w is the width (diameter) of the end circle of the cone.
h is the height of the cone.
w = 5cm, h = 5cm
w = 10cm, h = 5cm
w = 20cm, h = 5cm
They were dropped from a height of 1m and 2m and the time taken to fall measured with a stopwatch and recorded:
From an experiment viewpoint, the last cone, w = 20cm, h = 5cm, d = 2m would be best as it has the largest time to fall and hence the smallest percentage uncertainty in the measurement of its time. However it required the construction of a paper cone from two pieces of A4 so the w = 10cm, h =5cm, d = 2m was chosen as a simpler alternative.
Determining a maximum mass for the weighted paper cone
The paper cones are weighted using Bluetak, and their mass established using an electronic balance.
A trial of three different masses was attempted to establish the viability of weighted cones at 5g, 15g and 30g. Each cone was dropped from a height of 2.00 m and its descent time measured using a stopwatch.
The mass of the cones was adjustable to the nearest 0.5g without substantial time or effort. This gives a worst case uncertainty in the measurement of mass of ±10%.
The 30g weighted cone gave the smallest time of 0.80s ± 0.08s and this gives a ±10% uncertainty. The worst case uncertainty (for m=5g) in the t is more significant at ±25%.
Range and number of readings
Keeping the time above 1s should keep the uncertainties in the time below 50% and so results will be taken from m=3g incrementing in 1g intervals until the time of descent approaches 1s. This should provide between 10 and 20 results.
- A mark is made on a white board at a height of 2m (±10mm). The measurement is made using a plumb line pre-measured at 2m using a metre rule.
- A cone is constructed from A4 paper by cutting out a circle with a radius of 71mm and folding over an angle of 105O to make a cone with a width of 100mm and a height of 50mm.
- This cone is then weighted with Bluetak until it has a mass of 3g±0.5g as measured on an electronic balance.
- The tip of the cone points downward and is lined up with the mark on the whiteboard. This cone is then dropped and a stopwatch started.
- When the tip of the cone hits the floor the stopwatch is stopped.
- This measurement is repeated another two times.
- The mass of the cone is then increased by 1g and steps 4-6 repeated
- Steps 1 to 7 are repeated until the time taken to fall drops below 1s.
Bluetak (approx 50g)
Metre rule (giving a resolution of ±1mm appropriate for measuring, h.)
Stopwatch (giving a resolution of ±0.01s appropriate for measuring t.)
(giving a resolution of ±0.001g appropriate for measuring m.)
Raw results tables including absolute uncertainties
Tables of calculated values e.g. means, percentage uncertainties
Raw graphs of results
The trend shown here confirms a non-linear relationship between time and mass. It is consistent with a inverse relationship of some form.
There are no anomalous results to be dealt with.
By drawing a graph of t2 versus 1/m the relationship can be confirmed:
The processed results hence look as follows:
The graph was plotted and a best fit line added:
The trend on this graph confirms that the theory outlined earlier is still plausible. t2 is linearly related to 1/m with a small y-axis intercept.
The gradient of this line should be equal to .
Hence this gives a value for
The equation of the straight line can be used to find the difference between the data points and the best fit line:
The worst case difference between the data point and the best fit line is 23%.
This can be accounted for by reference to the % uncertainty in t, which from the measurements was found to be at least ±16%. Hence the uncertainty in t2 could be expected to be up to about ±32%.
Sources of error (in order of decreasing significance)
Time to drop a fixed distance
The measurement of time was the most significant uncertainty in this experiment (t = ±16%). This random error was largely due to human variation in starting and stopping the stopwatch, on launch and landing. To improve the measurement the drop of the paper cone could be electronically detected by switching off an electromagnet holding the paper cone. An electronic timer would then run until the paper cone struck a piezo sensor which would be configured to stop the timer. It is anticipated that the uncertainty could be reduced to a sub ±10ms value by this method.
Mass of the paper cone
Mass of the paper cone had a worst case uncertainty (m=3g) of 17%. This was due to the ±0.5g precision to which the paper cone mass was prepared. It would, with care and extra time, be possible to reduce this uncertainty down toward the precision of the electronic balance which is around ±0.01g.
The uncertainty in the drop distance (±10mm) was mainly due to the hand-release nature of the experiment, variation is likely to be random and could easily be reduced by the use of a fixed electromagnetic release. By this method the uncertainty in the measurement of drop distance could be reduced to the resolution of the metre rule used (±0.5mm).
Shape of the cone
Although the same paper cone was dropped on each occasion, the shape of the cone can be seen to deform. It is likely that the shape of the cone deformed differently for different cone masses due to the increased velocities at higher masses. Although it has not been possible to quantify this uncertainty it could be made insignificant by the use of a stiff plastic cone.
Range of measurements and the number of readings taken
The range of measurements was limited to masses of a few grams. Clearly this limits the confirmation of the prediction to within this range. A larger range of masses would be required to confirm the relationship outside of the readings taken.
The number of readings taken (3 per independent value) was sufficient to enable an indication of the uncertainties in the measurement of time. A larger number of repeats (>30) would be required for a statistically significant sample to be taken to confirm the random nature of the uncertainties and their distribution.
Combined Uncertainty for the experiment
Using a simple sum of percentage uncertainties we get a worst case value of :
±32% (time) + ±17% (mass) + ±5% (drop distance) = ±54%
Note: the time uncertainty of ±16% is doubled because of the t2 nature of the relationship.
Validity of the results
The largest percentage difference between a data point and the best fit line for t2 versus m-1 was ±23%. This is more than accounted for by the combined worst case uncertainty for the experiment of ±54%.
Hence the results do indicate that the predicted relationship is plausible.
Here's what a star student thought of this essay
Quality of writing
Spelling and grammar are of a high standard. The presentation of the report is clear and done to a really high standard.
Level of analysis
The language that the writer used is of a high standard. The area that the writer has investigated is not fully covered within the syllabus, but it's good that the writer was able to gain suitable information on the area within the research that they did. The writer used the correct terminology with regards to experimentation and the scientific background.
Response to question
The writer has approached the investigation well. As a result, the writer has produced a well written report. All of the writers objectives have been clearly covered, therefore making their report clear and easy to follow. It was good that the writer was able to explore the theory behind the experimentation taking place. It was also good that the writer was able to explore ways to reduce error by carrying out a preliminary investigation, this being explained during the writers conclusion. Overall, this is a high quality report.