16-March-2015: Coffee Filter Lab

The purpose of this lab was to find a relationship between the air resistance force and terminal speed of falling coffee filters. Along with this, we were to input this relationship into Excel to try to model the fall of our coffee filters with air resistance.

Part 1: Gathering Data
The goal of the first part of this lab was to find the terminal velocity and air resistance force for some bound coffee filters. We will be basing our data off the assumption that F(resistance)=kv^n, where F(resistance) is the air resistance force, k is a term that accounts for the shape and area of an object, v is the terminal velocity of the object, and n is some unknown numerical value. F(resistance) and v are the more practical values we can find.

Procedure:
The way we conducted this lab was with 5 coffee filters. We went to the Design Technology building at Mt. SAC to conduct the experiment. There, we would drop the desired amount of coffee filters, record the falling with video capture, and find the terminal velocity of the filters using LoggerPro. There were several steps to accomplishing this:

Fig. 1
This was the point of view of the webcam
with which we were going to perform
our videos capture.

1. We brought a laptop with us to the Design Technology building. We set it up on the stairs in such a way that its webcam would be facing the balcony from which we would be dropping the coffee filters. We opened up a LoggerPro video capture window. One person would be controlling the laptop.
2. We had a second person go to the balcony, where they would first drop one coffee filter alone, the two coffee filters stuck together, then three coffee filters stuck together, and so forth all the way to five coffee filters. (Idea: adding more filters increase both the mass, air resistance force, and terminal velocity). This person would also be holding a meter stick visible to the camera. (see Fig. 1)
3. Once we were all set up, we dropped the coffee filters and recorded the fall for all five runs. Once completed, we returned back to class to analyze the data. 

Obtaining Data:

On the laptop, we had 5 different videos for five different drops. In order to find the terminal velocity of the coffee filters, we would need to create a position vs time graph for the fall. To do this, we scaled the video and inputted a position axis into it. The meter stick that was held during the drop was used to scale the video. (Note: we flipped the position axis 180 degrees. This way, the graph we would be obtaining would be positive and rising as the object physically fell.) No that the video was prepared, we began to plot points onto the video of where the coffee filter was every 1/30th seconds. We did this all the way down as it fell, which resulted in our position graph for that specific filter drop (see Fig. 2). We repeated this for all he coffee filters we dropped.

Fig. 2
There was special preparation for the video we captured to ensure that we got fairly accurate results.

Once we had our position vs time graph for each of the filters, we gave a linear fit for the portion of the graph that seemed to have the most consistent slope. This consistent slope would indicate that the coffee filter(s) had reached their terminal velocity. The linear fit would provide us with the slope of that line, which in turn was the terminal velocity (see Fig. 3).

Fig. 3
The only important part of this position graph is where the slope starts to even out. This is where we will find our
terminal velocity.

We now had to find the air resistance force for each of the coffee filters. To do this, we simply used Newton's 2nd Law. When the filters reached their terminal velocity, the net force acting on the system was now zero. With only air resistance force and weight acting on the system, we could then find that F(resistance)=mg. Thus, we needed to find the mass of a single coffee filter. Since each filter is so light, we measured the mass of 50 coffee filters, and then divided by 50 to find the mass of 1 filter.One coffee filter had an approximate mass of 0.9 g. We now knew the air resistance force and terminal velocity for each of the filter drops. With this data, we created a table of air resistance force vs terminal velocity (see Fig. 4) We used this data table to create a graph of air resistance force vs terminal (see Fig. 5) We input a power fit through the points which yielded us the result of F=0.01348v^1.886.

Fig. 4
As we added more and more filters,
the mass of the system rose, as did the
air resistance force and terminal
velocity.

Fig. 5
This linear fit gave us our unknowns of
k and n.

Part 2:

Now that we had our mathematical model, we wanted to try to predict the terminal velocity of the coffee filters by creating a model in Excel. To do this we opened up a new spread sheet and inputted our known values of k, n, m(mass of 1 coffee filter), ∆t, and g(9.8m/s^2). Beneath this, we created 6 columns: t(time), ∆v(change in velocity), v(velocity), a(acceleration), ∆x(change in position), and x(position). The initial value of each of these variables was zero, except for acceleration, which started at 9.8 m/s^2. For our time column, we inputted our equation for time, which was previous t + ∆t. For our ∆v, our equation was the product of the previous acceleration and ∆t. For our velocity column, our equation simply added the previous velocity with ∆v. For our acceleration, we derived our expression to be a=g-((kv^n)/m) (see Fig. 6). Our equation for ∆x was simply average velocity multiplied with ∆t. Finally, for our position column, we simply added ∆x to the previous position.

Fig. 6
We had to derive an expression for acceleration that we could input into Excel.

Our goal was now to fill down until acceleration was relatively zero and the velocity was consistent. This constant velocity would be the terminal velocity for that specific coffee filter. To test the other coffee filters, we would simply adjust the mass. (see Fig. 7-1 and Fig. 7-2)

Fig. 7-1
This was the general look for our spread sheet modeling the terminal velocity of our coffee filters.

Fig. 7-2
Once the acceleration and ∆v became zero, we knew we had reached our terminal velocity, which
for one coffee filter was 0.81 m/s.

We adjusted the spread sheet five times, each time changing the mass to the new amount of coffee filters we wanted to model. Overall, Excel was able to model us fairly good results. Our percent error from our obtained value for terminal velocity and Excel's value is seen in Fig. 8.

Fig. 8

09-March-2015: Propagated Uncertainty

The purpose of this lab was to determine the uncertainty in measuring the density of certain objects, as well as the uncertainty when calculating other unknown values.

Part 1: Copper, Steel, and Aluminum Masses:
For the first part of this lab, we sought out to find the propagated uncertainty when calculating the density of three different cylindrical masses. The three masses where copper, steel, and aluminum.

Procedure:

Fig. 1
No device is flawless. We had to account for the error in each 
instrument as well as place the appropriate amount of significant
figures within our data table. 

To calculate the density of these solid cylinders, we first needed to find their individual masses and volumes. To do this, we used a set of calipers and a mass scale. However, these instruments would not give us EXACT results. Thus we had to account for the uncertainty that came with each instrument. The calipers had an uncertainty of  0.01 cm while the mass scale had an uncertainty of 0.1 g. We had to account for this when we measured the mass, height, and diameter of each cylinder (see Fig. 1).

Once we had found our values, we could then plug them into our density equation. For density, we substituted volume with the actual equation for the volume of a cylinder (see Fig. 2).

Fig. 2
The equation for density was required to be broken down into the variables we had measured.

Once we had this, we had to find a way to account for the uncertainty in our density calculation. After all, if the values that we input into our equation are uncertain, then the value the equation will yield will also be uncertain. To do this we had to take the derivative of our density equation with respect to three variables (mass, height, and diameter). We did this using partial derivatives (see Fig. 3).

Fig. 3
Since density is in terms of three variables, in order to find the derivative of density, we needed to take the partial
derivative of each variable.

Once we had this, we were able to input our values for each cylinder to find their calculated uncertainty. This value would then be recorded together with our calculated density in order to show the level of uncertainty in our findings. For Copper, we found the density to be 9060 kg/m^3 with an uncertainty of 236 kg/m^3. For Steel, we found the density to be 7.60 x 10^3 kg/m^3 with an uncertainty of 149 kg/m^3. Finally, for aluminum we found the density to be 2740 kg/m^3 with an uncertainty of 50.0 kg/m^3 (see Fig. 4).

Fig. 4
Our final density calculations and uncertainty.

Part 2: Finding the Mass of a Hanging Object.

We again used this tool of calculated propagated uncertainty when calculating the mass of two hanging objects. 

Procedure:

Fig. 5
Thankfully, the system was already set up
the day of the lab. We just need to record
we saw.

There were models set up across the classroom with a mass hanging in the middle of some stretched string (see Fig. 5). Our goal was to calculate the mass of the object on the string for two of the set ups, while still accounting for our uncertainty. To do this, we needed four pieces of information from each set up. First, we needed to know the tension within the two parts of the string. To do this, we measured the force scale set up on each string. Second, we measured the angle at which the tension force was being applied. To do this, we used a type of inclinometer to measure the angles (see Fig. 6). Once we had all of our data, we created a free body diagram, as well as net force equations, for each of the setups (see Fig. 7). We found the general expression for the mass of the object to be m=F(1)sin(θ(1))+F(2)sin(θ(2))/g. Now we could come with our combined uncertainty for mass. 

Fig. 6
To use the inclinometer properly, we need to place its base
parallel to the string whose angle we are measuring.

Fig. 7
Creating the free body diagram gave us a clear
indication of what was occurring within the
system.

The Uncertainty:

To find the uncertainty in our calculation of mass, we once again found the derivative of the mass equation. We again did this by partial derivation of all the variables (see Fig. 8). 

Fig. 8
Once again, in order to find the uncertainty in mass, we needed to take the partial derivative of all the variables.

NOTE: When inputting the uncertainty in our measured angle, we converted the degree to radians.

This is because the calculus portion of this lab is based only in radians.

For our mass #1, we calculated the mass to be 0.846kg with an uncertainty of 0.068kg. For our mass #2, we calculated the mass to be 0.933kg with an uncertainty of 0.060kg (see Fig. 9).

Fig. 9
Final calculations for our mass and uncertainty.

07-March-2015: Free Fall Lab

The purpose of this lab was to determine the value of the gravitational acceleration constant "g."

Procedure:
In order to calculate "g", we would simply need to let some type of object free fall and determine the acceleration at which it did so. In order to measure this, an apparatus was set up. This apparatus consisted of a 1.86 m column that would allow an object to free fall 1.5 m. A long paper strip was attached to the apparatus, and a wooden cylinder with a metal ring around it would free fall next to the paper. As it fell, a spark going at 60 Hz would hit the metal ring as it went down. The paper strip would record the markings between the spark and metal ring. Since we know the time interval between each marking was 1/60th of a second, we could calculate the rising velocity of the wooden cylinder as it fell.

Fig. 1
Using a ruler, we began
measuring from one dot
and continued to measure
relative to that dot.

For this lab, however, we were already given a strip of paper which had markings. The marking were obtained using the same procedure as stated above. To determine the acceleration of the falling wooden cylinder, we needed to find the distance between each of the markings. Since acceleration should be constant, the starting dot we chose was irrelevant. So we began recording the distance from one certain dot, to 15 dots away from that one. We recorded our data in an Excel spreadsheet. (see Fig. 1) We measured the distance relative to the starting point. Once we had all our data, we then input columns for time, change in position, mid interval time, and mid interval velocity. The time change was 1/60th of a second, the change in position was the difference of the distance between each point, and the mid interval velocity was the change in position over the time interval. The mid interval time was used to graph the velocity graph, since the velocity we found was simply the average velocity. We filled these columns down to create a spreadsheet that we could graph. (see Fig. 2).

Fig. 2
This spreadsheet served as an organized data table that Excel could use to graph and give us equations for position and velocity.

Our first graph was a position versus time graph. We plotted the distance column vs the time column.(see Fig. 3) The next graph was the velocity vs time graph, We plotted the Mid Interval Velocity column vs Mid Interval Time column to form this graph. (see Fig. 4)

Fig. 3
Using Excel, we were able to construct a graph that could
illustrate the relationship between velocity and time, and thus
lead us to find acceleration.

Fig. 4
Using Excel, we were able to construct a graph of position
vs time that would also allow us to find acceleration through
derivation.

Questions/ Analysis:

1. For constant acceleration, the velocity in the middle of a time interval will always be the same as the average velocity of that time interval. This is because the relationship between velocity and time is linear if acceleration is constant. The slope is continuous and straight. The average velocity between two time intervals will be the same if you follow the slope of that graph to the mid time interval between those two. (see Fig. 5)      

   

   Fig. 5

   
   

                             Since the velocity graph is linear, the average of two points can also be found at their midpoint.                                                                                                                                                       

2. Finding acceleration through a velocity vs time graph is very simple. Since we know that acceleration is a function of velocity/time, we can assume that the slope of a velocity graph is acceleration. Thus, if we have an equation for the line created by a velocity graph, the slope of that line is the acceleration.
3. Finding the acceleration through a position graph requires some derivation. If we have a constant acceleration, a position graph will look parabolic. Given the equation for position, the first derivative of this function would be the velocity equation, and the second derivative would be our acceleration value.

Our Value for "G":

From second derivative of our position graph, we calculated the acceleration to be 930.08 cm/s^2. From the slope of our velocity graph, we found the acceleration to be 930.46 cm/s^2. Taking the average of these two values, our final calculated acceleration for gravity was 937.77 cm/s^2. Or 9.38 m/s^2 for us physicists. Our percent error was 4.31%, which can be mostly accounted for our lack of proper equipment. (using a cm ruler) For calculations, see Fig. 6

Fig. 6
Calculations that led us to our value for gravitational acceleration.

09-March-2015: Non-Constant Acceleration Problem

The purpose of this lab is to illustrate the utility of the computer program "Excel" in complex physics problems that involve a non-constant acceleration.

The Problem:
For this lab, we were given a word problem involving a non-constant acceleration. The problem regarded a 5000-kg elephant on friction-less roller skates with a velocity of 25 m/s once it reaches the bottom of some hill. On the elephant is a rocket that produces a constant 8000 N thrust opposite the elephant's direction of motion. However, the problem continues, adding the fact that the mass of the rocket is a function of time m(t)= 1500kg - 20kg/s(t). Our goal is to find how far the elephant moves before stopping.

Fig. 1
Using integration leaves room for
far too much error.

Why The Problem is Difficult:
What makes this problem difficult is that all of our simple kinematic equations are now obsolete. Previously, we had always assumed acceleration to be constant. Thus, in order to find the distance traveled, we must use calculus to derive some function that relates position and time. We know that velocity is the derivative of position, and acceleration is the derivative of velocity. Thus, in order to find position, we have to integrate the acceleration function twice. However, this proves to be time consuming, tedious, and extremely vulnerable to multiple errors in calculation (See Fig. 1).


Numerical Integration:
To complete the problem, we are going to use the program "Excel" to manually integrate our values. In order to do this, we needed to open up a new spread sheet and enter the following at the top of the first 6 columns: time(t), acceleration(a), average acceleration(a_avg), change in velocity(Δv), velocity(v), average velocity (v_avg), change in position(Δx), and position(x). Above these, in cells A1 and B1, we indicated the time interval(Δt) to be 0.01 seconds (see Fig. 2)

Fig.2
These columns will allow us to recognize what each number means once we begin to input our equation and values.

The goal of making this spreadsheet was to find a relationship between all these different features of motion. If we provided "Excel" with the basic equations that described these things, then the program would be able to calculate thousands of them in seconds. All we have to do is input the simple things about motion we already know.

We Know:

• With this given problem, acceleration = -400/(325-t). 
• The change in velocity is the product of average acceleration and the time interval. 
• The velocity at any time "t" is just the initial velocity added with the change in velocity. 
• The change in position is the product of average velocity and the the time interval.
• The position at any time "t" is just the initial position added with the change in position.

To input this information, we have to provide each column with their own equations. For time(t), we simply start at zero and add the time interval(Δt) to each previous time. For acceleration(a), we input the formula -400/(325-t). The average acceleration(a_avg) will be calculated by the previous two accelerations. For the change in velocity(Δv), we multiply the average acceleration(a_avg) and the time interval(Δt). For the velocity(v) at that specific time, we simply add the change in velocity(Δv) to the previous velocity and in the case of this problem, we are given an initial velocity of 25 m/s. The average velocity(v_avg) is again calculated by the previous two velocities. To find the change in position(Δx), we multiply the average velocity(v_avg) by the time interval(Δt). Finally, the position(x) is calculated by adding the change in position(Δx) to the previous position and in the case of this problem, we start at a position of 0m. We input all these equations into excel, and from here we can now calculate the values we are looking for. 

For the problem, once the velocity was zero, we found the position to be 248.697m with a time interval(Δt) of 0.05s. (See Fig. 3)

Fig. 3
Once we had our time interval, we continued down the spread sheet all the way until velocity was
as close as it could be to zero. We then looked for the position at that particular time.

Conclusions:

1. When doing the problem analytically, we found the exact answer for the problem, which was 248.7m when rounded. Our value of 248.697m had a percent error of 0.001%. If we had chosen an even smaller interval, our answer would have been even more exact. 
2. The issue is now knowing when the time interval is small enough to yield you a usable answer. The only way to know this is to continuously input different values for Δt until the difference between the answers you are looking for become negligible.        

28-February-2015: Finding the Relationship Between Mass and Period for an Inertial Pendulum

The purpose of this lab was to come up with an equation to describe the period of an inertial balance when different masses were set on it.

Materials:
C-Clamp
Inertial Balance
Masking Tape
LabPro- power adapter, USB cable, plug adapter, photogate
Computer with LoggerPro Software
Multiple Different Masses (.1kg - .8kg)

Setting Up the Experiment:

Fig. 1

The C-clamp secured the 

inertial balance to the table, ensuring 

it wouldn't move during oscillation.

To begin, my lab partners and I needed to set up the inertial balance in such away that we could measure its period of oscillation. To do this, we used the C-clamp to secure the inertial balance to the edge of our table (See Fig. 1). Once secured, we needed to set up the LabPro equipment at the other end of the inertial balance. Using a metal pole stand, we placed the photogate to the opposite end of the inertial balance and made sure it was also level with the balance. We also placed a piece of masking tape at the end of the inertial balance. (See Fig. 2). The photogate was then hooked up to the LabPro, along with the power adapter and plug adapter. We then finally hooked up the LabPro to the computer and opened up the LoggerPro Software. The file that we used was the Pendulum Timer.cmbl. (We searched this via the magnifying glass at the top right corner of the macbook). We were now ready to begin collecting Data.

Fig. 2

The photogate produces a light 

beam when turned on. The masking tape

will be used to break that beam during 

oscillation.

      

This is what the final setup looked like. 

Collecting Data and Making Graphs:

  

Fig. 3

We continuously added different masses,

securing them with tape each time. 

To begin collecting data, we intended to measure the oscillation of the inertial balance under a multitude of different circumstances. First, we began with no mass on the balance. We pulled the balance back slightly, released, and collected data of its oscillation. We then added 100g to the balance and repeated. We continuously added 100g after each trial, all the way until we reached 800g (See Fig. 3).  We then measured the period of oscillation of the balance when we added two objects whose masses were unknown. For this experiment, we used a tape dispenser and car keys. Once we had all our data, we recorded it into the data table provided in the lab handout. (See Fig. 4).                                            

Fig. 4

Fig. 5

We created the data table to mimic the 

y=mx+b form. TM is short for "Total

Mass" which is what (m+Mtray) is.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          Once we recorded all the data into our data table, it was time to make a graph of period vs time. From the handout, we were given the initial equation:T=A(m+Mtray)^n. With this equation, we had three unknowns: A, Mtray, and n. To make a more reasonable looking equation, we took the natural logarithm of each side, which gave us: lnT=nln(m+Mtray)+lnA. This equation mimicked that of y=mx+b, in that lnT is y, n is m, ln(m+Mtray) is x, and lnA is b. To begin graphing our data, we first opened a blank LoggerPro document, which allowed us to plot data in an "x" and "y" table. In this table, we labeled the "x" column "Mass (kg)" and the "y" column "Period (sec)." We then created three new columns for (m+Mtray), lnT, and ln(m+Mtray) by first clicking the "Data" tab, and then "New Calculated Column" (See Fig. 5) Finally, we had to create a parameter Mtray, which would be our initial guess for the mass of the inertial balance by itself. To do this, we clicked the "Data" tab, and then "User Parameters."  We first guessed 0.3 kg.

When making the graph, we set the vertical axis as lnT and the horizontal axis as ln(m+Mtray). This gave us a linear graph with a slope "m" (which is n) and a y-intercept "b" (which is lnA). We then had to adjust the parameter Mtray several times to get a correlation coefficient that was as close to 1 as possible. We found that our best range for Mtray was from 0.295kg to 0.299kg (See Fig. 6).

Fig. 6

0.9997 was the closest correlation coefficient we managed to obtain. The fit parameter for Mtray was essentially our best guess for what the mass of the inertial balance on its own would be, since we couldn't actually measure the mass of the part that was oscillating by itself. 

Deriving The Final Equation

Fig. 7

Once we had our originally unknown values for n, lnA, and Mtray, we were now able to complete our equation of T=A(m+Mtray)^n using algebra (See Fig. 7). When testing the equations, we found that we were roughly 0.005 seconds off from the actual period for most of the masses. When evaluating the mass of the tape dispenser (T=0.611s), we found it to be 0.606kg with the equation using Mtray as 0.295kg, and 0.612 with the equation using Mtray as 0.299kg. The actual mass of the tape dispenser was 0.605 kg. As for the keys (T=0.358s), we found its mass to be 0.104kg when using both equations. The actual mass of the keys was 0.105kg. The equations we derived gave us very accurate measurements of our unknown masses. The relationships we found between mass and the period for our inertial balance from this lab are as follows:
T=0.6517(m+0.295)^0.6518 and T=0.65005(m+0.299)^0.6559