Paper cones and drag

by Zac Sayle - Year 12 Physics 2024

Introduction

Drag on objects is affected by several factors, namely the density of the fluid they are travelling through ( $ρ$ ), their relative velocity ( $v$ ), and their cross-sectional area ( $A$ ).

These are related by the formulas

F_{D} = \frac{1}{2} ρ v^{2} C_{D} A

and

ρ = \frac{p}{R_{specific} T}

To determine how the size of a paper cone affected its drag, cones were made from circles of equal radii, but with different central angles for removed sectors. The time to fall a set distance was measured, which could then be used to calculate the average velocity, $v$ , drag force, $F_{D}$ , and drag coefficient, $C_{D}$ . For the other variables not already stated, $p$ is the absolute pressure of the air, $R_{specific}$ is the specific gas constant of air, and $T$ is the absolute temperature of the air.

While there have already been investigations into the most aerodynamic shapes for flight, this investigation could theoretically contribute to ensuring that those results remain valid for conical nose shapes.

Hypothesis

It is hypothesised that as the angle removed from the circle increases, fall time will decrease, because this change decreases the cross-sectional area, $A$ . This also follows logical reasoning based on previous experience with letting things fall, especially considering the design of parachutes.

Methodology

Equipment

Paper (approx 73.8 g·m^-2 was used)
Scissors
Sticky tape
Protractor
Compass & pencil
Scales
Stopwatch & video camera
Ruler
Thermometer
Barometer

Steps

Circles with equal radii were cut from similar pieces of paper
A sector was cut from each circle, with the angle recorded. The remaining part of the circle was bent and taped into a conical shape. Six unique angles were used for this experiment, with the sector angles listed in the results.

Circle dimensions and cone construction diagram

Diagram of circle measurements — Measurements used in the creation of the cone from the circle.
$θ$ represents the angle of the removed sector, as referred to in the data.
The red sector is removed, and $r_{1}$ and $r_{2}$ are joined to form the cone.

A small (known) mass was placed into the first cone
The first cone was dropped (tip-down) from the set 1.5 m drop height (based on its tip's position). The time for the cone to touch the ground from release was measured with a stopwatch, and the drop was filmed with a video camera, providing an alternative method for finding the drop time. This was repeated for at least three trials with each cone
Steps 3 & 4 were repeated with the remaining cones
After all trials were completed, the videos were analysed in a video player, by counting the number of frames between the release of the cone and it first making contact with the ground, and then multiplying by the frame rate recorded in the metadata

Experimental setup diagram

Maintaining controlled variables

Jump to data collected about controlled variables

In order to maintain the controlled variables, the following steps were taken:

Care was taken to drop the cones from a stationary position (so $u = 0$ , and $F_{↓} = m g$ )
Each trial was conducted within the same room to ensure that the conditions of the air remained consistent, controlling $p$ , $R_{specific}$ , $T$ , and thus $ρ$
An apparatus consisting of a ruler and a perpendicular object was set up to ensure that a consistent 1.50 m drop height was observed, ensuring $s = 1.50 m$
The gross mass of the cone was kept consistent as far as possible by using consistent paper, although the masses of each of the cones were measured in order to be able to account for these changes (because, naturally, given that different amounts of paper were used, this would vary slightly). Similarly, a consistent additional mass was used between all trials.

Results

All result videos can be accessed here. They are organised by the angle of the remaining sector used to create the cone.

Overall, there was some resemblance of a proportional relationship between the cross-sectional area of the cone and the the time which it took to fall to the ground, however this relationship became less obvious with larger areas.

Full results table for data collected with stopwatch

Sector angle	22.5°	47°	90°	180°	226°	269°
Trial 1	0.93 s	0.91 s	0.66 s	0.50 s	0.47 s	0.69 s
Trial 2	1.06 s	0.75 s	0.66 s	0.63 s	0.69 s	0.66 s
Trial 3	0.88 s	0.84 s	0.66 s	0.56 s	0.37 s	0.40 s
Trial 4	(null)	(null)	(null)	(null)	(null)	0.56 s
Mean	0.96 s	0.83 s	0.66 s	0.56 s	0.51 s	0.58 s

Full results table for data collected with 59.9 Hz video camera

Sector angle	22.5°	47°	90°	180°	226°	269°
Trial 1	63 frames = 1.05 s	62 frames = 1.04 s	49 frames = 0.82 s	39 frames = 0.65 s	35 frames = 0.58 s	35 frames = 0.58 s
Trial 2	65 frames = 1.09 s	62 frames = 1.04 s	51 frames = 0.85 s	40 frames = 0.67 s	35 frames = 0.58 s	35 frames = 0.58 s
Trial 3	64 frames = 1.07 s	59 frames = 0.98 s	51 frames = 0.85 s	40 frames = 0.67 s	35 frames = 0.58 s	44 frames = 0.73 s
Trial 4	(null)	(null)	(null)	(null)	(null)	36 frames = 0.60 s
Mean	1.07 s	1.02 s	0.84 s	0.66 s	0.58 s	0.63 s

The data collected with the video camera is being treated as the main data used for analysis, as it is believed that it is more accurate, due to there being less room for introduced human error based on reaction times for stopping the stopwatch. The third trial for the 269° cone should also be interpreted cautiously for two main reasons:

the mass in the cone fell out during the drop, a consequence of not properly following the original method, which would have included better securing it
the results between the stopwatch measurement and the camera measurement differ significantly: for the stopwatch, the time is well below the mean, while for the camera, the time is well above the mean.

Note that this third trial for the 269° cone was kept in the means for completeness. It is hoped that the additional trial undertaken for this cone will help to remove the error to some extent.

Considering this, although the times from the video camera have been deemed more accurate, a "super mean" has been calculated by taking the means of both sets of results.

Combined mean results

Sector angle	22.5°	47°	90°	180°	226°	269°
Mean (stopwatch)	0.96 s	0.83 s	0.66 s	0.56 s	0.51 s	0.58 s
Mean (video)	1.07 s	1.02 s	0.84 s	0.66 s	0.58 s	0.63 s
Combined mean	1.01 s	0.93 s	0.75 s	0.61 s	0.55 s	0.60 s

Plot of the mean time to fall vs. the angle cut to make the paper cone, including linear trendlines.
The mean from all three methods (the stopwatch, the video and the mean of both of those) is shown.
Error bars represent the inherent measurement error in the equipment used (0.005 s).

Notably, in the data, the hypothesis is supported until the 269° cone, as the average time to fall was longer than for the 226° cone. This applies based on all three means.

Other data was also collected for the purposes of allowing drag force and co-efficients to be calculated more effectively, based on the air conditions in the room which the experiment was conducted in. This is included, alongside any other data collected, for completeness and reproducibility purposes below.

Other data

Base circle radius	75 mm
Mean paper area density	73.8 g·m^-2
Mass of additional mass	1.98 g
Drop height	1.50 m
Air temperature	21°C
Air pressure	1017.3 hPa
Specific gas constant of air	287.052874 J·kg^-1·K^-1 (not measured)

Intermediate calculations

Cross-sectional area

To calculate the cross-sectional area, $A$ of each cone from the angle of the sector removed, $θ_{removed}$ and the radius of the circle, $r_{ci}$ , the following formula was used:

A = π {(\frac{r_{ci} (360° - θ_{removed})}{360°})}^{2}

Results

Angle removed	Cross-sectional area
22.5°	0.0155 m²
47°	0.0134 m²
90°	0.00994 m²
180°	0.00442 m²
226°	0.00245 m²
269°	0.00113 m²

Air density

Applying $ρ = \frac{p}{R_{specific} T}$ with the data relating to the air above gives $p = 1017.3 hPa$ , $R_{specific} = 287.052874 J \cdot {kg}^{-1} \cdot K^{-1}$ and $T = 21 °C$ , and thus that $ρ = 1.2 kg \cdot m^{-3}$ , which is approximately air's density, according to the International Standard Atmosphere (Wikipedia contributors, 2024a).

Drag

The drag force can be calculated from the SUVAT equations and Newton's second law (assuming that drag and gravity are the only forces operating, and they are operating parallel) as follows:

Formula and derivation

F_{{net}_{av}} = m a_{av} = F_{↑} + F_{↓}

Let F_{↓} = m g

Let F_{↑} = F_{D} = \frac{1}{2} ρ v^{2} C_{D} A

\begin{matrix} s = u t + \frac{1}{2} a t^{2} & but u = 0 \\ ∴ & s = \frac{1}{2} a t^{2} \\ ∴ & a = \frac{2 s}{t^{2}} \\ ∴ & F_{D} = \frac{2 m s}{t^{2}} - m g \end{matrix}

Results for drag force

Using the video results

Angle removed	Drag force
22.5°	0.0227 N
47°	0.0214 N
90°	0.0164 N
180°	0.00781 N
226°	0.00251 N
269°	0.00496 N

Using the combined results

Angle removed	Drag force
22.5°	0.0218 N
47°	0.0195 N
90°	0.0132 N
180°	0.00478 N
226°	-0.000548 N
269°	0.00350 N

Results for drag co-efficient

Formula used for drag co-efficient: $C_{D} = \frac{2 F_{D}}{ρ v^{2} A}$

Using the video results

Angle removed	Drag co-efficient
22.5°	1.11
47°	1.01
90°	0.684
180°	0.490
226°	0.227
269°	1.17

Plot of the calculated drag co-efficient vs. the angle cut to make the paper cone.

Using the combined results

Angle removed	Drag co-efficient
22.5°	1.06
47°	0.925
90°	0.551
180°	0.300
226°	-0.0494
269°	0.828

Plot of the calculated drag co-efficient vs. the angle cut to make the paper cone.
The negative value is likely due to random experimental error, particularly in measurement.

Discussion

Overall, particularly with the video results, there was a clear downward trend within the data. It is difficult to tell whether this fits a linear model based on the data collected, considering the significant gap in the data between 90° and 180°. If the experiment were to be run again, it would be worth collecting data for at least one point within this interval.

Errors

There were two significant errors which occurred during the running of the experiment, which were clearly visible in the data:

226° cone negative drag

The drag co-efficient for the cone from the 226° sector was negative, which should not be possible. This negative drag co-efficient is a result of the drag force being calculated as -0.000548 N. Practically, this result means that the measured acceleration was greater than $g$ , 9.8 m·s^-2, and so the drag force had to work downwards to provide a great enough downward force to cause that level of acceleration. This does not follow expectations of drag, and points strongly towards error, especially considering that this was only observed for one cone, in one measurement system, and only three trials were undertaken, limiting the effectiveness of multiple repetitions for reducing random error.

Given that the error was only found in the combined mean, significant measurement error while using the stopwatch is strongly suggested, owing greater authority to the video results, given their apparent greater accuracy.

269° outlier

By both measurement systems, the 269° cone presents itself as an outlier, with its drag coefficient being significantly higher than expected for such a pointy "streamlined" design. Again, it is likely that this was caused by some form of error in the dropping process or during the measurement, particularly considering that there were already potential problems identified with this cone's drop process. Again, although an additional trial was conducted with the cone, it was insufficient at negating this error in the data. Aside from the already identified problems, this outlier could be caused by the cone rotating during the fall, and hence increasing its cross-sectional area during the fall. This possibility was not accounted for in the calculation of the drag co-efficient, as it was assumed to be consistent throughout the fall.

Aside from running additional trials (the preferred remedy), it may be possible to remove the most problematic data point, however this may not be an ideal solution as the trial was technically performed correctly, and thus otherwise usable data would be discarded. This could be especially problematic with such a small dataset.

Conclusion

In line with assumptions and the hypothesis, the experiment makes it evident that pointier shapes experience less drag than flatter shapes. This knowledge may be useful in designing things which need to travel through fluids in some way, whether they demand more (e.g. parachutes) or less (e.g. planes) drag. Despite this, the additional instability introduced with more streamlined shapes (i.e. those with sharper noses) may outweigh the inherent reduced drag due to the rotation which could occur. Further research on this subject may be beneficial, however this suggests overall that there may be a limit to how much drag can be theoretically reduced for some applications without introducing excessive instability to the object.

References

Wikipedia contributors. (2024a, June 11). Density of air. In Wikipedia, The Free Encyclopedia. Retrieved September 3, 2024, from https://en.wikipedia.org/w/index.php?title=Density_of_air&oldid=1228461341
Wikipedia contributors. (2024b, August 12). Drag (physics). In Wikipedia, The Free Encyclopedia. Retrieved September 3, 2024, from https://en.wikipedia.org/w/index.php?title=Drag_(physics)&oldid=1239858611
Wikipedia contributors. (2024c, May 31). Terminal velocity. In Wikipedia, The Free Encyclopedia. Retrieved September 16, 2024, from https://en.wikipedia.org/w/index.php?title=Terminal_velocity&oldid=1226583260