FINAL VERS NOAH N Energy Conservation Report 1g
Stage 1 Physics
Conservation of Energy Practical Investigation
Figure
1Energy can be neither created nor destroyed, but only changed in
form.
Name: Noah NishiharaHome Group: E05
Report of
experiment on:
How the Speed
of A Pendulum Bob At The Lowest Centre of Gravity Proves The Law of Conservation
of Energy
Completion Date of Experiment: Monday,
September 04, 2017
Work Finalised: Monday, September 18,
2017
Table of Contents
Results for the Photogate
Timer Section of the Experiment
Theoretical Calculations
for Comparison with the Photogate Timer Section of the Experiment
Experimental Calculations
for Comparison with Theoretical Results
Percent Error of the
Photogate Timer Section of the Experiment
Summary of All Photogate
Results
Results for the Tracker
Video Section of the Experiment
Efficiency from the
Gold-coloured Pendulum Tracker results
Background
To
investigate the conservation of mechanical energy, an experiment is designed to
allow accurate analysis of circularmotion by a pendulum. Thevelocity at the
lowest point of the pendulum is calculated and compared with the experimental
results. Similar velocity from both proves that mechanical energy is indeed
conserved.
A
simple pendulum consists of a mass attached by a cord to a fixed point. When it
is raised and released, gravity accelerates it to its equilibrium position
whereby the accumulated momentum causes it to rise again to its original
height. Its mass is concentrated at a point.
Energy
is a concept used to explain change. Given that all matter possesses energy,
and the energy in the universe is constant, energy cannot be created or
destroyed. To prove the Law of Conservation of Energy, a small-scale system is
created, in which energy is stored or transformed. Using the knowledge of
gravitational PE at the top of the pendulum’s swing being equal to its KE at
the bottom of the swing,the theoretical velocity is calculated and compared with
the velocity determined experimentally.
Aim
To show that energy is conserved
in the motion of a simple pendulumusingvelocity measurements of a pendulum at
its lowest point.
Figure 2 Power Functions
Hypothesis
|
Figure 2 Power Functions |
If theoretical velocities
derived from the Conservation Law equal the experimental velocities, then total
energy transforms but is conserved (without acting external forces). Pendulums releasedat
different heights will change their velocity at the lowest centre of gravity like
a power function graph.
Outline of Procedure
Apparatus
·
2 Retort Stands
·
Protractor
·
Brass mass hanger
·
Wooden rulers (Plastic recommended - 50 cm) or Metre sticks
·
Colour A3 paper
·
Data logger interface (VernierLabquest Mini) - 12-bit
interface
·
Vernier Photogate timer
·
Tracker software (PC)
·
Pendulum bob – (gold colour, mass: 64.1 g) and (brown colour
of 500 grams, mass: 147.9 g)
·
1mm width tape
·
Tape
·
Blu-tack
·
2 Threads of string
·
Metre ruler
·
2 clamps
·
Wooden block
·
Computer
·
Video Camera/Mobile Phone
·
Scissors
·
Stand/tripod
·
Phone holder for tripod
Note for Apparatus:All equipment should be
properly calibrated. The metre rulers should be plastic for better results.
Attach to the wooden block for accuracy in measurements in the range of 5 to 20
cm.
The pivot point did not
allow smooth swinging, as friction acted upon the pendulum and caused it to
have an unequal swing. An inelastic string was required and a fixed point
(clamp). The brass mass hangerwas clamped securely, but greater heightsof
release caused the stand itself to rock back and forth.
The brown coloured (500
gram) pendulum bob was 3.5 centimetres above the desk and had a diameter of 3.2
centimetres along the line indicating centre of mass.The brown pendulum is not
aero-dynamic. It was assumed that it has a symmetrical shape.The bob also rotates
and loses momentum/energy. Slower stopping times will result from this. The bob
may swing in an ellipse pattern. It does not hang straight either.Its mass may
be unequally distributed. Its centre of gravity would not have been in the
middle because it was noticed that it did not hang perpendicularly during its
swing.
The gold coloured mass was
2.0 cm above the desk and had a diameter of 2.4 centimetres. The diameter of the
bob must be measured to calculate the speed from timer results of the
Photogate. In this report the gold coloured mass is only used for Tracker
analysis. It is advisable, to minimize systematic error, for the pendulum to be
covered with tape which holds a thin white line (masking tape cut to 1mm) at
the centre of mass.
Modern photographic
devices should be able to capture Full HD 1080p, which was utilised for this
experiment. Some phones are slow-motion capable and are recommended. An Xperia
XA Ultra was used which by design has its lens on the corner, further
increasing the error of parallax. Its maximum frame rate is 30fps.
Due to safety concerns,
the original method involving a razor cutting the string of the pendulum is not
used. Its setup is shown below and involves measuring the distance from the
razor to the carbon paper.
The results attainable
through the procedure below are either the time of entry and exit for the
Photogate, or the videos for later analysis in Tracker software.
Procedure Description
1. Using a balance, measure
the mass of the pendulum bob and string, and record values.
2. Set retort stand and clamp
at appropriate heights and attach string(s).
3. Attach bob to string so
that the centre of the mass of the bob is 3.5 centimetres above the desk,
adjusting the string length as required.
4. Set the bob so that it is
directly covering the photogate timer (Gate) diode. (Place on a suitable wood
block)
5. Connect the photogate
timer to the Vernier interface.
6. Clarify that the photogate
is functional.
7. Fix the gate with tape on
both sides and turn on.
8. Lift the bob so that the
centre of gravity is H cm (height) from the desk, also pulling the string back
to form a straight line, and ensure that the string has a firm tension.
9. Let go of the bob and once
it passes the Gate, cease its movement by catching it to prevent another swing.
Otherwise, simply take the video and allow a maximum of ten swings to each
side. Using the Gate, the H heights were from 5 to 35 cm increasing at 5 cm
increments.
10. Copy the data from the
Logger Pro application to separate sheets in different Excel spreadsheet files
where applicable.
11. Repeat three times for
either method and calculate an average of the time difference values (if using
Gate).
12. Change the H height in
five different ways and measure the velocity.
Diagrams: Original
Method Setup and ActualSetup Drawing
|
Figure 3: Original Method Setup |
Figure 4 Diagram of Set-up/Configuration of Equipment |
Source of Figure 2 and 3: (四天王寺高等学校・中学校, n.d.)
|
Figure
5Snapshot of the Video showing setup method in
Tracker |
Figure 6The Second Experiment with the Brown-coloured
Mass |
Safety and ethical issues overview
By keeping all parts of
the stand secure, accidents due to drops did not occur. The clamps must be
tightened on the retort stand and computers should be kept away from the
experiment to minimise possibility of the screen being hit. Correct use of
apparatus should be ensured to prevent accidental damage of property.
The pendulum bob should only
be swung when others have been notified to prevent injury. Also, keep
electronic products running for a minimal amount of time to save energy.
For other types of
conservation of energy experiments, safety glasses should be worn as the
possibility of a projectile hitting the body is greater.No injury or harm is to
be caused on others during the experiment.The method above will always be
followed to ensure this.
The apparatus should be
kept safe to avoid losing any equipment. All apparatus should be returned to the
designated area for others to use. No safety gear will be necessary for this
pendulum practical.
Principles
The
data was collected directly asdecimal values from measuring devices. The
Photogate timer devices measured time from the first disruption of the diode to
the end of the disruption (the covering of the diode). The Logger Pro software
reported the time values with an accuracy to six decimal places. For simplicity
in the calculations, the number of decimal places was reduced to three. Calibration
was not carried outfor the sensor devices as the digital readingsweredeemed correct
before use.
The length of the Blue sheet (background) acts as the
calibration stick (0.64 metres) in the Tracker software. This is done by
zooming into the video frame and dragging the measuring line to the ends of the
sheet’s bottom side. Any velocity values shown afterwards will be accurate to
two significant figures.
Formulas Used
This is
found by equating the initial potential and kinetic energy with the final
potential and kinetic energy. This is because only gravity (an internal force)
does work and so the total mechanical energy is conserved according to the law.
The initial
kinetic energy however, is zero, while the final gravitational potential is
zero at its lowest point of the swing.
The
velocity values derived experimentally will be compared with the theoretical
velocity from
Used for
calculating the velocity from time values from the Photogate.
Variables
Independent
Variable:Height
of the pendulum bob from desk (H cm)
(excluding the distance
from the bob to the desk which is h cm)
How the Independent Variable is changed: Changed by measuring
accurately the release height of the pendulum bob
Dependent
Variables:Pendulum
bob speed as it passes its lowest position
How the Dependent Variable is Measured: Using Vernier sensors and
Video cameras
Height of release and
speed are both variables resembling real life situations.
Other factors held Constant in
the Experiment:
|
|
|
|
|
|
|
·
The
position of the video camera |
Results
The
results are divided into sections. Theoretical predictions are shown first,
then Photogate Timer results are introduced, followed by Tracker results.All
tables are arranged in order.A discussion involving errors such as energy lost
due to air resistance or frictionfollows the Calculations section.
Results for the Photogate Timer Section of
the Experiment
Remember
that the Mass of the Pendulum bob is actually 147.9 grams (500 grams is written on it) and that the diameter is 3.2 centimetres.
Approximately four trials were attempted, with some having only three or one
valid result(s). For the calculations, it was noted that the bob is 3.5 centimetres above the desk (denoted by h) and the total height (denoted by H) covers heights
from 5 centimetres to 40 centimetres (8 different total heights of
release). These values are in centimetres and must be converted into metres
before they can be used to calculate velocity.
The
Formula used for the Photogate timer calculations is V
Theoretical Calculations for Comparison
with the Photogate Timer Section of the Experiment
The
other results are listed in the table below:
Table 1
|
Total
Height (H) (metres) |
Velocity
of the Bob at its lowest point (m/s) |
|
0.05 |
0.542 |
|
0.10 |
1.129 |
|
0.15 |
1.501 |
|
0.20 |
1.798 |
|
0.25 |
2.053 |
|
0.30 |
2.279 |
|
0.35 |
2.485 |
|
0.40 |
2.675 |
Theoretical Calculations for Comparison
with Velocity Values Derived from Plotting in Tracker using Videos of Pendulum
Swings (Section of the Experiment involving Gold-coloured Bob)
The
theoretical calculations for the gold-coloured bob with a height (h) of 2.0 cm
between the bob and the desk are outlined below for use later:
Table 2
|
Total
Height (H) (metres) |
Velocity
of the Bob at its lowest point (m/s) |
|
0.10 |
1.084 |
|
0.15 |
1.596 |
|
0.20 |
1.878 |
|
0.25 |
2.123 |
|
0.30 |
2.343 |
|
0.35 |
2.543 |
To
compare the experimental results with the theoretical results above, the
experimental results must be substituted into the
It
is believed that the range of the diode can lead to random errors provided the
object passes it slowly.
Figure
7 Point of Entry and Exit
The
average time in seconds recorded for each trial is shown in separate tables in
Appendix 1 for each height H (metres). These averages are used instead of a
single measurement, which would likely be the time taken for the first swing to
pass by the Photogate. Air resistance and friction would cause the bob to slow
down. However, averaging was found to be an effective way to achieve the best
value from theinconsistent Photogate. Random errors caused by the pendulum bob
hitting the Photogate or curving away during its swing and entering the gate at
an angle were also reduced. Note that the Photogate may not function properly
when the angle of the swing is sideways instead of perpendicular to the gate.
Several
outlier values were observed,with several being possibly wrong measurements, however
only the later measurements wereremoved and the analysis was continued for the
remaining data. Theseoutliers are extreme but possible. There may be wrong
assumptions involved. Since all the outlier results were obtained under the
same conditions as the other values, they were included.
Experimental Calculations for Comparison
with Theoretical Results
Using
the formula V
The
other velocities of different heights (H from 10 to 40 centimetres) derived
experimentally are outlined in tables found in Appendix 2.
Percent Error of the Photogate Timer
Section of the Experiment
Using
the percent error formula, the results of the experiment (velocity values at
the lowest point of pendulum) were compared with the theoretical calculations
of the velocity values at the lowest point of pendulum.For full calculations,
refer to Appendix 3.
Summary of All Photogate Results
All results from the Photogate and Brown-coloured Bob sectionare
summarised in the tables below:
Table 3
|
Theoretical
Speedfor 5 cm |
Experimental Speedfor
5 cm |
% Error for 5
cm |
Percent
uncertainty (%)for 5 cm |
|
0.542 |
|
|
170.1064 |
|
0.542 |
|
|
191.4286 |
|
0.542 |
|
|
183.1618 |
|
Theoretical
Speedfor 10 cm |
Experimental Speedfor 10 cm |
% Error for 10 cm |
Percent uncertainty (%)for 10 cm |
|
1.129 |
1.094 |
|
2.833638 |
|
1.129 |
1.266 |
|
9.581359 |
|
1.129 |
0.970 |
|
14.51546 |
|
Theoretical
Speedfor 15 cm |
Experimental Speedfor 15 cm |
% Error for 15 cm |
Percent uncertainty (%)for 15 cm |
|
1.501 |
1.375 |
|
6.101818 |
|
1.501 |
1.230 |
|
14.6748 |
|
1.501 |
1.152 |
|
20.18229 |
|
1.501 |
1.253 |
|
13.18436 |
|
Theoretical
Speedfor 20 cm |
Experimental
Speedfor 20 cm |
% Error for
20 cm |
Percent
uncertainty (%)for 20 cm |
|
1.798 |
1.149 |
|
31.41862 |
|
1.798 |
1.745 |
|
1.690544 |
|
1.798 |
1.424 |
|
14.60674 |
|
1.798 |
1.612 |
|
6.414392 |
|
Theoretical
Speedfor 25 cm |
Experimental Speedfor 25 cm |
% Error for 25 cm |
Percent uncertainty (%)for 25 cm |
|
2.053 |
1.595 |
|
13.98746 |
|
2.053 |
1.598 |
|
13.86733 |
|
2.053 |
1.843 |
|
5.550733 |
|
2.053 |
1.620 |
|
13.01852 |
|
2.053 |
1.822 |
|
6.174533 |
Table 4
|
Theoretical Speedfor 30 cm |
Experimental Speedfor 30 cm |
% Error for 30 cm |
Percent uncertainty (%) for 30 cm |
|
2.279 |
1.737 |
|
13.69027 |
|
2.279 |
1.764 |
|
12.81179 |
|
2.279 |
1.968 |
|
6.935976 |
Table 5
|
Theoretical
Speedfor 35 cm |
Experimental
Speedfor 35 cm |
% Error for
35 cm |
Percent
uncertainty (%)for 35 cm |
|
2.485 |
1.859 |
|
13.5503 |
Table 6
|
Theoretical
Speedfor 40 cm |
Experimental
Speedfor 40 cm |
% Error for
40 cm |
Percent uncertainty
(%)for 40 cm |
|
2.675 |
2.454 |
|
3.365933 |
|
2.675 |
2.254 |
|
6.983141 |
|
2.675 |
2.238 |
|
7.301162 |
|
2.675 |
2.140 |
|
9.345794 |
Following on from the
Percent Error Formula calculations which indicate the ratio of the observed and
true values, the 6th equation is used to find the percent
uncertainty. The experimental result is compared with the theoretical result in
% error, while % uncertainty is found to give a better idea of how well a value
can be determined. This uncertainty value also reveals how precise the
experiment was. The values for uncertainty are in the fourth column. Many
values have large uncertainty ranges showing discrepancy. These values do not
prove energy conservation.Thus, the measured values do not
agree well with the calculated values.
There
is an uncertainty in all measurements. Errors are caused by human judgement
including during the plotting of points in Tracker. It provides extra precision
but does not reduce error as human accuracy is fallible. The Photogate timer
had possible calibration errors. Human error in using the metre stick is a
random error foreseen but not eliminated. The metre stick could not be attached
to a wooden block due to its mass and length. Therefore, it swayed in human
hands and resulted in inconsistent lengths. A shorter ruler would reduce this
error. The actual measurements of height H may have been shorter than the
theoretical height H. Hence the speed is slower for the majority.
Only
six attempts proved the Conservation Law by being within a 10% error range. The
use of a metre stick to measure small distances affected these results as a
systematic error. With my partner during the experiment we could have exchanged
our roles to investigate the other large error which is the way the bob is
released.The release of the pendulum involves friction which causes
inconsistent release heights.The releaser may alsoinputunbalanced forces and
change the overall energy of the system. By comparing the results from both of
our release methods, the extent of the effect of release friction can be
determined.
Also,
longer periods of time recorded from the Gate would lower the speed considerably
as observed from the 5 cm results.
Results for the Tracker Video Section of
the Experiment
Tracker
results for the same pendulum setup i.e. the brown coloured bob with same
height and mass configurations as the Gate section are in Appendix 4.
The
gold-coloured bob, on the other hand,was at a height (h) of 2.0 cm above the
desk. Only the first velocity of the bob at its lowest point (m/s) which can be
found from the first swing will be considered.
The speeds were observed to decrease over time during
the swings. This is apparent from the graphs. Since only the first swing is of
use for the comparison with the theoretical velocity, graphs were drawn for
each Excel sheet data set but are not included here. The graphs were drawn to
check and ensure consistency in the velocity values. The maximum speed of each swing
was also compared to explain any changes that occurred.
Table 7
|
Velocity of the Bob at its lowest point (m/s) |
Theoretical Velocity of the Bob at its lowest
point (m/s) |
% Error |
|
|
0.10 |
1.269 |
1.084 |
17.0664 |
|
0.10 |
1.454 |
1.084 |
34.1328 |
|
0.10 |
1.324 |
1.084 |
22.1402 |
|
0.15 |
1.700 |
1.596 |
6.5163 |
|
0.15 |
1.831 |
1.596 |
14.7243 |
|
0.15 |
1.603 |
1.596 |
0.4386 |
|
0.20 |
Undisclosed due to Mix-up in file storage |
1.878 |
Omit |
|
0.25 |
2.091 |
2.123 |
1.5073 |
|
0.25 |
2.170 |
2.123 |
2.2138 |
|
0.25 |
2.144 |
2.123 |
0.9892 |
|
0.30 |
2.378 |
2.343 |
1.4938 |
|
0.30 |
2.310 |
2.343 |
1.4085 |
|
0.30 |
2.266 |
2.343 |
3.2864 |
|
0.35 |
2.495 |
2.543 |
1.8875 |
|
0.35 |
2.489 |
2.543 |
2.1235 |
|
0.35 |
2.562 |
2.543 |
0.7471 |
From the table above, the percent error shows a much
lower range of error for the heights from 25 to 35.
A line graph was drawn for the three velocities in
which the Gate and second Tracker results (for brown coloured bob) were
compared. With the total height as the independent variable, the velocities
should be the same. This is because the
The mass is cancelled out from the first equation
above (which were used in the process of proving energy conservation.Note that
the period (swing) of a pendulum depends on the length of string instead of the
mass. Therefore it can be inferred that objects fall at speeds unaffected by
mass.
It can be seen in the graph above that the Gate results
have the greatest error. With greater precision is the Tracker (Brown) results
but the direct proportionrelation outlined above cannot be proven due to
inaccurate velocities.
Graphs are drawn for both Gate and Gold-coloured
Pendulum resultsseparately to discover the source of error. The graphs have an
x-axis of the initial height and a y-axis of the velocity at the lowest point.
The lines of the best fit are drawn and compared.
The reason for the greater velocity with the 10 and 15
centimetre heights is unclear. This may be an example of the tension force
acting on the string bringing about an increase in speed. Where is the extra
13.4% of the inputted energy from? See the calculations section below.
From the graph above the relationship between increasing
heights and the resulting velocity at the bob’s lowest centre of gravity is
indeed a power function. It is predicted that there is a limit to the maximum
velocity possible. It will not increase indefinitely. Also, the clear
correlation between height and velocity based on the high correlation
coefficient is misleading. Errors have caused the experimental velocity to
become zero only when the height becomes negative. When the string is zero, the
velocity of the bob would still be a little over 0.1 m/s. The data relationship
is depicted well, for instance in the theoretical velocity being generally
lower for heights less than 20 cm but higher for those above 20 cm. The absence
of 20 cm data may have caused this.
Slow velocities may cause errors for both Gate and
Tracker values. From v=s/t we know that either the height may have measurement
errors or the Gate may have difficulty in recording data for slow speeds.
The inconsistency is apparent. Comparison of the
theoretical and experimental speeds’ lines of best fit show that the velocities
found were lower than the expected values.The metre stick may have been used to
measure shorter total distances through incorrect placement of the ruler on the
table and other factors.
Because of the scatter observed, the results are not
precise. The results are also inaccurate as they are not close to the
theoretical values. The line of best fit does
begin at the (0,0) origin. A different factor may have caused this.For example,
the scatter may allow this to happen. Note that the bob may travel longer
towards one side than the other due to different string tying methods. Also,
the stringstretches for heavier masses (brown bob is heavier than gold bob).
The string stretches more at higher speeds. This is not the case in the graph
above.
Only the last three heights for the Tracker (the
second graph above) have high accuracy. Precision was also high onlyfor those three heights.
Calculations
Efficiency from the Gold-coloured Pendulum
Tracker results
Efficiency measures how much energy is conserved. The
efficiency is the energy output divided by the energy input, and is expressed
as a percentage. For reference purposes in this report: output is kinetic
energy, and input is potential energy or the height that we raised the bob.
97% efficiency means this data is nearer to the
expected real result. For more, refer to Appendix 5.
Discussion
The
results from this experiment align with the energy conservation law. The
experiment was conducted to find the maximum velocity of a pendulum to compare
with the theoretical calculated value. It was hypothesized that theoretical
velocities would overlap with the experimental velocities to prove the law. The
results of this experiment are affected by errors;there is
discrepancy compared toexpected findings.Overall, they show that for all
masses, regardless of method, heights above 10 cm give better results.
While
the pendulums did change their velocity at a rateof apower function, errors skewed
the graph of the Tracker version. The relationship between release height and velocity
is a direct proportion limited by the stability of apparatus.
Although
the hypothesis was only proven by a minority of attempts, two methods were
tested and the Photogate method found to be more susceptible to random errors.Three
heights gave highly accurate and precise results. Since only a minority agree
with previous studies, the methodology must be reviewed.
The
design controlled variables by ensuring that focus was on measuring the release
height; incorporating video-taking helped. All other variables were kept
constant by limiting interactions with apparatus.
This
experimentis limited by parallax caused by the camera not being perpendicular
to the apparatus. The distance from the viewpoint of the smartphone to either
side of the pendulum (x,ycoordinates) is skewed. Longer distances between the
camera and the apparatus can reduce this systematic error. Also, measurements
of the independent variable using unsuitable apparatus caused a random error.
Shorter devices are necessary for small measurements.
The
zoom function was utilised during the analysis of the videos. However, precise
coordinates of points arelimited by the accuracy and consistency of the plotter.
To address this issue, the pendulum bob could have been slowly plotted to
ensure precision. Autotrackerwas more accurate.
The experiment could be
improved by utilising video-taking only. Accurate height measuring is aided by
marking bob’s centre of mass. Barriers could keep the swing straight to prevent
accidental collisions with the gate. A device to release at the same height is
necessary to achieve consistent data.Care is necessary for configuration of future
experimentsfor greater accuracy in measurements.
The
results apply to any system without external forces doing work. The
significance of these results centre around methodology issues but also give
insight into how all energy is interchangeable.
Conclusion
Energyconservation has been verified.The results were on average, within 21.5% of the calculated values. The discrepancies are due to parallax and measurement errors. A bifilar pendulum could be usedwith clamps instead of hooks as the pivot.
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