13-Mar-2017: Modeling the fall of an object falling with air resistance

Lab 5: Modeling the fall of an object falling with air resistance

13-Mar-2017

Idea (Part 1): 

Determine the relationship between air resistance force and speed, by modeling the terminal velocity of falling coffee filters at various increments of mass.

Summary Part 1:

1. First, we took coffee filters and a computer webcam to the Design Technology building where we recorded coffee filters being dropped from the balcony using Logger Pro.
2. The filters were dropped first, with just one, then two, three, four, and five stacked on top of each other.
3. Using Logger Pro, we scaled the video to the height of the balcony, and positioned the filter's position on a position vs time graph at an increment of about 6 frames.
4. We obtained the terminal velocity by fitting the last few points of the position vs time graph with a linear fit. The graph subsequently showed us the values for k and n.

Part 1 Analysis:

By doing a linear fit on the position vs time graphs in Logger Pro, especially near the end of the graph, we could find where the velocity (the slope) of the filter becomes constant. I.E, reaching terminal velocity, for all five of the filter sets. As we noticed, terminal velocity increased the more coffee filters we added. Therefore, we suspected that the force of air resistance would also increase.

1 Filter

 2 Filters

 3 Filters

 4 Filters

 5 Filters

The terminal velocity of the five test cases, as previously mentioned.

Next, I calculated the k and n constants, using the values I obtained from my previous graphs. This allowed me to graph terminal velocity versus the force of air resistance. In this case, we primarily used 

Using the formula for the force of air resistance, I created a terminal Velocity vs Force graph in Logger Pro with a power fit, using my measured values for V, and the calculated values of F for the various test cases. As I could observe, my data did not fit along the curve as well as I had thought.

Idea (Part 2): 

Use the mathematical model obtained from Part 1 of the lab in order to mathematically predict the terminal velocity of the coffee filters.

Summary Part 2:

1. Next, we opened up Excel. We first created cells to store our values for ΔT, m, g, k, and n, in column B.
2. Then, column A7 was time, B7 ΔT, C7 was velocity, D7 was acceleration, E7 was ΔX, and F7 was X. For the values of ΔV, ΔX, and a, I calculated them by inputting the formulas for acceleration,  ΔV, ΔX into their respective columns in row 8:

Afterwards, I dragged the columns down until my ΔV gave values close to zero. In order to be more accurate, my group played around with the value of Δt, until Isettled on .0050s.

Part 2 Analysis:

With my spreadsheet adjusted with appropriate $ signs so that I could adjust my values of ΔT,k, and n, I predicted earlier that the terminal velocity should increase as more coffee filters were to be added, as my experimental data had already shown. The following data calculates terminal velocity for the first coffee filter, with .0050 as the smallest time interval that would give me a change in velocity close to zero, meaning that I have reached terminal velocity.

Case 1

As I saw, the experimental terminal velocity for the first coffee filter was 1.174 m/s, whereas my mathematical model gives 1.183 m/s. My experimental data was off by a hundredth in the case of the first filter drop.This was a pattern that I would see for the other cases as well: each of the cases are off by approximately 1-7 percent.

 Case 2: Experimental: 1.567 Model: 1.652

Case 3: Experimental: 2.094 Model: 2.007

Case 4: Experimental: 2.331 Model: 2.305

Case 5: Experimental: 2.533 Model: 2.566

Conclusion:

Because everything in this lab was purely based off our initial experimental measured data, such as our values of k and n, there were bound to be several sources of uncertainty.

1. When recording the dropped filters, my particular webcam was out of focus. Thus when I began recording the filters' positions, especially when the change in position was not very significant, I noticed we had made flawed recordings as a direct result of not being able to ascertain the bottom of the filter very clearly. This definitely affected our graphs in this section, and as a result, our calculations for K and n.
2. In addition, when scaling the video to the height of the balcony, again, the slightly out of focus webcam meant that we were likely off by more than a few pixels all things considered. But ultimately, considering that our value was off only by a hundredth, these were errors that did not change our result too significantly. 
3. With that in mind, I calculated the percentage error for all of my 5 cases using the data from my tables and experimental values:

Case 1: (.76%)

Case 2: (5.1%)

Case 3: (4.3%)

Case 4: (1.13%)

Case 5: (1.05%)

15-Mar-2017: Trajectories

Lab 6: Trajectories
8-Mar-2017
Idea: Predict the impact point of a ball on an incline using projectile motion.

Summary: 
1) In regards to materials, we set up an incline connected to a ramp using a stand, a clamp, two metal v-channels, white+black carbon paper, and lastly, a steel ball. 

2) We then taped a sheet of paper with black carbon paper at approximately the spot where the steel ball would land, so the ball would leave a mark on the paper.

3) Next, we measured the height of the launch point, and released the ball five times.



4) Then, we measured how far from the edge the ball landed. The marks that our ball left on the carbon-paper were more or less consistent.



5) Using the height, I calculated the time it took for the ball to land. Then, I used the time and the measured distance to calculate the launch velocity of the ball.



6) Then, we propped an inclined wooden board right under the v-channel. Using the launch velocity I previously calculated, I set up an equation that would allow me to predict the landing point on the inclined board with α, the angle the board is at with respect to the ground. I attached the paper+carbon paper to the edge of the board



7) Then, after actually measuring α, I attached the paper+carbon paper to the edge of the board and again rolled the steel ball five times.



8) To get the uncertainty for this measurement, we made our measurement for x the middle mark, and our uncertainty was the range of the farthest and closest mark.

Analysis:

As I noticed early on with the first test case, the marks left by the carbon paper were surprisingly consistent. When solving for v0, I knew that the vertical component of the launch velocity would be zero, due to the horizontal launch ramp, which let me solve for time, which let me solve for v0 knowing the horizontal distance traveled in the first case. In addition, as all the measurements for height and length were done with a meter stick, they ha an uncertainty of around tenth to a fifth a centimeter.

When calculating/predicting the landing spot, I used the obtained v0, and our measured angle alpha and set the equations for distance for both the x and y directions:

Then, I actually did the second test and recorded the length, as already mentioned on step 8. As I could see, even with the uncertainty in mind, my prediction was off by 1-2 cm.

Conclusion:

In calculating my uncertainty, I had to take into account the uncertainty of my height+length measurements, alpha, and their subsequent calculations that give the values of time, and launch velocity.

Therefore, with a propagated uncertainty of distance of 6.82 cm, my calculated prediction was definitely more accurate than it could have been.

8-Mar-2017: Non-Constant acceleration

Lab 4: Non-Constant acceleration
8-Mar-2017
Idea: Find how far a 5000 kg elephant traveling 25 m/s goes before coming to rest when a 1500 kg rocket mounted on its back generates a 8000N thrust opposite to the elephant's direction of motion.
Summary: In Excel, we first created a column where we could enter the given values, and be able to adjust them later on.

1. Column A stood for my intervals of time, Column B for the acceleration of the rocket, column C for the average acceleration, Δv for Column D, V for Column E, Δx  for Column G, and X for Column H. Because the acceleration isn't constant, I had to calculate Δv, v, a, Δx, and x separately, as shown below. In the first case, acceleration was calculate by dividing the net Force (8000N) by 6500-20t. Thus a= (400)/(325-t) 
   
   
   
   
   
   
2. Then, I dragged the columns down until my value of V in Column E was zero.

Analysis

When I stretch the columns, for each of the intervals of time I set (1s, .1s, and .05s) I look for values of t where V is zero. As I saw when ΔT was 1, V is zero in between T = 19 and 20. As I decreased ΔT from 1 to .1, to .05, the value of V was narrowed down to be most likely to between T=19.65 and T=19.7s.

Data (Time Interval 1)

Data (Time Interval 0.1)

Data (Time Interval 0.05)

Conclusions/Questions

1) Analytically, the value for of T where ΔV was 0 is 19.69075s. My numerical result was a value between 19.65 and 19.7s. My numerical value was rather accurate.

2) I know if my interval is "small enough: once the values of V are similar to a few decimal places. If  I didn't have the analytical approach, I would have to keep decreasing my ΔT values and make a judgment as to when I would accept the margin of error.

3) If the elephant's initial mass were 5500 kg, the fuel burn rate is 40kg/s and the thrust force is 13000N, the new acceleration equation is:

Plugging this formula back into Excel along with our new data,

The new V=0 in an interval of .05s gives me a value of time between 12.95-13s.

6-Mar-2017: Propagated Uncertainty in Measurements

Lab 3: Propagated Uncertainty in Measurements
1-Mar-2017
Idea: I have to calculate the uncertainty in the calculation of density of two objects following their measurements.
Summary: First, with regards to materials, in this lab we used a cylinder of aluminum and zinc. We also used an electronic scale, and a caliper that gives accurate measurements to a tenth of a cm.

1. We first took the mass of our zinc and aluminum. Then, we also measured the diameter and height of the cylinders, using the calipers.
   

   

   zinc (cm)
   

   

   zinc (cm)
   

   

   aluminum

   

   aluminum

2. After that, we were able to start our propagated uncertainty calculations.

Analysis:

1) We knew that because the electronic scale gave us a measurement to one decimal place, we had an uncertainty of ± .1 grams.

2) For our measurements of height and diameters, the calipers gave us an uncertainty of only ±.01 cm.

3) Thus, when calculating uncertainty, I knew that if my density were to be off, it would most likely be due to the larger uncertainty in the measurement of my masses.

Data

Calculations

Conclusion

As expected, the largest source of my measured uncertainties was from my masses. Another source of uncertainty was also in the calipers, but because they were accurate, yet calculated for volume, it did not contribute heavily to any kind of larger propagated uncertainty.

1-Mar-2017: Free Fall Lab

Lab 2: Determination of g and some statistics for analyzing data
1-Mar-2017

Part 1

Idea: I have to demonstrate the idea that a falling body will accelerate at 9.81 m/s^2 when there are no other external forces except gravity.
Summary: First, with regards to materials, in this lab we used a special paper tape with markings already recorded by a dropped spark generator (shown below), as well as a meter stick to record the distance between each mark, and lastly Excel to plot our data.

1. Then, we used our meter stick to record the position of each dot from the 0 cm mark.
2. We opened up a data set in Excel and set column A to record our time. We set A1 to be Time, A2 = 0, and A3 = A2+1/60.
3. Column B recorded our distance, as B2 was 0 (for the 0 mark) and we went all the way down to B16, filling in our measured distances in each cell starting with the position of the second dot.
4. Column C was simply our change in distance, so from C2 downwards we set C2 = (B3-B2)
5. Column D would give the time for the middle of each 1/60th second interval, so from D2 we entered =A2+1/120
6. Lastly, column E gave the nid-interval speed, so from E2 down we entered =C2/(1/60)
7. Using this data table, we selected columns D and E, and used a linear fit, and obtained the equation of this line.
8. Subsequently, we graphed columns A and B, abut instead used a polynomial fit of order 2, and got the equation for that as well.

Measured Data

This is the data based off the recorded distances in cm for each mark on the paper tape, and subsequently steps 2-6 in the previous section in which we found the mid-interval time & speed.

Analysis: 

1) In the first graph, we plot the mid interval speed vs time because doing so will give us the slope, which is also the acceleration of the free falling object. If our measurements were accurate, then there is reason to believe that the slope should read a gravitational acceleration close to 9.8 m/s^2.
2) The second graph, distance over time, will show simply the slope, also the velocity, of the falling object.

Graph/Calculations:

This graph depicts Columns D and E with a linear fit

As I could see, the slope of the linear graph depicts a downward acceleration of the object at 958.15 cm/s^2. This was obviously off from the expected gravitational acceleration of around 980 cm/s^2.

This graph depicts Columns A and B, with a power fit of order 2

The slope of this graph depicts the velocity of the falling object.

Questions/Analysis

1)Velocity in middle of time interval vs average velocity for time interval

2) From my velocity/time graph, I can also get my acceleration by

compared to the accepted value of 980 cm/s^2, the value is also off by around 2.2 cm/s^2.
3) From a position.time graph, I can again get my acceleration by

compared to the accepted value of 980 cm/s^2, the value is also off by around 2.5 cm/s^2.

Conclusions for Part 1

1) There were clear patterns to the data. First, the increase in X in column C of my data table remained noticeably consistent all the way through, and of course even without a line of best fit, the graph points that columns A and B, D and E gave showed definitive patterns. Linear and power, respectively.

2) As the lines of best fit showed, the patterns I expected to find were all there.

Experimental uncertainty:

I found the relative difference of my resulting acceleration to that of the accepted value, how much they differ in percentage:

Part 2

Idea: I have to analyze the class data for gravity and find the Standard Deviation of the Mean of the class data.
Summary:

1. I opened up a new file in Excel, and entered the class values for g from cells A2 to A11. In cell A12, I average these values with =average(A2:A11)
2. Column B is for my deviation from the mean, so from B2 to B11, I enter and fill =A2-$A$12

Data Table

Based on the class data, the chart gives me the average, average deviation squared, and lastly, the standard deviation

Analysis: 

Thus the standard deviation for g was 6.11, meaning that within the entire probability range of the entire class, from (961.5-6.11)=955.2 to (967.6+6.11)=973.7, the range of values from 961.5 to 967.6 would be the most likely.

Conclusion/Questions

1) We observed that for the values of g, the higher its deviation, the lower the value of g.

2) Our average, 961 cm/s^2, was less than the accepted value of 980 cm/s^2 by approximately 19 cm/s^2.

3)  In all of our values of g, none of us ever got to or exceeded the accepted value. All of our data for g pointed to values less than 9.8 m/s^2.

4) For the discrepancy between my measurements and those of the class, the fact that there is likely to be a  ±1mm of uncertainty in the measurements of the marks (due to simple human error in judging the distance) could certainly be a factor. This would directly affect my entire data table, and subsequently the velocity/time and position/time graph.This would be a random error.

The second would be in possibly the marked paper itself. As the object falls, and creates sparks that leave marks on the paper, it makes slight contact with the paper as it falls, introducing an external force apart from gravity, friction, to the object. This would lead to inaccuracies in the marks themselves, that we would not be aware of, so it is subsequently a systematic error.

5) In this part of the lab, we were to analyze multiple results of the same experiment in order to firstly be able to analyze the standard deviation, and in addition, distinguish possibilities in conditions that may have led to discrepancies in each of our values of g. We accomplished this by finding the average of our values, the average deviations, which allowed us to visualize the probability range of our values within 68%. Subsequently, we were able to notice a trend in all of our data being lesser than the accepted value of g, which allowed us to conclude that there must have been a systematic error in the experiment that affected everyone on top of individual random errors.

27-Feb-2017: Finding a relationship between mass and period for an inertial balance

Lab 1: Finding a relationship between mass and period for an inertial balance
27-Feb-2017

Idea: I have to determine the period of an inertial balance first for various known masses, then the period of various unknown objects, and be able to calculate those masses based on the data obtained from the graph of mass/period of the known masses.
Summary: First, with regards to materials, in this lab I used an inertial pendulum with tape for a marker, and a photogate that measures the period of the oscillation which was connected to the Logger Pro application, which records the period of the oscillations over a length of time. In addition to a stand, various known masses, and two random objects (a Samsung S6 phone and this clamp we found in the room), as shown here:

1. Then, we loaded the inertial pendulum's tray with masses ranging from 0 (nothing on the tray)  to 800 grams in increments of 100 grams, measuring the period over 4-5 seconds with Logger Pro.
2. We then opened up a data set (x and y) and set Mass in grams to be X, and Period in seconds to be Y and created 4 new columns, Mtray, m+Mtray, lnT, and ln(m+Mtray).
3. Based upon the graph that these parameters gave, we adjusted the value of Mtray until the graph gave us a straight line of best fit.
4. Simultaneously, we also experimented with different values of Mtray that would give us the range of uncertainty for Mtray, that is, we found the minimum and maximum values of Mtray that would still give us the best possible straight line. 
5. In addition, we recorded the slope and y-intercepts that resulted from these different values of Mtray, for each different value of Mtray.
6. These values would allow us to find the value of constants A and n.
7. Subsequently, we measured the period of two different objects with unknown masses, and using the values of Mtray and their respective values for A and n in the previous step, determined the range of uncertainty of unknown masses 1 and 2.
8. Finally, I used Excel in order to graph my raw data.

Measured Data

Part 1: Data Table for Period of the various known masses

Part 2: Data Table of the two unknown masses. Shows data from the lowerbound, middle, and upperbound values of Mtray.

Graph/Calculations: These are the graphs and calculations for the known masses based off our data.

Graph of ln(t) and ln(m+Mtray) for known masses

Graph for lower value of Mtray (240 g). This shows the graph with the lowerbound value of Mtray that would give me the best possible correlation closest to 1. For me, the best I could get to was .9996.

 Graph for likely value of Mtray (270g), which was closer to its lowerbound and subsequently shown in the graph as its slope is most similar to the Mtray lowerbound graph.

Graph for upper value of Mtray (300 g), similarly, with 300 grams being the absolute maximum that would allow the graph to keep the best possible correlation of .9996.

Calculations

Values I needed to calculate

Finding the mass of the added object

Resulting data

Graph of Resulting Data

The topmost line is the graph of the second mass (the clamp), and the bottom line is of the phone, the first mass, and both for the middle value of Mtray. As I could see, the bottom-most graph (of the phone) is noticeably far more accurate, as its slope is closer to 1 than mass 2's.

Analysis: 
In the first three graphs, line of best fit allowed us to find the slopes (n), y-intercepts, and best correlation for the three variations of the Mtray value. As the value of Mtray increased, I observed that the slope of the line also increased.
Then, in order to find the unknown masses, I used the A and n constants (for the three values of Mtray) I found based on the previous known masses, which allowed me to later rearrange the equation of period in order to solve for the two unknown masses. I repeated the calculations for both objects, with the various values of Mtray.
Afterwards, in graphing my raw data, I again set my set my parameters and columns exactly the same as before, but instead entered in the calculated masses of my two unknown masses. In order to get a line of best fit I utilized Excel, which allowed the comparison to the other graphs a lot easier. The graph essentially showed me that the experiment had been more accurate with the first unknown object (the phone) rather than the second, as its constants (slope, for example) is far more accurate than those of object 2 with a slope of 1.0086.

Conclusion: I was able to graph the data obtained from the two known masses, and calculate the mass of two unknown objects, as 179 grams and 1008 grams, respectively. Weighing the two objects on a scale gave us their actual mass, 184 grams (with a phone case) and 1300 grams (with the clamp), showing that the margin of error varied greatly between the two throughout my calculations.

• In hindsight, an area of uncertainty in this case was possibly (though not likely) first in the photogate, as the condition of the equipment is something we could not ascertain. 
• Then, in the inertial pendulum itself, as we observed that the metal connecting the tray would twist/bend if the mass in the tray was too heavy, which could certainly affect the oscillation of the pendulum, and subsequently its period. The clamp we used in the second part of the lab was much heavier than the other masses in the lab, and as a result I did observe a slight twist to the pendulum. In the end, this could explain the discrepancy between the constants of my last graph.