Cd = drag co-efficientp = fluid density (air)V = velocityA = surface area affected

Figure 1 - Two forces acting on a sphere falling vertically from rest

Calculating the Co-efficient of Air Resistance in Blu-tack

Blu-Tack is a substance engineered to partially attach two surfaces together, one of which often being paper or card; to allow this to happen the substance needs to be tough, adhesive and effective. If Blu-Tack were to be thrown or dropped through air, its physical properties would be factors in the way it travels. The magnitude of air resistance acting on Blu-Tack is largely determined by its physical composition, amongst other factors. The extent to which a substance is affected by air-resistance/drag can determine its rate of acceleration, terminal velocity and in some cases the path of flight.

In order to make calculations to analyse the effects of air resistance on Blu-Tack, and hence find a value for its ‘co-efficient of drag’, I have observed its flight only through free-fall. By doing this, the substance experienced only one form of acceleration: vertical acceleration due to gravity. Consequently there were no horizontal factors affecting the motion. For an object falling vertically through the atmosphere, it is subjected to two external forces: gravitational force and aerodynamic drag. This is shown in Figure 1:

Gravitational force is expressed as weight ‘W’, which is defined by the weight equation as: mass ‘m’ x gravitational acceleration ‘g’. The value for gravitational acceleration is 9.806654 m/s² (to 6 d.p) at the altitude of the Physics laboratory; close to the approximate surface of the earth. In actual fact, gravitational acceleration decreases with the squared distance from the centre of the earth, but for most practical problems we can assume that this factor is constant.

Aerodynamic drag is present during free-fall and affects any sort of motion within the atmosphere. This force opposes the gravitation force/weight and is expressed by the drag equation:

The aim of this investigation was to calculate an approximate value for the co-efficient of drag within the material Blu-Tack.

Drag / Air-resistance

Gravitation Force / Weight

Mass

Throughout this experiment I tested the Blu-Tack as a spherical object; this simplified my calculations. Since I could not ensure that the Blu-Tack was entirely spherical, this will have caused negligible invalidity in my experiment.

Cd = 2D / (V²A)

In order to have found a value for the co-efficient of drag, I re-arranged this equation to attain the following:

This equation became the basis of my experiment; this is what I was ultimately trying to find. Therefore in order to conduct valid tests to calculate this co-efficient, I would need to have a range of corresponding values for the force of air resistance, density of air, velocity of the object and the surface area of the Blu-Tack affected by the air; my idea was as follows.

When an object is held stationary at a position above the ground, it has gravitational potential energy (G.P.E). When it is released this energy is conserved, it is converted mostly into kinetic energy (K.E) and falls towards the ground through free-fall. However, some of this energy is lost; the only significant factor to consider in this case is air resistance. I therefore made the assumption that:

E grav = E kinetic + E lost E lost = E grav – E kinetic

Energy lost through Air Resistance = E grav (G.P.E) – E kinetic (K.E)

In conclusion, if I could calculate the kinetic energy of a piece of Blu-Tack, having fallen over a measured distance, I could subtract this value from its change in gravitational potential energy. This would essentially leave me with an approximate value for the energy lost due to air resistance (for that length of free-fall). I used this idea and applied it to my experiment so that it was conducted over a range of lengths travelled by free-fall.

This practical experiment involved the repeated task of dropping a piece of Blu-Tack from various heights. This abided by the health and safety regulations that were set, and involved no hazard to me or other others around me in the physics laboratory. I remained aware of where the Blu-tack was falling so that its landing was safe and remained controlled.

Finding values for G.P.E can was the first step of my experiment:

G.P.E (J) = Mass (kg) x Gravitational Acc’ (m/s²) x Change in Height (m)

Using a fixed range of heights, I attained a set of values for the change in G.P.E. To do this I needed to know the mass of the spherical Blu-Tack that was to be used throughout the experiment. Using precise digital scales, the mass of this ball was measured at 0.01155 kg.

Since the digital scales gave a value for mass to five decimal places, the uncertainty of the mass of blu-tack was ±0.000005 kg. This gave a percentage uncertainty of 0.043% for the 0.01155 kg Blu-Tack. I measured the height intervals using a ruler with a 1mm resolution (0.001m), this gave an uncertainty of ± 0.0005m. This was calculated as a percentage uncertainty for each height interval and was added to the percentage uncertainty of the mass; I then converted this overall uncertainty back into the unit uncertainty for G.P.E. Figure 2 shows the table of data I attained:

Experiment 1

Experiment 2

I kept the potential energy values to 6 decimal places in order to keep as much accuracy as possible when demonstrating my calculations.

Throughout this experiment I repeated each step twice, as shown in Figure 2. These two experiments are named Experiment 1 and Experiment 2. The reason for repeating this experiment was to attain a wider range of values to represent my investigation, making my results and conclusions more reliable.

The next part of my investigation was to create a third column of data, this column would be to state the kinetic energy of the Blu-Tack after free-falling (from rest) over the varying range of distances.

Figure 2 - Values for Gravitation Potential Energy including uncertainties for the height interval and mass

In order to work out the kinetic energy of a moving object, I used the formula:

K.E (J) = 0.5 x Mass (kg) x Velocity² (m²/s²)

Using the same piece of Blu-Tack (with the same mass and spherical shape), I essentially needed to conduct an experiment where I could measure its velocity at a particular point during free-fall. By doing this, I could simply release the Blu-Tack from the given range of heights, and attain a corresponding set of readings/velocities.

To measure the velocities I set up a simple arrangement of equipment, this involved fixing an electronic light gate to a stand, using a clamp and a boss. The light gates measured the velocity of a moving object to six decimal places, which meant that the uncertainty per reading was 0.0000005 m/s. For each height interval, I decided to take five readings for the velocity; I then found an average for these velocities. Figure 3 shows my results:

The values attained for these velocities often varied to different extents, therefore different uncertainties will have occurred in the average velocity values that I calculated and used later in the experiment. Because the uncertainty within the average velocity was substantially larger than the uncertainty due to precision of instruments, it was more reliable to calculate uncertainties using the range of velocities.

For each height interval I found two average velocities, from experiment 1 and experiment 2, which each have an uncertainty. The average uncertainty of these values comes from the original readings and can be found as follows:

Maximum reading – minimum reading2

Using the table of raw data taken from the experiment, I can take the minimum and maximum velocity values for each height interval and calculate the uncertainty. This uncertainty will then apply to the average velocities previously calculated.

Figure 3 - Velocity readings including uncertainties for the average values

These uncertainties can be represented on a graph, where average velocity is plotted against the distance of free-fall:

Here are two graphs to show my average measurements of the velocities of Blu-tack in free-fall, over varying distances. Included in these graphs are the uncertainties previously calculated for each set of readings.

Using this range of average velocities, I used the K.E equation to convert this into a range of energy values corresponding to each height interval. To calculate the uncertainties, I converted each uncertainty into a percentage and then added them according to the equation as follows:

±% (Kinetic Energy) = ±% (Mass) + ±% (Velocity) + ±% (Velocity)

In the next stage I calculated a fourth column of data that stated the average energy lost over each period of free-fall. As mentioned previously, this can be calculated by subtracting values of K.E from the change in G.P.E as shown in Figure 5.

The uncertainties for the energy loss were calculated by adding the unit uncertainties, as follows:

±J (Energy Lost due to air resistance) = ±J (G.P.E) + ±J (K.E)

However, to apply this data to the equation for the drag co-efficient, the values for ‘energy loss’ needed to be in the form of forces acting on the Blu-Tack. A force is essentially the ‘work done’ or energy on an object per unit of distance; I therefore found approximate values for the force of air-resistance acting on the Blu-Tack during each free-fall; this was done by dividing the value for ‘energy loss’ by its corresponding height. Here, I attained my fifth column of data shown in Figure 6.

Figure 4 - Calculated results for Kinetic Energy, including uncertainties

Figure 5 - Values for average energy lost due to air resistance, through each height interval, including uncertainties

Cd = 2D / (V²A)

The final stage of this experiment was to use this data to find an approximate value for the co-efficient of drag.

‘D’ (the force of air resistance) - This is the corresponding value in the fifth column of my data table.

‘ρ’ (the density of surrounding fluid) – This value is dependent mainly on the temperature of the room. I measured the room temp at 20 degrees, which gives an approximate density value of 1.2041 kgm−3 (value taken from wiki article – Density of Air); this value was constant for all measurements.

‘V’ (average velocity) – This value is unique for each height interval, therefore using the previous table of velocities, I took the average corresponding values for each fall.

‘A’ (the surface area affected by air resistance) – This value is half the surface area of the Blue-Tack ball, as only half of the ball is making contact with the oncoming air. This value was also constant. I measured the diameter of the ball as 2.2cm (0.022m), this equates to a spherical surface area, which is halved to attain a value of 0.0029044 m².

The diameter of the Blu-Tack was measured using a ruler with resolution of 1mm (0.001m). The diameter was therefore 0.022 ± 0.0005m (percentage uncertainty of ± 2.3%). For the surface area of the sphere (4ηr2) this percentage uncertainty becomes ± 4.5%.

Since the values for ‘Cd’, ‘ρ’ and ‘A’ are constant, the formula can be rearranged as follows:

K x Cd = D/V2

Where K = ρA/2

Therefore by plotting a graph of D against V2, there should be a linear relationship. Due to the uncertainties that have persisted throughout this experiment, the linear relationship is likely to be approximate and can be best represented by a line of best fit, whose gradient is equal to ‘K x Cd’. This is shown in Figure 7 (experiment 1) and Figure 8 (experiment 2):

Figure 6 - Average force of air resistance for each height interval, including uncertainties

Force of Air Resistance / D (Newtons)

(Average Velocity)2 / V2 (m2/s2)

(Average Velocity)2 / V2 (m2/s2)

Force of Air Resistance / D (Newtons)

Figure 7 - Experiment 1: A graph of D against V2, including the line of best fit.

3 4 5 6 7 8 9 10 11 120.000

0.005

0.010

0.015

0.020

0.025

3 4 5 6 7 8 9 10 11 120.000

0.005

0.010

0.015

0.020

0.025

Figure 8 - Experiment 2: A graph of D against V2, including the line of best fit.

The average uncertainty for the gradient of the line of best fit can be calculated as shown in Figure 9. These percentage uncertainties were calculated from the previous data shown in figures 1 – 8.

Average ±% for D Average ±% for V ±% for V2 ±% for D/V2

Experiment 1 19.422% 9.318% 18.636% 38.058%Experiment 2 16.370% 9.716% 19.433% 35.803%

Figure 9 - Percentage uncertainties for the gradient of line of best fit

Using the data from experiment 1, the gradient of the line of best fit (D/V2) was 0.001917 (± 38.058%). Re-arranging the formula and inputting the measured values for ρ and A, my calculated value for the co-efficient of drag was 1.096441.

Using the data from experiment 2, the gradient of the line of best fit (D/V2) was 0.001958 (± 35.803%). Re-arranging the formula and inputting the measured values for ρ and A, my calculated value for the co-efficient of drag was 1.120772.

The percentage uncertainties for these co-efficient values are as follows in Figure 10:

±% for D/V2 ±% for ρ ±% for A ±% for CdExperiment 1 38.058% - 4.5% 42.558%Experiment 2 35.803% - 4.5% 40.303%

Figure 10 - Calculating percentage uncertainties for the co-efficient of drag

My results for the co-efficient of drag were therefore as follows:

Experiment 1 – 1.096 ± 0.466 Experiment 2 – 1.121 ± 0.452

Throughout this experiment, there were many forms of uncertainty, which persisted throughout my measurements and calculations. The uncertainties due to the limited precision of the apparatus used were accounted for in my results and calculations, as shown in figures 1 – 10. These uncertainties were also accounted for in my overall results. These uncertainties were dealt with by representing them when stating all of my calculations, but due to the limited precision of the equipment used, it was difficult in the school environment to reduce these uncertainties and factors affecting my results. In terms of the procedure of my experiment, I could not have reduced the uncertainties by choosing an alternative method. The readings taken from the electronic light gates were given to 6 decimal places; this was the most precise method of measuring the velocity of a falling object.

However, natural uncertainties occurring in the environment around me were not necessarily accounted for due to the high complexity of calculating them. When finding the density of air, I measured the room using a digital thermometer (reading to 1 decimal place). However, the uncertainty behind this single decimal place could not be accounted for when stating the air density. The density of air could involve many uncertainties that could derive from either or all of the following:

Air pressure Altitude Temperature of Air Humidity Potential drafts within the environment

In order to deal with this unavoidable uncertainty, I decided to crosscheck the values for the density of air across two sources, this was to ensure that the value was as reliable and accurate as possible; Wikipedia Article: Density of Air and HowStuffWorks article: Air Pressure. I consequently displayed no uncertainty for air density when evaluating my results; this was the greatest limitation of my experiment and is something that could be improved using more advanced and able equipment.

Although attaining two similar results to fairly high precision, my results were not conclusive due to the unknown uncertainty in the environment in which the Blu-Tack was dropped and recorded. Therefore to extend and progress in my investigation, I decided to carry out the similar experiment in different fluid, water.

The progression on my experiment was influenced by a previous investigation

carried out by engineers and researchers of Em-solutions. Here, they tested different shapes within different fluids in order to prove that co-efficient of drag was in fact a constant.

Following Source: http://www.engineeringtoolbox.com/drag-coefficient-d_627.html.

Engineers behind the organization ‘em-solutions’ investigated the drag co-efficient of various objects of varying shapes and sizes. When publishing their findings, they quoted that “The drag coefficient expresses the drag of an object in a moving fluid. Any object moving through a fluid experiences drag - the net force in the direction of flow due to pressure and shear stress forces on the surface of the object.”

The drag coefficient is a function of several parameters like shape of the body, Reynolds Number for the flow, Froude number, Mach Number and Roughness of the Surface. The characteristic frontal area - A - depends on the body.

Objects’ drag coefficients are mostly results of experiments. Em-solutions applied the experiment in two fluids, air and water. Drag coefficients for some common bodies are indicated below in Figure 11:

The formula and calculations behind progressing my experiment were similar to the first method. The co-efficient of drag is essentially the same in all fluids, as stated by

Figure 11 - Drag co-efficient of a rang of objects

em-solutions when composing their findings, hence the density of fluid being taken into account in the equation.

The idea behind this method was to drop an almost-spherical piece of Blu-Tack down a clear, glass measuring cylinder (containing water) from varying heights. The cylinder had to be clear so that the light gates could pick up readings through the water and glass. The measurements and calculations were the same from this point onwards; essentially the only difference in this experiment was the fluid in which the Blu-tack fell through, which consequently altered the value for fluid density (ρ) when calculating the co-efficient.

This experiment involved refilling a cylinder after each fall; it was therefore important that I ensured the environment around me was safe after transporting water. I kept a towel near to the area at all times. This was to avoid others or me in the laboratory from the risk of an accident. Apart from this hazard, my experiment was safe and abided by health and safety regulations.

In this experiment the mass of Blu-Tack was 0.00699kg, which gave a percentage uncertainty of ± 0.072%. Using the same ruler as before, the measurements for height had the same uncertainty of ± 0.0005m. The values that I calculated for G.P.E, including uncertainties were as follows in Figure 12:

To calculate kinetic energy, I used the same previous method when measuring the velocity of the Blu-Tack falling through the water; my results are shown in Figure 13 along with the average velocity uncertainty:

Like before, I converted these velocities into values for kinetic energy. The uncertainties were also calculated in the same way, as shown in Figure 14:

Figure 12 - Values for G.P.E including uncertainties

Figure 13 - Velocity of the Blu-tack after varying heights of free-fall, including average velocity and uncertainties

Cd = 2D / (V²A)

Proceeding with the experiment using the same methods previously used, I attained a set of values for the force due to air-resistance during each period of free fall. My results were as follows, including uncertainties:

I could then re-apply these values to the equation to find the co-efficient of drag:

‘D’ (the force of air resistance corresponding to a particular height interval) ‘ρ’ (the density of surrounding fluid) – The density of water is 1000 kg/m3. ‘V’ (average velocity corresponding to a particular height interval) ‘A’ (the surface area affected by air resistance) – This value is again constant.

I measured the diameter of this ball as 1.8cm (0.018m), this equates to a spherical surface area, which is halved to attain a value of 0.000537605 m².

The diameter of the Blu-Tack was measured using a ruler with resolution of 1mm (0.001m). The diameter was therefore 0.018 ± 0.0005m (percentage uncertainty of ± 2.8%). For the surface area of the sphere (4ηr2) this percentage uncertainty becomes ± 5.6%.

Again, putting this data into a graph of Force against Velocity2 (D/V2), there should be a linear relationship due to the other values of the equation being constant, shown in Figure 16:

Figure 14 - Values for kinetic energy, including uncertainties

0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.240.05

0.06

0.07

0.08

0.09

Figure 15 - Forces of air resistance, including uncertainties

Force of Air Resistance / D (Newtons)

(Average Velocity)2 / V2 (m2/s2)

Figure 16 - A graph showing force against the square velocity, including line of best fit

Using the data from this progressed experiment, the gradient of the line of best fit (D/V2) was 0.3620441 (± 27.681%). The uncertainty was calculated using the same method as the initial experiment. Re-arranging the formula and inputting the measured values for ρ and A, my calculated value for the co-efficient of drag was 1.34687.

The percentage uncertainties for these co-efficient values are as follows in Figure 17:

±% for D/V2 ±% for ρ ±% for A ±% for CdWater experiment 27.681% - 5.6% 33.281%

Figure 17 - Calculating percentage uncertainties for the co-efficient of drag

My results for the co-efficient of drag were therefore as follows:

Experiment with water – 1.347 ± 0.448

In conclusion, my results were as follows:

Experiment 1 – 1.096 ± 0.466 Experiment 2 – 1.121 ± 0.452

0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.240.05

0.06

0.07

0.08

0.09

Experiment 3 (using water) – 1.347 ± 0.448

Firstly, the consistency of my uncertainty values (33.2% - 42.5%) suggests that the uncertainty errors in my investigation were consistently down to the limitations in precision of apparatus; human error appeared negligible overall when carrying out practical measurements.

Co-efficient of drag should be a constant value for any one material, whichever substance or fluid it falls through. This is portrayed in my results, showing a fairly consistent value between air and water. The co-efficient of the Blu-tack’s drag through water has a slightly larger value that it does concerning air; however, when testing the Blu-tack through water, its shape became a more noticable factor in the direction of its movement. Although the ball was shaped as spherical as possible, the fall of the Blu-tack in water was not exactly vertical, it also moved slightly in the horizontal direction. Essentially this makes the length and distance of the fall greater, which can consequently increase the overall force of air resistance acting on the substance over the set distance; hence a greater co-efficient of drag,

Experiment 1:

Average Velocity (m/s)

Average Velocity (m/s)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Experiment 2:

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Change in height (m)

Change in height (m)