pendulum in meters
and g is the gravitational acceleration constant. The oscillation period is the time it takes for the pendulum
bob to complete a full back-and-forth motion starting from the equilibrium position.
Aim
This IA aims to determine the degree of accuracy with which the gravitational acceleration constant (g)
can be calculated by measuring the oscillation period of the pendulum against different lengths.
Hypothesis
If the oscillation period of a pendulum is inversely proportional to the square root of its length as
𝑠
described by T = 2π√𝑠, then by measuring the oscillation periods against different lengths of pendulum,
the gravitational acceleration constant g will be determined to a high degree of accuracy.
Variables
Independent Variable
Dependent Variable
The independent variable is the length of the pendulum measured
in centimeters (cm). It is the distance from the center of the
pendulum bob to the end of the string.
The dependent variable is the oscillation period of the pendulum,
measured in seconds.
2
Controlled
Variable
Size and mass of
pendulum bob
Type of string
Method / Potential Effects
-
-
Air Condition
-
-
Although the mass and size of the pendulum bob theoretically will not affect the
oscillation period, it was controlled as it could have a minor impact in real life. As an
example, the oscillation period could increase if the size of the pendulum bob were to
be greater because more air resistance would act upon it due to a greater surface area.
Method: The same pendulum bob was used throughout.
Different types of string have different physical properties such as weight, rigidity,
and tensile strength. Differences in these factors are likely to affect the motion of the
pendulum by stretching its length, therefore affecting the oscillation period. The type
of string was controlled by using the same type of string throughout the experiment.
Method: Wool string was chosen because it had an insignificant weight while having
enough tensile strength to withstand the weight of the pendulum bob.
Arguably, wool is susceptible to stretch, hence more trials (5 for each) were taken, and
new strands of string were used for each trial.
Random forces from the air condition (ex. wind) acting upon the pendulum will affect
the motion of the pendulum, therefore affecting the oscillation period. Also, the
temperature was kept constant at room temperature (approximately 298 K) since the
collision of particles in the air with the pendulum bob could affect its motion. Higher
temperature means that particles have more kinetic energy, which causes them to
collide frequently with the bob. Moreover, temperature changes can affect the length
of the pendulum due to thermal contraction or expansion.
Method: To prevent this, the experiment was done in a room-temperature closed lab
without any fluctuations in the air condition.
Equipment
-
Strand of wool string (10 meters long)
Ring stand with a horizontal bar at the top (80cm tall in height) x1
Pendulum bob (mass: 50.5g) x1
Stopwatch x1
Ruler x1
Tape x1
Safety goggles
3
Apparatus
Figure 1 (self-taken)
Methodology
The setup of the experiment is as shown in Figure 1, a self-taken image of the apparatus.
The procedure for the experiment was as follows:
1. Wear a safety goggle
2. Set the length of the pendulum, starting from 10 cm. Length will go up to 60 cm, with an
increment of 10 cm.
3. Firmly tie the pendulum to the bar of the ring stand using tape
4. Without exerting any force, release the pendulum from a pivotal position. (the angle at which the
pendulum is released will not impact its oscillation period; thus it does not need to be controlled)
5. Using a stopwatch, start recording the time as the pendulum passes the equilibrium position.
4
6. Stop recording when the pendulum returns to the equilibrium position after completing a full
oscillation
7. Write down the time measured in the raw data table
8. Repeat step 2~7, taking 5 trials for each length of the pendulum
Safety Procedures
If the string of the pendulum were to be disconnected while oscillating, there are likely to be potential
dangers as the pendulum will be launched in a random direction. To prevent this, the pendulum was
firmly tied to the ring stand in addition to extra layers of tape, to ensure that it would not get
disconnected. In addition, safety goggles were worn to minimize potential damage.
Data Collection and Processing
Table 1. Raw Data for Oscillation Period (T) measured against different Lengths of Pendulum (L)
L ±0.05 (cm)
Trial 1
T ±0.01(s)
Trial 2
T ±0.01(s)
Trial 3
T ±0.01(s)
Trial 4
T ±0.01(s)
Trial 5
T ±0.01(s)
10
0.65
0.71
0.73
0.66
0.55
20
0.90
1.03
1.00
0.93
0.93
30
1.19
1.13
1.08
1.14
1.13
40
1.26
1.31
1.37
1.24
1.29
50
1.45
1.48
1.35
1.46
1.48
60
1.65
1.51
1.55
1.53
1.53
The length of the pendulum holds an uncertainty of 0.05 cm, which is half the smallest reading of the
ruler.
The oscillation period holds an uncertainty of 0.01s, which is the smallest reading of the stopwatch.
Table 2. Processed Data
Average
%
L ±0.05 T ±0.01 Uncertainty
(cm)
(s)
in T
Absolute
Average T² % Uncertainty Uncertainty in
(s²)
T²
T² (s²)
Max T²
(s²)
Min T²
(T²)
10
0.66
13.6
0.4
27.3
0.1
0.6
0.3
20
0.96
6.78
0.9
13.6
0.1
1.0
0.8
30
1.13
4.85
1.3
9.70
0.1
1.4
1.2
40
1.29
5.02
1.7
10.0
0.2
1.8
1.5
50
1.44
4.50
2.1
9.00
0.2
2.3
1.9
60
1.55
4.50
2.4
9.01
0.2
2.6
2.2
Column 2: Average T
5
The average value of oscillation period (denoted as T), was obtained by dividing the sum of the five
measured periods from each trial by 5, which is the total number of trials. The equation is as follows:
Sample Calculation:
When L = 10cm, 𝑠𝑠𝑠𝑠 =
0.65+0.71+0.73+0.66+0.55
𝑠
5
= 0.66 s
Column 3: % Uncertainty in T
The percentage uncertainty in the average T was calculated by the following equation, where 𝑠𝑠𝑠𝑠𝑠 is
the absolute uncertainty calculated by dividing the range of T by 2.
𝑠𝑠
% 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝑠 𝑠𝑠𝑠 × 100
𝑠𝑠𝑠
Sample Calculation:
When L = 10 cm, % uncertainty is:
(
0.73−0.55
)
2
0.66
× 100 = 13.6363%
≈ 13.6%
2
Column 4: Average 𝑠
Average 𝑠2were calculated by squaring the average oscillation period.
Sample Calculation:
When L = 10 cm, 𝑠2= 0.662
= 0.44
≈ 0.4 (1 𝑠𝑠)
Column 5: % Uncertainty in 𝑠2
According to the rules of propagation of uncertainty, when a value is raised to a power, its percentage
uncertainty should be multiplied by the power. The percentage uncertainty in 𝑠2 𝑠𝑠𝑠 was calculated by
doubling the corresponding percentage uncertainty of 𝑠𝑠𝑠𝑠 .
Sample Calculation:
For L = 10 cm, % uncertainty in 𝑠2 𝑠𝑠𝑠 = 13.64×2 %
= 27.28%
Column 6: Absolute Uncertainty in 𝑠2
The percentage uncertainty in 𝑠𝑠2 𝑠𝑠𝑠 was converted back in the form of absolute uncertainty, as they
were essential to determine the maximum and minimum values of 𝑠2. The number of significant figures
is limited to one because it is an absolute value of uncertainty. For conversion, the following equation was
used:
𝑠𝑠2 𝑠𝑠𝑠 =
% 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠 𝑠2 𝑠𝑠𝑠
× 𝑠2 𝑠𝑠𝑠
100
Sample Calculation:
27.3
When L = 10 cm, 𝑠𝑠2 𝑠𝑠𝑠 = 100 ×0.4
= 0.1092
≈ 0.1 (1sf)
Columns 7&8: Maximum and Minimum 𝑠2
The maximum and minimum possible oscillation periods were calculated by adding and
subtracting𝑠𝑠2 𝑠𝑠𝑠 from 𝑠2 𝑠𝑠𝑠, respectively.
6
Sample Calculation:
When L = 10 cm,
0.3 𝑠2
Max 𝑠2: 0.4+0.1 = 0.5
Min 𝑠2: 0.4-0.1 =
≈ 0.6 𝑠2
The reason why Max 𝑠2 calculated (0.5) differs from the value of Max 𝑠2 in Table 2 (0.6) is that
intermediate values were used by Google Spreadsheets for the calculations, although the displayed value
is different due to the limit in the number of significant figures. Also, even though T was initially timed to
the hundredth decimal place, 𝑠2 was rounded to the tenth place following the decimal place of their
uncertainties.
Graphing
Squaring both sides of the equation for a simple harmonic motion of a pendulum results in the following:
𝑠
𝑠2= 4𝑠2 𝑠
Although the unit of L is in meters for the above equation, since the length was measured in centimeters,
L is replaced with L/100, where L is now the pendulum length in centimeters:
𝑠
𝑠2= 4𝑠2 100𝑠
Since 𝑠2 is proportional to L, 𝑠2 is plotted against L for a linearized graph in the form of y = mx + b,
where m is the gradient and b is the y-intercept.
Table 3. Plotting data collected on a graph
𝑠2 (𝑠2)
L ±0.05 (cm)
P1
10
0.4
20
30
40
50
60
P2
P3
P4
P5
P6
Maximum Line of
Best Fit
(maxLOBF)
Minimum Line of
Best Fit
(minLOBF)
0.3
0.6
2.6
2.2
0.9
1.3
1.7
2.1
2.4
Table 3 shows the coordinates of the points (P1~P6) plotted in the graph, in increasing order of the length
of the pendulum. Error bars were added for each point, based on the absolute uncertainty calculated
previously in Table 2.
Graph 1 (Oscillation Period Squared 𝑠2 vs Pendulum Length L).
7
A clear linear relationship is shown when T² is plotted against L, suggesting their positive proportionality.
1
m corresponds to 4𝑠2 100𝑠, which is the coefficient of L. According to the equation of Average T² (line of
best fit) generated, gradient m is shown to be 0.0394. The following equation can be set up:
0.0394=4𝑠2
1
100𝑠
Rearranging to make g the subject,
1
g = 4𝑠2 100×0.0394
≈ 10.0 𝑠𝑠−2 (3 sf)
The gravitational acceleration constant g is calculated to be 10.0 𝑠𝑠−2
The uncertainty in g can be calculated by finding the uncertainty in m (the gradient). The uncertainty in m
can be calculated by:
𝑠𝑠𝑠𝑠 − 𝑠𝑠𝑠𝑠
, where 𝑠𝑠𝑠𝑠 and 𝑠𝑠𝑠𝑠 are the gradients of the line of maximum and
2
minimum fit, named maxLOBF and minLOBF respectively. The unit of m is 𝑠𝑠−1 𝑠2 , which other
than being the reciprocal of the unit of acceleration, does not have any meaning.
Calculation of uncertainty in m
Absolute Uncertainty
(0.0463-0.0329)/2 = 0.0067
Percentage Uncertainty
100×(0.0067/0.0394) = 17.01%
In the process of making g the subject of the equation,
4𝑠2
was divided by 0.0394. Following the rules of
100
propagation of uncertainties, the percentage uncertainty should be added. Therefore, g also has a
percentage uncertainty of 17.01%. Converting this into the absolute form of uncertainty,
8
17.01
10× 100 = 1.7𝑠𝑠−2
≈ 2𝑠𝑠−2
Final Value
g = 10±2𝑠𝑠−2
Conclusion
In summary, the gravitational acceleration constant was determined through a hands-on experiment
involving the simple harmonic motion of a pendulum. The oscillation periods were measured against
different lengths of the pendulum, which showed a clear positive correlation, as shown in Table 1 (Raw
Data). Then, to produce a linearized graph in the form of y=mx+b, 𝑠2 was plotted against L in
𝑠
accordance with the equation T = 2π√𝑠. As the final step, equating the slope of the line of best fit
generated by Google Spreadsheets with m enabled the calculation of g. The uncertainty in g was
calculated with appropriate methods of propagation of uncertainty.
The hypothesis of this investigation was “If the oscillation period of a pendulum is inversely proportional
𝑠
to the square root of its length as described by T = 2π√𝑠, then by measuring the oscillation periods
against different lengths of pendulum, the gravitational acceleration constant g will be determined to a
high degree of accuracy.” According to the calculated value of g, the hypothesis was supported as the
calculated value of 10𝑠𝑠−2 is reasonably close to the literature value of g, 9.81𝑠𝑠−2. This is
0.19𝑠𝑠−2 less than the true value, meaning the time taken to complete oscillations (T) took longer than it
should have -it was due to the errors, such as air resistance slowing down the bob. In fact, the calculated
value is within the range of uncertainty propagated, ±2𝑠𝑠−2 . The uncertainty originates from errors
that were present during the experiment due to imperfection and provide a range of possible values that
the true value of g can fall within. Despite the numerous sources of error in the experiment, how the
literature value is within the range of uncertainty of the calculated value, reinforces the validity of the
hypothesis. One thing that should be noted is the y-intercept, which has coordinates of (0, 0.0903). This
logically does not make sense, as it signifies that an oscillation period exists when the length of the
pendulum is equal to zero. Thus, it is an indicator of systematic errors that were present during the
experiment, which will be discussed in the evaluation section.
Evaluation
Type/Source of
Significance/Impact
Improvements
9
Error
Random Error:
Reaction time
Random errors were caused by the
imperfection in the way oscillation periods
were measured during data collectionmainly the inconsistencies in the reaction
time of the data collector. Variations in the
experimenter's reaction time during the start
and stop of time - whether it be too early or
too late- are likely to have impacted the
magnitude of uncertainty. Moreover, because
the percentage uncertainty in T was doubled
as 𝑠2 was plotted against L, the impact of
the reaction time error was doubled on the
final value.
A motion sensor could be used for a
more accurate measurement of
oscillation periods. The stopwatch
could be started and stopped by the
motion sensor’s sense of the
pendulum bob completing a full
oscillation. Although this was not a
viable option due to budget
concerns, it is a way to mitigate
reaction time errors, thereby
arriving at a value with higher
accuracy.
Another way to minimize this error
is by increasing the average
oscillation period by using strings
of longer lengths in general. Then,
the uncertainties caused by the
reaction time would become less in
terms of the percentage uncertainty,
ultimately resulting in a more
accurate result.
Systematic Error:
Air Resistance
Rounding Error
Ideal pendulum assumes that
1) The pendulum is a point mass
2) Air resistance is negligible
However, as the experiment was done in
normal air conditions, the air resistance is
likely to have slowed down the pendulum for
all the trials. Thus, it is a systematic error.
This is a significant error that is likely to
have caused relatively large uncertainties,
because the consistent exertion of a braking
force directly led to the underestimation of
the true oscillation period. As a result, the
linearized graph was shifted upwards, having
a y-intercept of (0,0.0903).
The experiment can be done in a
vacuum air condition to eliminate
this error. However, it was not a
viable option due to budget
concerns.
During the processing of data, data were
often rounded to adjust to the appropriate
This rounding error is closely
related to the previous random error
Another option is to use a
pendulum bob that has a smaller
surface area. This would reduce the
air resistance acting on the
pendulum during the motion, hence
reducing the systematic error.
10
number of significant figures. As an
example, although the oscillation periods
were measured in seconds to the hundredths
decimal place, its uncertainties were too
large, resulting in the absolute uncertainty in
𝑠2 being in the tenth decimal place. To
appropriately adjust the number of
significant figures in 𝑠2, they were rounded
to the tenth place, which caused numbers to
be lost in the intermediate steps.
This rounding error occurred again later on,
which affected the final value of g to have
only one significant figure. Thus, it is a
significant error.
of inconsistent reaction time. This
is because the main source of the
rounding errors are relatively large
uncertainties, which result from
other significant errors. To
minimize this, data should be
collected with a high degree of
precision, to prevent any loss of
intermediate values caused by large
uncertainties. Thus, the same
improvement as the reaction time
error (using motion sensors) could
be applied here as well.
Systematic Error:
Precision of
Ruler
A systematic error caused by the limited
precision of the ruler was present, as a
consistent error in reading was made
throughout.
However, this is insignificant -half of the
smallest scale of the ruler is 0.5 mm -which
in the form of percentage uncertainty - is
only 2.04% of the length on average. Thus, it
was not taken into consideration during
calculation, including when plotting the error
bars in the graph.
A ruler with smaller scales could be
used for a more precise
measurement of pendulum lengths.
As this is also an insignificant error,
improvements are unlikely to result
in a significantly different result.
Systematic Error:
Precision of
stopwatch
This is also an insignificant error caused by
the systematic error of the stopwatch. The
stopwatch has an uncertainty of 0.1s, which
is insignificant compared to the
inconsistencies in the reaction time which
ranged from 4.50 to 13.64 percent of the
oscillation period.
As it is an insignificant error,
improvements are unlikely to make
meaningful changes to the results.
Bibliography
11
Segun, Oyetade. “Experiment 04: The Simple Pendulum.” Experiment 04: The Simple Pendulum,
Practical Physics, 26 June 2016, johnwellphy1.blogspot.com/2016/06/experiment-04.html.
“Graphical Representation of Simple Harmonic Motion.” BYJUS, BYJU’S,
byjus.com/jee/graphical-representation-of-simple-harmonic-motion/. Accessed 4 Dec. 2023.
12