Lab 3 - March 7, 2015 - Non-Constant Acceleration Problem/Activity with EXCEL

Lab 3:
Brandon Elder
March 7, 2015

Non-Constant Acceleration Problem/Activity with EXCEL

Purpose: To find how far an elephant on roller skates will travel, with a rocket (strapped to his back) firing above his head in the opposite direction, before he his direction changes (See Fig. 1). We will investigate these distance questions analytically (physics) and numerically (Excel).

Fig. 1 How far will the rocket let the elephant travel
before he changes direction?

Problem: A 5000-kg elephant on frictionless roller skates is going 25 m/s when it gets to the bottom of a fill and arrives on level ground. At that point a rocket mounted on the elephant's back generates a constant 8000 N thrust opposite the elephant's direction of motion. The mass of the rocket changes with time (due to burning the fuel at a rate of 20 kg/s) so that the: 

m(t) = 1500 kg - 20 kg/s*t.

Find how far the elephant goes before coming to rest.

Fig. 2 Acceleration as a function of time.

Fig. 3 Finding the change in velocity in order to
derive an equation for v(t).

Fig. 4 At the bottom of the picture you will see the equation
for x(t). Thanks to the professor!

Fig. 5 The result of plugging the time into the postion
equation. Result: 248.7 meters.

Analytically: We were given the acceleration of the elephant plus the rocket as a function of time (See Fig. 2). From the acceleration function we can integrate from 0 to t to find the change in velocity and then to derive an equation for v(t) (See Fig. 3). Following the same logic, we can integrate the velocity function from 0 to t to find the change in X and then use this to derive an equation for x(t) (See Fig. 4). Next, we find the time at which the velocity is zero. As this will be the time we can plug into the distance formula to solve for the length the elephant travels before turning around. Plugging in the result of 19.69075 seconds results in a distance of 248.7 meters (See Fig. 5). Next, we will analyze the same problem numerically by using Excel.

Numerically: Open up Excel and set up the column headings as seen in the pic below (See Fig. 6).

Fig. 6 Column Headings are in row 2. The Change in Time factor is in cell A2.

Input the following formulas: All formulas should be dragged down throughout the rest of the rows.

B3: Input the formula to be used for acceleration: "=-400/(325-A3)"
C3: Average acceleration: "=(B3+B4)/2"
D3: Change in velocity: "=C4*$B$1"
E3: Velocity at the end of time interval: "=E3+D4"
F3: Average Velocity: "=(E3+E4)/2"
G3: Change in Position: "=F4*$B$1"
H3: Position: "=H3+G4"

For intervals of 1 for time, at time = 20, the position is (see Fig. 7). When you decrease the interval for time, making it closer to zero, the position at t = 20 should be exactly what was calculated using the integrals above. That answer was 248.7 meters. If you set the time interval in excel to .1, the position at t = 20 is a little less than 248.7. However, look at the position at t = 19.7, the answer is exactly what was calculated earlier. This is because the time that was calculated from above was actually 19.69 seconds, closer to 19.7 than to 20. Therefore, the position that the elephant changed directions is at 19.7 seconds or 248.7 meters (see Fig. 8).

Fig. 7 At time equal to 20, the position is close to the
analytically calculated number of 248.7 meters.

Fig. 8 When time is set to intervals of .1 seconds, observe how the position
is exact at 19.7 seconds. The elephant turns around at 248.7 seconds,
confirming the numbers from earlier.

Conclusion:
Imagine the graphs of the above functions for acceleration (see Fig. 9), velocity (see Fig. 10), and position.

Fig. 9 Plot of acceleration vs time. The area
under this graph, between t (initial) and
 t1 represents the velocity. 

Fig. 10 Area under the graph of velocity vs time is
the change in
position.

The reason that we made the time intervals closer and closer to zero in order to get the most accurate position in Excel was because, if you look at the graph, the smaller that the values of t are, the smaller the distance between t (initial) and t1 will be. The integral is the most accurate when the line on the graph is as close to linear as possible, and that is what reducing t from 1 to .1 did for us in our data.

Conclusion Questions:

1. The results from doing the problem numerically varied depending on the interval that was set for our delta t. If the number was set to a low interval, such as .1, our results matched. If the t value was set to 1, there was some slight variance. The smaller the values the more accurate the number would be because this represents the amount of squares used to calculate the area under the graph, the integral.

2. If the analytical value wasn't available you would be able to tell the correct time interval when you see the max distance value peak out and the amount of difference in the values before and after the peak were negligible, or as close to zero as possible. The time interval is small enough for t when the values reach the peak and then start to become less again. The values will max out at the correct position. When comparing the two values, the time interval is correct when the position numbers match. Obviiiiii..... 

Lab 2 - March 3, 2015 - Free Fall Lab

Lab 2: Free Fall Lab

Brandon Elder

March, 5, 2015

Calculation of Gravity using Excel to Analyze Position and Velocity Data

Fig. 1 The l.5 meter column.

Purpose: The purpose of this free fall lab is to analyze the motion of a free falling body as it drops a known distance and then to record distance-time and velocity-time graphs in order to determine the acceleration (gravity) of the object, which should be 9.8 m/s^2.

Setting Up the Experiment:

The apparatus was set up for us in class already, it is a sturdy column that drops an object 1.5 meters (See Fig. 1). The free-fall body is held at the top by an electromagnet, and when released drops straight down. There is a spark generated that marks the location of the free-fall body every 1/60th of a second on a piece of paper hanging behind the object.

The drop of the object was performed in class by the professor. Each team was then given a length of paper that had the measurements from the spark generator. Each measurement was a dot that corresponded to the position of the falling mass every 1/60th of a second.

Procedure:

Fig. 3 Taped down paper

from free fall apparatus with

dots marking location of object.

The distance was measured in 

centimeters.

Step 1: Each team took the length of paper and taped it down on the desk (See Fig. 2). From there we measured the distance from the start to each dot in centimeters (See Fig. 3). We recorded the measurements for the first 15 or so measurements. All of our recordings were written down so we could transfer them to Excel next.

Fig. 4 Excel columns with
time and distance inputted
and calculated.

Fig. 2 The entire strip taped down.

Step 2: Open up a new Excel document and start inputting the data. The inputted data should look like the following table with the appropriate column headings added (See Fig. 4). The time cells (column A) will have a formula inputted. The first entry (A2) will be 0. The next row (A3) will be the formula: "=A2+1/60". Copy this formula down for as many rows as you have recorded distances. For our group, this was a total of 16 measurements. Column B is the Distance column and will not have a formula. The distance in centimeters will need to be inputted for all measurements (shown below).

Step 3: Collecting the Data: The next step is to set up Excel so that we can eventually plot the data on a graph to observe the graphs of position-time and velocity-time. As seen in Fig X, label the data columns respectively. There will be a formula inputted for the remaining columns as was done for column A. The formulas are summarized below. Column C represents the change in position between two measurements. Column D represents the mid-interval time, or the time every 1/120th of a second. Column E is the speed at the mid-interval times. (Fig. 5) below has the data that we collected.

Column B: Enter: "=A2+1/60" - Fill this formula down into all remaining rows in column B.
Column C: Enter: "=(B3-B2)" - Then fill this cell down through the end of your rows.
Column D: Enter: "=A2+1/120" - Fill down through remaining rows.
Column E: Enter: "=C2/(1/60)" - Fill down all remaining rows.

Fig. 5 Entire Excel data table with all formulas inputted into each column.

Step 4: Graphing your Data: The first graph to put together, using Excel, is the velocity vs time graph. This graph will allow us to calculate a best fit line that will go through the average of the data (See Fig.6). The slope of this line will be the acceleration, which should be a figure close to gravity, 9.8m/s^2 or 980 cm/s^2.

Fig. 6 Chart of  Mid-Interval Time (sec) vs. Mid-Interval
Speed (cm/s). The slope given is our experimental value
for gravity.

In this case, you can see that the slope of the line was 934.71 cm/s^2 (See Fig. 6) which represents the gravity in our experiment. In the graph of position vs time (See Fig. 7), the gravity is half of this number. So, once doubled, it becomes the gravity expression.

Fig. 7 Chart of Position (cm) vs. Time (sec) with best fit line.
The equation of the line is displayed and is our value for
gravity (acceleration) in our experiment.

Step 5: Analyzing the Data: The graphs of mid-interval time as well as the graphs of the regular time interval have the same acceleration. The experimental value of our acceleration, or gravity, is lower than the expected value of gravity, which is 981 cm/s^2. There are multiple reasons why these numbers that we come up with are smaller. Our experiment was not the most accurate experiment to test gravity. For example, there was no factor for air resistance in the experiment. There was also no factor to counter the friction force that is applied on the free falling mass. Lastly, there are some systematic errors introduced into the experiment from the column set up, such as calibration of the equipment.

Determining the Relative Difference:
Regardless of the above errors, we must come up with a way to estimate these errors and to estimate how big these uncertainties will be. One way of estimating these errors is by evaluating the relative difference. This can be done by performing the following calculation:

[(Experimental Value - Accepted Value) / Accepted Vale ] * 100% = Relative Difference %

In the above experiment, our value was: 

[(934.71 - 981) / 981] * 100 = 4.72%

Standard Deviations:

Lastly, we combined all the class data onto one spreadsheet in order to calculate the standard deviations of all our data. These calculations will help determine the accuracy of our returns and to see how reasonably close our experiment's results were to each other. We took the standard deviation of the mean. We collected all our g values that we took off of the graphs. We calculated the standard deviation from the mean by subtracting all the deviations from the mean of all the g values.

Fig. 8 Formula used to calculate
standard deviation of the mean.

952 in the below graph is the mean g value. The dev from mean column is a formula that subtracts the g value from the mean and puts the result into the dev from mean column. Then, to make all the data positive, the data is squared and entered into the Dev^2 column. The average of the this column is 1016.6. The square-root of this result is the standard deviation value (See Fig. 8) of 31.884 (See Fig. 9)

Fig. 9 Excel data table displaying entire class's g values and the
standard deviation value for all our data.

Conclusion:

This standard deviation tells us how spread out all of our data is. One standard deviation was 31.884. According to theory, 68% of the measured values should be within one standard deviation from the mean, while 95% of data falls within two standard deviations from the mean.

In review, we measured the acceleration of a free-falling object by putting together graphs of position vs time and velocity vs time. In both of these cases, the acceleration given was the amount of gravity acting on the object. We then collected all the class results and put together a table to determine what one standard deviation from the mean value would be. We used this to determine if our data was as precise (spread-out) as is customary, and yes, our data was in line with theory. 

The pattern among all g values was the same, they were all lower than the accepted value of gravity. As discussed above, the reason that these values are lower are due to systematic errors in the equipment, friction in the falling of the object, and air resistance. 

If these errors and assumptions were corrected, I am confident that the values of g would be much closer to 981 cm/s^2. 

Lab 1 - February 23, 2015: Finding a relationship between mass and period for an inertial balance.

Finding a Relationship Between Mass and Period for an Inertial Balance

Physics 4A, Lab 1:

Author: Brandon Elder

Date of Lab: Feb 23 & 25, 2015

Purpose Statement:   To determine the inertial mass of an unknown object by using the relationship between mass and period on an inertial balance.

Procedure:

Picture 1: C-Clamp is holding the inertial
 balance in place on the table. The piece of tape 
attached to the end of the balance is shown. 
The photogate is set up accordingly as per 
Step 1 in above procedure.

Step 1: The first step in this lab involves the set up of all equipment that will be used for the purpose of eventually measuring an object with an unknown mass. First, assemble the inertial balance to the edge of the table with a C-Clamp, then place a piece of tape on the end of the balance. Additionally, a photogate will be assembled at the same height as the balance so that when the balance is oscillating the piece of tape placed on the balance will pass through the beam of the photogate (Picture 1).

Step 2: Next, connect the computer with LabPro installed. Open the Pendulum Timer.cmbl file and connect the photogate sensor. When all equipment is connected properly the lab experiment will be ready for collection. The green button in the LabPro software is the collect data button.

Picture 2: LoggerPro recording a series of periods from the oscillating balance. 

Step 3: Data collection starts next. Start with a base line test of the equipment and software. The first test will have no mass. Hit the collect button and pull the balance to set it oscillating. The software begins to record the time for each period and plots the data on the given graph (Picture 2).

Picture 3: A total of 800 g added to the
inertial balance.

Picture 4: Recorded results from the various
mass used with the inertial balance.

Next, we added mass in increments of 100g (Picture 3) and collected the results. All results were recorded in the handout provided (Picture 4). After all the required mass measurements were recorded, we added our unknown object which was a cell phone, with 300g on top of it. The measured period for our unknown object was: .531381 seconds.

 

Picture 5: Steps from the handout with information on
adding the three new columns.

Step 4: Once all data was collected with the various mass sizes we added three new columns into LoggerPro. One column was for the mass plus the mass of the tray. One column was for the ln of T. The final column was for the ln of the mass plus the mass of the tray. Picture 5 details out the instruction for adding these rows and the purpose.

Step 5: Analyze the graph of ln T vs. ln (m + Mtray). Adjust the value of Mtray, through the "User Parameters", until the "Correlation (below) is close to 1.000. This step makes it evident to us that a range of mass values inputted into the settings will result in a correlation close to 1.000. In fact, there will be a minimum value and a maximum value that will yield the correct results. Picture 6 shows the values of the max Mtray range.

Picture 6: The graph of  ln T vs. ln (m + Mtray). The value of Mtray was at it's maximum here. 

Picture 7: The graph of  ln T vs. ln (m + Mtray). 
The value of Mtray was at it's minimum here. 

 



In our experiment, we came up with the results in Picture 7 for the minimum Mtray values.

Picture 8: Min and Max values for Mtray. The power-law
equation used to determine the range of mass for the
unknown object that was measured.

Step 6: It's clear at this point that the mass is related to the period by some power-law type of equation. Picture 8 below has the base equation, under the min and max values for Mtray. A and n are constants in this equation. After taking the natural log of both sides and rearranging you have the equation that will allow us to solve for the mass. T represents the period of the unknown object. A represents the value of e to the y-intercept. n is the slope of the line. Both values, A and n, can be taken from the graphs in Pictures 6 and 7 for the min and max values, respectively.

Step 7: With this information, the minimum and maximum range values for mass can be calculated. The math results in a minimum value of .498kg and a maximum value of .498kg. The actual mass of the phone was 197g plus the 300g weight for a total of .497kg. Our estimated weight was within .001 of the actual mass.



Summary: In conclusion, we were able to relate the period of on an inertial balance to the mass that was set upon it. Before we even performed any calculations, we made an assumption that the phone was very close to 200g. We had measured the period of the phone plus the 300g and noticed that the period recorded was very close to the period observed for the 500g mass. Upon further investigation, by performing the math, we concluded that we were correct.