Zaid ShammasIB Physics Hl 1: Period 1November 16, 2010 Freefalling Cupcakes Varied by FormationIntroduction: In this investigation, the relationship between the amount of cupcakes in a formation and thechange they have on the coefficient of drag as they free fall through the air will be studied. Before theinvestigation can be discussed, an understanding of the concepts behind this investigation must beattained. The most prominent concept in this investigation is terminal velocity. Terminal velocity is astate achieved when the force of drag on an object is equal to the force of gravitywhile free falling, as shown in figure 1.01. Terminal velocity is governed by the equation . Where: Vt is terminal velocityg is the force of gravitym is the mass of the objectA is cross sectional surface areaρ is the density of the fluid which the objects is falling throughCd is the coefficient of dragAlso, it is important to understand that when an object reaches terminal velocity, it is no longeraccelerating nor decelerating (negative acceleration) but in fact, its acceleration is equal to zero. Otherfactors to consider are the forces that affect the terminal velocity of an object. The driving force of theobject in terminal velocity is the force that tries to cause the motion of the object to accelerate, such asgravity, while the resistive force of the object is the force that attempts to decelerate the object and whenthese two forces balance out, terminal velocity is achieved. In this investigation mass and cross sectional surface area, two factors that influence terminalvelocity, will change. In addition, by adding cupcakes to the free falling formation of cupcakes, thecoefficient of drag, or the dimensionless number that is used to quantify the amount of resistance anobject has in a fluid environment such as air, will change1. As more cupcakes are added, the mass of thesystem of cupcakes will stay proportional to the cross sectional area. This is because each cupcake beingadded is similar. As cupcakes are added, it is expected that the coefficient of drag will increase in a squarepower function. If equation  is manipulated to give a function of coefficient of drag, it will appear asshown below.
This equation is expected to yield a linear relationship if a graph is plotted, where the y axis will bedefined as the (coefficient of drag)2., and the x axis will be defined as the number of cupcakes in aformation.Design: The research question of this investigation is to understand what relationship a falling formationof cupcake has to the system’s coefficient of drag, and what happens to the coefficient of drag whencupcakes are added to the formation. The independent variable of this investigation the amount ofcupcakes there are in a formation. The dependent variable is the coefficient of drag. There are multiplecontrol variables. Mass must stay proportional to cross sectional surface area to remain as a control, sosimilar cupcakes must always be added to the formation, shown in figures 1.0 and 1.1Figure 1.0 Figure 1.1 Control variables such as air disturbances and air temperature will be kept constant by turning offthe air conditioners in the room. This allows for less temperature change, and less changes in the flow ofair around the room. The placement of tape connecting the cupcakes together must be consistent. Eachcupcake must be gently handled, and tape connecting the cupcakes together must be applied consistently,where it does not obstruct airflow through the ridges and holes around the cupcakes. A height that thecupcakes are dropped from must be established, to allow for the same time of free fall for everyformation. The cupcakes are only viable for a number of trials before their structure breaks down.Therefore, if the same formation of cupcakes is used for an extended amount of trials, the data becomesinvalid; the structure of the cupcakes must be maintained.
Around 30 cupcakes, a motion detector system, and tape are required for this investigation. Cupcake formations should be made first, as the cupcakes are connected gently with tape. For this investigation, cupcake formations of one, three, four, seven and thirteen were created. A set height of 1.4 meters was established, and the investigation was ready. Each formation of cupcake was dropped from the set height right above the motion detector three times. This provides three trials for five data points, to increase confidence in the research. The wide range number of cupcakes in the formation is to ensure a wide range of applicability, and to increase confidence within in research. The research setup should resemble figure 2.0. Figure 2.0 Data Processing and Analysis: Table 1: Trial data as a cupcake formation free falls, with mass measurements. Terminal Velocity (m/s) (± 0.01) Number of Total Mass Total Mass (kg) Average Speed Trial 1 Trial 2 Trial 3Cupcake Holders (g) (±0.01) (±0.00001) (m/s) (± 0.03) 1 0.51 0.00051 1.75 1.73 1.69 1.72 3 1.62 0.00162 1.56 1.52 1.52 1.53 4 2.18 0.00218 1.46 1.53 1.43 1.47 7 4.01 0.00401 1.39 1.33 1.41 1.38 13 7.65 0.00765 1.27 1.39 1.39 1.35 Table 1: Table of values that shows the total mass of each cupcake formation, along with trial data for each free fall. The terminal velocity was calculated with a motion detector, as the slope of the position time graph of the falling cupcake formation. The average speed is an average calculated column that reads the mean of the three trials of each data point.
Graph 1: Sample position-time graph of trial one, where a formation of one cupcake free falls.Graph 1: The linear fit is placed where the cupcake seems to experience terminal velocity. The slope ofthe linear fit determines terminal velocity. In this sample, the magnitude of terminal velocity is noted as1.75 m/s. Also, a starting height of 1.4 m is visible on the y axis, where the cupcake does not seem to fallyet.Table 2: Cross sectional surface area of each formation of cupcakes. Number of Cross-sect Surface Area Cross-sect Surface Cupcake (cm2) (± 1 cm2) Area (m2) Holders (± 0.0001m2) 1 50 0.0050 3 151 0.0151 4 201 0.0201 7 352 0.0352 13 654 0.0654Table 2: The cupcake holders used have an exposed radius on the bottom of the cupcake of 4 cm. Whenused to attain the surface area that is exposed while free falling, 50.3 cm2 is the result, from theequation . For multiple cupcakes in the formation, the cross sectional surface area for onecupcake is multiplied by however many cupcakes are present.
Table 3: Coefficient of Drag and (Coefficient of Drag)2 for each formation of cupcake holder # Cupcake Holders Coefficient of (Coefficient of Drag)2 in Formation Drag (±0.04) (± 0.04) 1 0.56 0.32 3 0.75 0.57 4 0.82 0.68 7 0.99 0.99 13 1.06 1.11Table 1: Table of values that shows the coefficient of drag for each formation of cupcakes. As shown, thecoefficient of drag increases as cupcake holders are added to the formation. This means that ascupcakes are added, the resistance the formation has to air resistance increases. Formations from oneto seven also show a linear relationship to the coefficient of drag 2, shown in graph 2.Graph 2: Coefficient of Drag 2 over the number of cupcake holders in a formationGraph 2: Final concluding graph that shows the relationship between the number of cupcake holders ina similar formation and their coefficients of drag. The y axis is the coefficient of drag, and it is squared toallow a linear relationship. This shows that the relationship between the coefficient of drag and multipleformations of cupcake holders is a square function. However, this is only applicable to a certain extent,since the formation of 13 cupcakes does not fall in the linear relationship.
Sample Calculations: i. Average Terminal Velocity (Trial 1 velocity + Trial 2 velocity + Trial 3 velocity)/3 (1.75+1.73+1.69)/3 1.72 m/s ii. Uncertainty of Average Terminal Velocity (Highest velocity trial-Lowest velocity trial)/2 (1.75-1.69)/2 0.03 m/s iii. Cross Sectional Surface Area (π*(Diameter/2)2)/1000 (π*42)/1000 (50.3 cm2)/1000 0.0053 m2 iv. Uncertainty of Cross Sectional Surface Area (π((diameter + 0.1)/2)2 – π((diameter-.01)/2)2)/2 (π(8.1/2)2-π(7.9 /2)2)/2 (51.53-49.02)/2 1.255 ± 1 cm2 v. Coefficient of Drag (2gm) / (Aρ(Vt2) (2*9.8*0.00051) / (0.005*1.19*1.722) 0.56 vi. Uncertainty of Coefficient of Drag ((2*g*highest m) / (lowest A*ρ*lowest Vt2)- (2*g*lowest m) / (highest A*ρ*highest Vt2)) / 2 ((2*9.8*0.00052) / (0.0049*1.19*1.692) – (2*9.8*0.0005) / (0.0051*1.19*1.752))/2 (0.611988 – 0.527269) / 2 ± 0.04
Conclusion and Evaluation: The research conducted is aimed to find the relationship between the amount of cupcakes in aformation while free falling and the change in coefficient of drag. By manipulating the data, it is foundthat the relationship is a square function, but supported only by values when the coefficient of drag isless than 1. This means that as cupcakes are added, the coefficient of drag decreases in an exponentialfashion. The data that was attained supports this conclusion with confidence, since the uncertainty onthe derived coefficient of drag is only around 7%.However, the last formation, with 13 cupcakes, doesnot support this conclusion. As a result, it is not included in the final graph, and shows that theapplicability of this relationship only lasts for a small range of cupcake holders. The coefficient of dragattained for the cupcake formation of seven cupcakes was 0.99. This figure, when squared, is also equalto 0.99. This shows that the range of applicability of this research ends at a formation of seven cupcakes,following the procedure carried out. When the coefficient of drag exceeds 1, the square function doesnot apply, as shown with the formation of 13 cupcakes, where the coefficient of drag is 1.06. When thecoefficient of drag exceeds 1, the square does not decrease the coefficient of drag, but rather increasesit, and it does not lie on the linear fit of Graph 2 anymore. While this conclusion holds true for this case,different types of cupcake holders and formations must be investigated to further this concept. There were many weaknesses in this investigation. As the cupcakes were handled and droppedover and over again, the structure deteriorated, and it was ultimately not controlled. Taping the cupcakeholders together was also a problem, as it added additional weight to the formation, and changed theshape of the cupcake holder. This affected the mass to surface area ratio, and skewed the data, becauseit was assumed that mass was proportional to cross sectional surface area. In addition, every cupcakeused was different. This weakness was not controlled, and cannot be controlled very well unless thecupcakes are made of a different material that will allow the structure not to deteriorate while beinghandled and attached together. Special cupcakes must be used where external sources are not used toattach them together. The mass to cross sectional surface must stay constant throughout furtherresearch for data to be reliable. ρ, or the density of the air, was a weakness in this investigation. A valueof 1.19 was used, but in reality, the temperature of the room in which the investigation was carried outfluctuated, since it took nearly an hour to compile all the data. Along with air temperature was the issueof wind or turbulence within the room. Slower moving cupcake formations had prolonged exposure tothe air in the room, which could have had an impact. Controlling this in future research may includedropping the formations down a tube, where the temperature and wind or turbulence is controlled.Further research may be carried out, varying the types of cupcake holders used, and the ways theformations are put together. This investigation only tested one type of cupcake holder, and insymmetrical formations. These variables can be changed, and observed to test whether they agree withthe conclusion from this investigation.