25 March 2015: Centripetal acceleration vs frequency

Purpose:

The purpose of this experiment was to measure different accelerations of a rotating disk and the corresponding  time it takes for one revolution (known as the period) to determine the relationship between centripetal acceleration and angular speed.

Apparatus:

                                                     Figure 1: Diagram of apparatus

The apparatus consisted of a heavy disk that is able to rotate about its center. A scooter wheel and a transformer were also used to cause the disk to rotate and give it centripetal acceleration at varying levels, depending on the amount of voltage run through the scooter. A wireless accelerometer was also placed on top of the disk to measure the centripetal acceleration of the rotating disk. In addition, tape and a photogate were used  to measure and record the time it takes for one rotation to happen. Tape was also used to hold down the accelerometer on the disk. Lastly, a computer with LoggerPro software was needed to connect the photogate and accelerometer to. 

Procedure:

The disk was rotated at different speeds using different voltages run through the scooter. Then, the centripetal acceleration was measured on LoggerPro every trial using the accelerometer. As expected, the higher the voltage run through the scooter wheel, the larger the centripetal acceleration would be. The number of rotations performed over a time interval was also measured. These measurements were made for five trials, with each trial increasing the voltage run through the scooter wheel and therefore increasing the centripetal acceleration of the rotating disk. The voltages used for each trial are listed below:

1. Trial 1: 4.4 V
2. Trial 2: 6.4 V
3. Trial 3: 8.6 V
4. Trial 4: 9.6 V
5. Trial 5: 10.8 V

The acceleration for Trial 1 was measured to be 1.557 m/s/s. The acceleration of trial 2 was 5.074 m/s/s. The acceleration of Trial 3 was 10.70 m/s/s. The acceleration of Trial 4 was 11.89 m/s/s. The acceleration of Trial 5 was 18.15 m/s/s. 

The radius of the rotating disk was then measured to be 13.8 cm. The radius is used to find the relationship between angular speed and acceleration.

Then, Excel was used to create a graph of acceleration vs angular velocity squared.

Discussion:

In order to find the angular velocity of the rotating disk, the period must be found first, since w = (2pi)/T, where w is angular velocity and T is the period, the time it takes for one revolution. The period was calculated by taking the time it took for "n" amount of revolutions to be done, then dividing that time by "n", which gives the time it takes for one revolution. For Trial 1, n = 8 revolutions. For the remaining trials, n = 10 revolutions. Once the period for each trial was found, the angular speed (w) was found by dividing 2pi by the period. Next, "w" was squared for each trial, and the graph of acceleration vs angular velocity squared was created. 

Voltage (V)	To (s)	Tf (s)	Period (s)	w^2 (rad^2/s/s)	a (m/s/s)
4.4	1.6716	16.461	1.848675	11.5515047	1.557
6.4	0.1789	10.5305	1.03516	36.84213	5.074
8.6	0.3783	7.57	0.71917	76.3302297	10.7
9.6	0.0534	6.7843	0.67309	87.1391697	11.89
10.8	0.377	5.865	0.5488	131.078508	18.15

Figure 2: Chart containing the period, angular speed squared, and acceleration for each trial

    Figure 3: Graph of centripetal acceleration vs angular velocity squared

Knowing that acceleration is equal to the product of the radius of the disk and the squared angular velocity, the slope of the graph above provides the radius of the disk, as shown below:

a = r w^2

y = m x

As found, the slope of the graph (experimental finding of radius) is 0.1383 m, or 13.83 cm. This is very close to the measured radius value of 13.8 m, which shows that this model was indeed very accurate for finding the relationship between centripetal acceleration and angular velocity. 

Conclusion:

One source of uncertainty in this experiment is the uncertainty in the measurement of the acceleration by the accelerometer and the measurement of the period by the photogate. However, these uncertainties are very small since, as shown, the experimental radius is very similar to the measured radius. Another source of uncertainty is the measurement of the radius with a ruler. This would have increased or decreased the percent difference, depending on whether the value was rounded down or up. 

Most sources of error, such as neglecting friction between the axle of rotation and the disk, assuming all power goes into rotating the disk, etc., were eradicated in this experiment since almost everything was measured via equipment and many assumptions that contain error, such as all power goes into rotating the disk, were not used in the data analysis and in the model since they have no effect on the relationship between centripetal acceleration and angular velocity.

23 March 2015: Trajectories

Purpose:

The purpose of this experiment was to experimentally investigate the properties of projectile motion and the independence of horizontal and vertical motion. Using the knowledge of projectile motion, a prediction was made for the impact location of a freely falling ball on an incline.

Apparatus:

                               Figure 1: The apparatus used in the experiment

The apparatus used in this experiment consisted of two aluminum v-channels, one sloped and one horizontal, as shown in Figure 1 above. The use of this equipment was to give a steel sphere momentum while only giving it motion in the horizontal direction before the ball gets into free fall. Blocks and a metal stand were used to hold up the slide, and tape was used to hold the pieces together as well as to mark the origin of the steel ball's location upon sliding. A plumb bomb, which consists of a weight tied to a string, was also used to determine the exact point on the ground the steel ball begins to fall freely from the apparatus.

                                Figure 2: Apparatus used in the second part of the experiment

As shown in Figure 2, a wood plank was used as the slope required for the second part of the experiment. In order to mark the place the ball came into contact with the slope (and the floor for the first part of the experiment), a carbon copy page was used, as shown on the plank above. A meter stick was also used to measure distances in both part 1 and part 2 of the experiment. Lastly, a magnetic protractor was used to measure the angle of the inclined board.

Procedure:

Once the apparatus was set up, the steel ball was launched off from the apparatus at the marked point on the inclined v-channel, taking into consideration its impact point on the floor. Once the impact point was found, a carbon copy page was taped onto the floor at the location of the impact point. Then, the steel ball was launched off five more times at the same marked point each time, taking into account the fact that the ball should land in nearly the same place each time. The height of the apparatus and the distance the ball had traveled in the horizontal direction before the impact were measured.

                           Figure 3: Measuring the horizontal distance the ball reached on impact

The horizontal distance was measured five times, once for each trial. Then, the average distance was calculated, with the uncertainty as the range of the measured distances from the average. The average distance the ball traveled was found to be 50.50 ± 0.30 cm. The height the ball traveled from the apparatus to the floor was measured to be 94.20 ± 0.10 cm. Using these dimensions, the initial velocity in the horizontal direction was measured.

                               Figure 4: Dimensions measured for the free fall of the steel ball

             Figure 5: Calculations used to solve for the initial velocity in the horizontal direction

The initial velocity in the x direction was calculated to be 115.2 cm/s. Obviously, the initial velocity in the y direction is zero.

In the second part of the experiment, an incline was used and it was predicted where the ball would land on the incline. The angle of the incline was measured to be 49º ± 2º. The distance from the end of the v-channel to the end of the board on the ground was measured to be 79.70 cm.

                               Figure 6: Dimensions of the second part of the experiment

Using the known dimensions and components of projectile motion (acceleration in y direction, velocity in x direction, etc.), an expression for the distance d the ball will travel along the board before impact was derived. The expression is shown below:

                               Figure 7: Expression for "d" as a function of Vo and alpha

With this expression, the theoretical distance the ball would travel relative to the board before impact was calculated and found to be 47.47 cm.

                   Figure 8: calculating the theoretical distance along the board the ball will travel

With the predicted distance in mind, the carbon copy page was pasted onto the board in the area the ball was expected to land. Then, with the apparatus set up as shown in Figure 2, the ball was launched off the v-channels in the same position as in part 1, and the experiment was run five times.Then, the distances the ball traveled along the board before impact were measured for each trial, and like in part 1, the average was calculated with the uncertainty as the range of the impact points from the average.

                       Figure 9: Measuring the distances "d" the ball traveled in the experiment runs

The average distance d the ball traveled was 48.02 ± 0.52 cm. That is a 1.15% difference from the predicted value, which proves that the prediction was very accurate and our model a good representation of projectile motion.

Discussion:

                                Figure 10: Uncertainty in "d"

Since the equipment used in this experiment contains uncertainty, it follows that our predicted value for d also contains uncertainty. To calculate the uncertainty, the expression of d must be changed into the variables that are the source of uncertainty. In this case, those variables are "x", "y", and alpha. To get d to be a function of x, y, and alpha, the expression for Vo (shown in Figure 10 above) must be substituted into the expression for d, as shown below:

                      Figure 11: Transforming the expression for "d" into a function of x, y and alpha

With this expression, the partial derivative with respect to x, y and alpha can be taken and the uncertainty can be solved for using the formula of "dd" shown in figure 10 above.

                                Figure 12: Deriving "dd"

As shown in figure 12, the partial derivatives with respect to x, y and alpha were derived, then multiplied by dx, dy and dalpha, respectively, and summed up together.Plugging in the values for x (79.70 cm), y (94.20 cm), alpha (49º), dx (0.30 cm), dy (0.10 cm), and dalpha (0.035 rad), it was found that the uncertainty in d (dd) is:
                                         dd = 0.564 - 0.0504 + 5.252 = 5.766
Therefore, the actual theoretical value for d is 47.47 cm ± 5.766 cm. Fortunately, the experimental value is in that range and quite close to the theoretical value for d.

Conclusion:

The sources of uncertainty in this experiment are from the equipment used for measurements. This includes the meter stick and the magnetic protractor. In addition, the range of the values in the five trials is also a source of uncertainty in the experiment. The uncertainty in d is fairly large (5.766; 12.1% of the value of  d) only because of the uncertainty in alpha. As shown in the calculation for dd above, most of its uncertainty comes from the uncertainty in dalpha. This is because the uncertainty of 2º (4.1% of the value of alpha = 49º) in the measurement of alpha is very large compared to the uncertainty in the measurement of other values. This uncertainty comes from the weak accuracy in the magnetic protractor. In order to lower the uncertainty, a more accurate protractor with a much smaller uncertainty must be used. Other than the use of the inaccurate protractor, the uncertainty was small and the accuracy of the experimental to the predicted value of d was strong.

The sources of error in this experiment include the ball not being let go from the same exact position each trial, resulting in a change of the initial horizontal velocity and therefore the horizontal distance the ball travels. Another source of error is the measurement of the initial position the ball begins to free fall relative to the ground. This results in either a smaller or large horizontal distance the ball travels, changing the calculated initial velocity of the ball. Another source of error is air resistance. This error affected the experiment only in measuring the initial velocity of the ball in the horizontal direction. Since air resistance was not taken into account, the calculated velocity is probably smaller than the actual velocity since air resistance causes the projectile to travel a smaller horizontal distance. Because the calculated initial velocity was used in the second part of the experiment, it resulted in a theoretical value for d smaller than the actual, which was seen in the experiment. However, the error is not very large since the projectile has a small attacking area (area of contact with the air as it travels through the medium) and it was not in free fall for a very long time either.

18 March 2015: Modeling Friction Forces

Purpose:

The purpose of this experiment was to explore the properties of static and kinetic friction and their effects on two systems in contact. Their application to different situations as well as their diverse dependence on different situations was investigated. Lastly, friction was used to predict the behavior and change in motion of an object due to external forces.

Apparatus:

                               Figure 1: Apparatus for Part 1 of the experiment

This experiment consisted of five parts. The first part of the experiment involved finding the coefficient of static friction by hanging a mass on the system that barely starts it to move. To do so, the apparatus consisted of five blocks, with one block containing red felt on the bottom side. The apparatus also consisted of a styrofoam cup that was used as the hanging mass; its weight was changed by adding and removing water. A balance was also needed to measure the mass of the blocks and the styrofoam cup. Lastly, string and a pulley were needed to allow hanging of the styrofoam cup.

                             Figure 2: Apparatus for Part 2 of the experiment

The second part of the experiment involved finding the coefficient of kinetic friction by measuring the pull force required to keep the object in constant motion. A computer with LoggerPro was required in order to measure the value of the pull force over an interval of time. A force sensor was also needed to allow the computer to measure the force of the pull on the system. A 500 gram mass was also used to calibrate the sensor by hanging it and getting a reading close to 4.9 N by the force sensor. Four different blocks were also needed, with one block containing the red felt on the bottom side. Lastly, a balance was needed to measure the mass of the blocks.

                                Figure 3: Apparatus for Part 3 of the experiment

The third part of the experiment involved finding the coefficient of static friction on a sloped surface by finding the maximum static friction on the system, which was done by creating a slope that causes the object to just begin sliding. The equipment needed for part 3 included a metal slide or plank, a magnetic protractor to measure the angle of the metal surface and a vertical stand with a bar perpendicular to it to hold the metal surface raised up. A block with red felt on its bottom side was also needed. Lastly, a balance was needed to measure the mass of the block.

                                           Figure 4: Apparatus for part 4 of the experiment

The fourth part of the experiment involved finding the coefficient of kinetic friction of a sliding block on an angled surface by measuring the acceleration of the block.The equipment needed for this part of the experiment included a motion detector to find the acceleration of the block, a metal slide, a block with red felt on its bottom and a magnetic protractor to measure the angle of the metal surface. A computer with LoggerPro software was also needed to record the motion of the block from data measured by the motion detector. Lastly, some sticky noted were needed to paste onto the block so that the motion detector is able to notice the block's presence.

                               Figure 5: Apparatus for part 5 of the experiment

Part 5 of the experiment involved predicting the acceleration of a block using the coefficient of kinetic friction calculated from the fourth part of the experiment, then comparing the expected acceleration with the acceleration recorded from performing the experiment with equal measurements and situations (i.e same masses, same situation with a hanging mass, etc). The equipment needed for this experiment included a pulley, a metal slide, a styrofoam cup containing 50 g and 10 g weights whose function was to act as the hanging mass, string, and a motion detector to record the motion of the system. Lastly, a block with red felt on its bottom side was used as the system, and sticky notes were pasted onto the block to allow the motion detector to detect the presence of the block and its change in motion. 

Procedure (Part 1):

Once the apparatus was set up, the mass of the block with red felt on its bottom side was measured. Then, water was slowly added to the hanging cup until the block just barely started to slide.Once the block began to just slide, the mass of the cup and water were measured and recorded. 

Next, an additional block was obtained and its mass was recorded. Then, it was added on top of the first block with everything in the same position as before. Again, more water was added until the blocks began to just move. The mass of the cup and water was then recorded again. The same procedure was performed again for three and four blocks.

With the masses of the blocks and the masses of the cup with differing amount of water, a graph of maximum static friction force vs. normal force was created. 

Discussion (Part 1):

Comparing the formulas of static friction and the equation of a line, :

1.

it can be seen that the force of static friction is the value for y, the normal force is the value for x, and the coefficient of static friction is the value for m, or the slope of the graph. 

The normal force was calculated by finding the weight of the block(s), since the normal force is equal to the weight of the blocks. The force of static friction was found by finding the tension in the string, which is equal to the weight of the hanging mass. The free body diagrams that display the method for finding the normal force and static friction force is seen below:

                          Figure 6: Free body diagrams and equations for part 1 of the experiment

Using this method, the data table of normal force and force of static friction was constructed, seen below:

                                     Figure 7: Data table of normal force and static friction

Using the data from the table above, the graph of static friction force vs normal force was constructed:

                              Figure 8: Graph of  static friction force vs normal force

As explained earlier, the slope of the above graph is the coefficient of static friction between the red felt and the table top. The coefficient of static friction is:

2.

There is a very small (0.9%) uncertainty in the coefficient of static friction, which shows that the modeling of the static friction force was accurately portrayed.

Procedure (Part 2):

Once all of the equipment needed was obtained, the force sensor was connected to the computer containing LoggerPro. The force sensor was then calibrated using a 500 gram hanging mass. The force recorded during the calibration should have been around 4.9 N. Once the force sensor was calibrated, it was placed horizontally onto the table and with no forces other than its weight and normal force, it was zeroed. Next the mass of the block with red felt was obtained, and the block was tied to the force sensor using a string. Then, the force sensor and block were pulled such that they were in constant motion. The force of tension in the string measured by the probe was collected by LoggerPro over an interval in time. The average tension force was then obtained. 

                                  Figure 9: Average tension forces (mean) of the four runs

Then, the mass of a second block was recorded and it was place on top of the first block. Next, the blocks and force sensor were pulled at constant speed and the force of tension was collected. The average tension force was then obtained. This procedure was repeated with three blocks and four blocks, as in part 1.

Using the masses of the blocks each trial and the average tension force, a graph of maximum kinetic friction force vs. normal force was created.

Discussion (Part 2):

Similar to part 1, comparing the equations of a line and kinetic friction,:

3.

it can be seen that the force of kinetic friction is the value for y, the normal force is the value for x, and the coefficient of kinetic friction is the value for m, or the slope of the graph. 

Using free body diagrams (shown below), it can be seen that the normal force is equal to the weight of the blocks. The force of kinetic friction can also be found because it is equal to the tension force, or the average force recorded by the sensor.

                              Figure 10: Free body diagrams and equations for part 2 of the experiment

Using this method, the graph of kinetic friction force vs normal force was made, and the slope was obtained. Using the same options in LoggerPro as in part 1, the data table and graphs were made:

                                        Figure 11: Data table of normal force and kinetic friction

                                  Figure 12: Graph of  kinetic friction force vs normal force

The coefficient of kinetic friction was calculated to be:

4.

The uncertainty in the coefficient of kinetic friction (3.8%) is fairly small, which shows that the model of the kinetic friction force was accurate.

Procedure (Part 3):

Once all of the equipment was obtained, the metal slide's angle was adjusted until the block with the red felt just began to slide. The angle was then measured and recorded and the mass of the block was obtained.

Discussion (Part 3):

Using the mass of the block and the angle of the surface, the coefficient of static friction between the felt and metal slide can be found through a free body diagram, summing the horizontal and vertical components, with acceleration in all directions equal to zero. The free body diagram is shown below:

                      Figure 13: Free body diagrams and equations for part 3 of the experiment

The coefficient of static friction was calculated to be 0.344. It makes great sense that the coefficient of static friction on the slope is larger than on the level ground since more friction is required to keep the system at equilibrium due to some of its weight wanting to push it down the slope. In addition, the metal slides are rougher than the table tops.

Procedure (Part 4):

Once all the equipment was set up, an angle higher than 19 degrees was needed in order for the block to accelerate downward. The angle was chosen at 30 degrees. Using the motion sensor, the acceleration of the block with red felt down the slope was measured by creating a velocity vs time graph , then taking a linear fit to find the slope of the graph. The mass of the block was then measured.

Discussion (Part 4):

The graph of velocity vs time of the block down the slope is shown below:

                               Figure 14: Velocity vs time graph of the block down the slope

The linear fit provides a constant slope for the graph, which also provides the average constant acceleration of the block down the slope. The acceleration of the block was found to be 1.08 m/s/s. 

Using the angle of the slope, the mass of the block, and the acceleration, the coefficient of kinetic friction can be determined using a free body diagram and summing the horizontal and vertical components, as shown below:

                    Figure 15: Free body diagrams and equations for part 4 of the experiment

The coefficient of kinetic friction was calculated to be 0.450. It may seem unusual at first that the coefficient of kinetic friction is larger than the coefficient of static friction in part 3. However, the reason for so is because the angle of the slope was increased by more than 1.5 times as much. As the angle is increased, more of the weight of the system is trying to cause the system to slide down, so a larger friction is needed to counteract the slippage.

Procedure (Part 5):

Once all of the equipment was set up, the acceleration of the block was measured by the motion sensor, like in part 4. Similar to part 4, a velocity vs time graph was constructed, and a linear fit was performed to find the acceleration of block. The acceleration of the block was recorded to be 0.193 m/s/s. 

Then, using the coefficient of kinetic friction from part 4, the expected acceleration of the block was derived and calculated using free body diagrams shown below:

                      Figure 16: Free body diagrams and equations for part 5 of the experiment

The calculated value of acceleration is 0.158 m/s/s.  There is a 19.9% percentage difference between the calculated and experimental value. 

Conclusion:

Most of the uncertainty and large difference in experimental and actual values in this experiment are caused by the equipment being not accurate enough. This can be seen in the uncertainty of the static friction  in part 1 versus the uncertainty in the accelerations in part 5. No measuring devices were used in part 1, resulting in a very small uncertainty (0.9%). However, most of the experiment in part 5 required the use of measuring devices, resulting in much larger uncertainties or differences (19.9%). If better equipment was used, such as more accurate motion detectors and newer versions of LoggerPro and measurement kits, a lot of the uncertainty that is present now would diminish. 

Also, in order to lower error in this experiment, more trials should have been performed for the last two parts. In order to do so, more time for this experiment should have been allotted.  

16 March 2015: Modeling the fall of an object falling with air resistance

Purpose:

The purpose of this experiment was to find an equation that provides the relationship between the speed of a falling object and the air resistance force on the object.

Apparatus:

    Figure 1: Simple setup of experiment: a meter stick, a computer with LoggerPro, and coffee filters

The apparatus of this experiment consisted of a computer that contains the LoggerPro software. LoggerPro was used for video capture and the creation of graphs and data tables. Five brown coffee filters and a meter stick were also used. The video capture program of the LoggerPro software was used to record the coffee filters in free fall, and then to track their position relative to time. The position and time data were then used to create a graph to find the velocity of the freely falling coffee filter(s). The meter stick was used to give the LoggerPro software a reference for the distance that the coffee filter(s) fall in an interval of time.

                                                          Figure 2: Brown coffee filters

Abstract (Part 1):

In order to create a model for the force of air resistance on the object, it was first guessed that the force depends on velocity, since the air particles under the freely falling object are accelerated by the object from "rest" in the downwards vertical direction to the speed of the falling object. Therefore, it was first guessed that the air resistance force is equal to to the mass of the air particles in the way multiplied by the acceleration, or change in velocity over change in time.

1

In other words, the air resistance force is proportional to the velocity of the falling object:

2

However, it was then deduced that if, for example, the speed of a falling object is doubled, then twice as many air particles are being accelerated to the speed of the object. Therefore, the guess was modified such that the air resistance force is proportional to the square of velocity, or:

3

But, even though more particles are accelerated if the velocity is increased, the mass of the air is not proportional to speed in that many air particles just move out of the path of the falling object. Therefore, it was finally decided that the air resistance force depends not only on the falling object's speed, bu also the shape of the object and the material it is moving through. In addition, the force is not directly proportional to the velocity, yet there is an unknown relationship between them. With these principles taken into account, the following equation was derived:

4

where A is a constant that takes into account the size and shape of an object, V is the velocity of the object, and n is a constant that provides the "proportionality" of the relationship between velocity and air resistance force.

Procedure:

Once all of the materials for this experiment were obtained,  a test for the video capture program on LoggerPro software was performed by dropping the coffee filters in class. Once the test was done and the video capture confirmed to be correct, the experiment was moved to the Design Technology Building, or building 13. This experiment was done indoors so as to minimize any other forces acting on the freely falling coffee filters (windage, for example). Then, using the meter stick as a reference for distance, video capture was used to record the fall of the coffee filters. First, one coffee filter was dropped and its fall recorded. Then, the fall of two coffee filters put together was recorded. Then the falls of three, then four, and then finally five coffee filters were recorded. Once the falls were recorded, a position versus time graph was formed for each of the five falls by using LoggerPro. An origin was selected and the position of the coffee filters was noted every few frames. The origin was selected as the coffee filters' start of fall at the top of the building 13 balcony. Using the position and time data, the graphs were formed. Using those graphs, the terminal velocities of each of the five trials were found by forming a linear fit of the data points at the near end of the graph, since the filters reach terminal velocity a bit before the end of their fall. Taking the slope of those points, the terminal velocity can be found. These graphs are shown below:

    Figure 3|: Position vs time graph of the fall of one coffee filter

Using the linear fit at the near end of the points, the slope of the linear fit provides the expected terminal velocity. The terminal velocity of the fall of 1 coffee filter is around 0.83 m/s. 

    Figure 4: Position vs time graph of the fall of two coffee filters

The fall of two coffee filters has a terminal velocity of 1.22 m/s.

    Figure 5: Position vs time graph of the fall of three coffee filters

Three coffee filters, when dropped together, had a terminal speed of 1.44 m/s.

    Figure 6: The position vs time graph of four coffee filters falling together

The fall of four coffee filters had a terminal velocity of 1.75 m/s

    Figure 7: Position vs time graph of the fall of five coffee filters

The fall of five coffee filters had a terminal speed of 1.85 m/s. 

With the terminal velocities of all five trials, a graph was constructed that displayed the force of air resistance vs velocity. The force of air resistance can be found easily since at terminal speed the falling object is not accelerating; its force of air resistance is equal to its weight. To determine the force of air resistance, the mass of 50 brown coffee filters were measured, and then that mass was divided by 50 to get the mass of each filter. Then, that mass was multiplied by the value of gravity to obtain the force of air resistance. The mass of 50 brown coffee filters was measured to be 46.3 g ± 0.1 g. Dividing by 50, the mass of one filter was found to be 0.926 g ± 0.002 g. With this value, the following formula was derived to find the air resistance force:

5

where n is the number of coffee filters. Using this formula, the air resistance force for all five trials were found and the data table shown below was created.

                                         Figure 8: Data table of terminal velocity versus air resistance force

The last data point had to be crossed out since the change in velocity from four to five coffee filters was not significant, skewing the graph. With the above data table, the following graph was made:

    Figure 9: Air resistance force vs velocity graph

As can be seen, the equation of the above graph fits the model equation derived that relates force of air resistance and velocity (see formula 4). From the graph, the constant A was found to be 0.009587 ± 0.001727. The value of n was found to be 3.568 ± 0.3429. 

Abstract/Procedure (Part 2):

       Figure 10: Solving for acceleration (the change in velocity with respect to time)

In order to test the model created in Part 1 of the experiment, Excel was used to model a falling object with air resistance containing the same mass, A, and n values, and then the terminal velocities found in Excel were  compared to the ones found in part 1. In the above figure, the acceleration of a freely falling object at terminal speed was calculated using Newton's Second Law (F = ma), where k is equal to A.

      Figure 11: Predicted layout for the model test on Excel

As seen in the figure above, six columns were made in Excel. The initial values for time, change in velocity, velocity, change in position, and position were all zero. Acceleration, however, began ta the value of gravity. The time interval between each data row (Δt) was 0.01 sec. Δv for the new row was then found by multiplying the acceleration of the previous row by Δt = 0.01 sec. "v" was then found by adding the change in velocity of that row and the velocity of the previous row. Acceleration was the only value based on the derived model. Using the equation for acceleration derived in figure 10, "a" was found by subtracting the value of gravity by (k/m)vⁿ, using the velocity of that row. Δx was then found by taking the average of the sum of the velocity of that row and the previous row, and then multiplying that value by Δt = 0.01 s. Lastly, "x" was found by adding the value of x from the previous row and Δx from that row. Using Excel, this process was repeated for all five coffee filters until v began to not change by much. This indicates that terminal velocity of the falling object has been reached. The Excel data tables for each of the five trials are found below:

                               Figure 11: Layout for the model test on Excel

                                   Figure 12: Excel data table for 1 coffee filter

From Excel, the terminal velocity of 1 coffee filter was 0.977 m/s.

                             Figure 13: Excel data table for 2 coffee filters

The terminal velocity of two coffee filters was 1.187 m/s.

                               Figure 14: Excel data table for 3 coffee filters

The terminal velocity of three coffee filters was 1.329 m/s.

                                Figure 15: Excel data table for 4 coffee filters

The terminal velocity of four coffee filters was 1.441 m/s.

                                Figure 16: Excel data table for 5 coffee filters

The terminal velocity of five coffee filters was 1.534 m/s.

The percentage difference of the terminal velocities for one coffee filter is 16.65%. The percentage difference of the terminal velocities for two coffee filters was 3.04%. The percentage difference of the terminal velocities for three coffee filters was 8.09%. The percentage difference for four coffee filters was 19.25%. Lastly, the percentage difference for five coffee filters was 18.84%.

Conclusion:

All of the percentage differences are in the range of the uncertainty displayed in the A and n values, roughly 10 - 20%. The percentage differences for two and three filters were even lower than that; however, there is large deviations between the percentage differences, and the uncertainty is very large. This shows that the model created to relate force of air resistance and velocity was not accurate enough to explain the real effects of the world on a falling object.

Most of the uncertainty in this experiment was caused by the poor terminal velocity values measured, which all traces back to the equipment used in the experiment. The equipment used was not good enough for an experiment such as this one that needs high accuracy. To lower the uncertainty in this experiment, a laptop with better processing power, a newer version of LoggerPro software, a camera that records faster (at around 60 frames per second) and has sharper recording quality (higher megapixels), and darker colored coffee filters (so that it is easier to see in the video capture, resulting in more accurate noting of position) of higher quality (so that they fall more efficiently i.e. less change in horizontal position, more accurate representation of air resistance on a falling object, etc.) would be needed.

In addition, in order to lower any human error (e.g. noting wrong positions on LoggerPro, dropping the coffee filters incorrectly so as to cause them to sway, fall faster that actual, etc), more time should have been given to video capture the falls of the coffee filters. More time would have allowed for a greater number of trials performed, testing the accuracy of the video capture software after each trial recorded, and allowing for more detailed video captures that could have resulted in more accurate values. Lastly, a better reference should have been used for the distance the coffee filters had moved. For example, a meter stick could have been taped onto the side of the balcony to allow for better recording of the reference. Also, the height of someone or something could have been used as well since it is easier to note something larger than a meter stick from a farther view.