Lab #5 (trajectories) Daniel Guzman

Daniel Guzman

Physics 4A.

Lab#5 Trajectories




The purpose of this lab was to understand the phenomena of projectile motion and the interesting components of it. For instance, we wanted to understand how the x and y components of the motion of a projectile are totally independent, and how they affect and describe the motion of the projectile.

The apparatus of this experiment consisted of track a sphere, a sheet of carbon paper, a sheet of white paper, a woof board, and a clamp. The set up of this experiment was somehow easy due that the same set up was used for most of the experiment. The first thing was to set up the track were the little sphere would roll, in order to set up the track we used a clamp to keep it in place, then on the track we used tape to mark where the sphere was going to start every time. After setting the track we proceeded and put the carbon paper underneath the white sheet of paper so the position of the sphere would be marked as it hits the floor. For the second part the set up was pretty much the same the only thing that would change was that instead of putting the carbon paper on the floor it was put on the wood board that laid against the table at some angle, so the ball would now hit the board instead of hitting the floor at the end of its trajectory.

Experimental procedure.
The first thing to do for this particular laboratory was to set up the apparatus, once the apparatus was set up, we proceeded and let the sphere slide down the slope and recorded 5 different times where the ball landed on  the white sheet of paper using the carbon paper. Once these recording were made we proceeded and measure the horizontal distance that the ball traveled. Once this is measured one also had to measure how high was the ball when it stating to move as a projectile, after these measurements are done one could proceed to second part where one would use the wood board, so the sphere would hit the board instead of hitting the floor. For this part some calculation had to be made so one would know where the sphere should hit the board and based on this one would know where to place the carbon paper and the white sheet of paper.


Calculation for first part

calculations to find the initial velocity in the x direction once the sphere starts moving as a projectile

Calculations for part two to find the distance where the sphere would hit the board, the angle was measured using a cellphone and the vox was obtained for the first set of calculations

comparing the average value to the theoretical value obtained

The average from the experimental value was 0.89978 meters and the theoretical found was 0.904. when comparing them I used the percent error and found that the percent error between them was 0.4%,

Calculated propagated uncertainties for the first and second cases


For the first case the propagated uncertainty was calculated for the v initial in the x direction and i calculated the uncertainty for delta x to compare to the uncertainty used when finding the propagated uncertainty for vo initial in the x direction.

for the second case the propagated uncertainty was calculated for the distance where the ball hits the board when it is moving as a projectile.

Conclusion : for this experiment the experimental distance and the theoretical distance where the sphere should hit the ball where very close to each other, one can support this because the percent error between the experimental result and the theoretical result was only 4 percent, which lets one assume that the experiment was carried out correctly due that the results obtained in the first part of it where very important when deriving the expression that would allow me to calculate the theoretical displacement. The sources of error in this experiment might come from how i measured the position of the sphere when it hit the sheet of white paper that was on top of the carbon paper. Another source of error might come from measuring the angle correctly due that the distance where the sphere is going to hit it depends on it.

Lab # 4 Modeling the fall of an object falling with air resistance (Daniel Guzman)

Daniel Guzman

Lab#4: Modeling the fall of an object with air resistance

Physics 4A.



In this particular lab we were trying to understand and model the fall of different filter papers,, and how air resistance play a very important role on the filter papers motion as they were falling down.


The apparatus of this experiment consisted of a laptop, which has a very interesting software called logger pro. This particular software has many different tools to model different phenomena. In this particular experiment we used the tool called video capture, which was used to record the free fall of different filters, to then analyze them and extract valuable data to obtain the results desired.

The experimental procedure for this experiment was very simple, but very interesting, the first thing that needed to be done was to learn how to use logger pro's video capture due that the entire lab would depend on the videos recorded. Once the class knew how to use the video capture feature of logger pro the entire class went to the technology and design building were the lab took place. In this building the professor hang a black rough from a high of approximately 6 meters, so the filter papers would be easily seen  as they were falling. After the rough was hang we run a trial just to see if we were able to record appropriately the motion of a coffee filter . Once each was able to record correctly the motion of the coffee filter, the professor started telling us how many filters were going to be dropped at a time and the time at which he was going to release them. First, we recorded the motion of one filter paper and saved the video on logger pro immediately to proceed to the next trial were two coffee filter papers were dropped. This process was repeated until six coffee filter papers were dropped. Once all the videos were recorded and saved we had to scale any portion of the wall from which the filters were falling down, so the software would approximate what was the real distance that the objects traveled to obtain the position vs time graph.


Data and Graphs

How it was mentioned before once the scaling on the video is done one would obtain a position vs time graph which is very important because one can obtain the terminal velocity of the coffee filter by setting a linear fit on the graph and this would give us a slope which is the terminal velocity of the filter.

Here is shown the linear fit that is set on the graph and the slope of it is the terminal velocity of the coffee filter.

After the linear fit has been set for every single video one would obtain the terminal velocity for 1, 2 ,3 ,4, 5 and 6 filters. then one would find the force which is mg. then these is plotted in logger pro.

Plot a point for this plot is ignore in order to obtain a better correlation.

when we set a power fit on this plot we obtain the values we wanted to obtain. on this graph the values are A and B. Once these values are obtained f resistance can be found because all the other values are known.

Part 2.

Excel data tables to model the fall of the coffee filters. using values found in the first part (red)

filter paper #1

filter paper #2

filter paper # 3

filter paper 4 was skipped, video recorded wrongly

filter paper #5

filter paper #6

comparing results and discussion

The results obtained when modeling it in excel using the values from part one were good for some of the filters the terminal velocities were somehow close enough, the percent errors range from 0.68 to 2.8, which lets one infer that there are some sources of error in the results. some of the sources of errors might come from using the wrong video with the wrong number of filter papers, which occurred to us so we had to eliminate a point. Nevertheless, i was very impressed when i obtained my final results because it was a very interesting a approach due that all we used was a video and we were able to come with a model that works and is somehow accurate when modeling the fall of an object, another interesting thing was that the only apparatus used was a computer with a very interesting software that provide one with really god tools to model different phenomena. the sources of error might come from using the wrong video, or miss calculating a values when using excel.

Lab # 3 Non constant acceleration problem/activity (Daniel Guzman)

Daniel Guzman

Lab#3 Non constant acceleration problem/ activity

Physics 4A.


The purpose of this problem activity was to approach a kinematics problem numerically, instead of solving it analytically, due that there are some problems, which are very tedious, time consuming , and sometimes very difficult when they are solved analytically;  due that the mathematical functions of this problems cannot be integrated sometimes. Nevertheless, when problems are solved using the numerical approach, one is likely to find or get the desired answer due that one is able to use the data provided in the problem in different ways until the answer is achieved. For instance, in this problem only a few values were given, and using them correctly and applying the correct logic we were able to solve the problem obtaining the same results as it was to be dome analytically.

There was not apparatus in this particular experiment, the only piece of equipment used in this activity was the computer, which has Microsoft excel.

Experimental Procedure: the first step in this experiment was to  analyze and read the problem carefully, so one would know what the problem was asking and what important values were given. After the problem was read and analyze we would look at the analytical approach of solving this problem in, which we would find out that solving it analytically is very tedious and time consuming. Once the problem is understood and is solved analytically, we could proceed and solve the problem numerically. To solve the problem numerically we had to use and understood how the data provided in the experiment could be used to solve it, after this is understood one can use excel and list all the values provided such as, the mass of the elephant  the mass of the rocket, the mass of the elephant, the initial of velocity of the elephant, the thrust of the rocket and the fuel rate of the rocket. Once this values are listed and organized I proceeded and made a table of other values which were : time, acceleration, average acceleration, change in velocity, average velocity, change in displacement and displacement. After this is done I put the correct values for vo and displacement and from here I started to make the respective calculations which allow me to get the desired answer. only the first calculations for the first values have to be done because since one is using excel, excel would do the other calculations as long as the calculations and equations for the first calculation are correct. for time I used the value established for the interval and added the previous number to it. for the acceleration I used the force/ (total mass- burnt rate(t)). for average acceleration I used  the average between the last two previous acceleration * time. for delta v I used the sum between the previous two average acceleration and multiply it by time. for velocity I used the change in velocity+ the initial velocity, for v average i used the sum of the previous two velocities and multiple by time, and for delta x I used the average of velocity and multiply it by time and for position i used the sum of the two previous values for delta x.

Data tables : the spread sheets were made making the time intervals : 1 second, 0.1 seconds and 0.05

Conclusion

1. compare the results you get from doing the problem analytically and doing it numerically?

Doing the problem numerically is much easier due that the integration sometimes can get really difficult and might be tedious. Another problem that one might encounter when doing it analytically is that the function cannot even be integrated. therefore approaching the problems numerically  is much easier. The results obtained doing analytically and numerically give one the same result.
2. How do you know when the time interval you chose for doing the integration is small enough ? how do you tell if you didn't have the analytical result to which you could compare your numerical result?
To know if the time interval is small enough one how to see if the change in velocity stabilizes or reaches a constant number, or if the change in this result is small enough were the changes are irrelevant or insignificant. when this is reached the time interval has been made small enough.
3. Determine how far the elephant would go if its initial mass were 5500 Kg, the rocket mass is still 1500 Kg, but now the full burn rate is kg/sec and the thrust force is 13000 N
when the time interval is 1 second the elephant goes 163.53 meters and when the time is 0,01 the displacement is 163.84 meters.

Lab#6 (Propagated uncertainty in measurements) Daniel Guzman

Daniel Guzman

Lab#6 propagated uncertainty in measurements

Physics 4A.


The purpose of this particular experiment was to measure the density of two different cylinders made out of different materials, to then calculate the propagated uncertainty of the measurements made, to then compare the propagated uncertainty of the densities of the cylinders to the accepted value.

Apparatus (description)
The apparatus for this experiment was very simple due that it only consisted of a caliber and a analytical balanced. The caliber in this experiment was used to make very specif measurements of the diameter and the height of the cylinders, and the analytical balance was used to measure the mass of the cylinders. These very specific measurements were used to find what was the density of the two different cylinders.

Experimental procedure

In this particular experiment the procedure was quite simple due that it consisted of making simple measurements. The first part of this experiment was to obtain two different cylinders, one made out of aluminum and the other made out of zinc. After having obtained the two different cylinders we had to use a caliber to make specific measurements on the cylinders. one of them was the height of the cylinders and the other one was the diameter. After having obtained these measurements on the cylinders, one had to record them in a data table, and at the same time write the uncertainty that the caliber gives when one uses it measure, in this case the uncertainty was +- 0.01. After having the measurements for the diameter and height we also had to measure the mass of the cylinders; therefore, we used an analytical balance where we put each cylinder separately and obtained a measurement for the mass, which had an uncertainty of +- 0.1. Once these measurements were collected we proceeded and calculate it the total partial differential of the density for the cylinders separately, due that the cylinders are made out of different materials, so is important to find the propagated uncertainty for the cylinders separately.

Data Table Cylinders(Aluminum and Zinc)

List of calculated results

For the first set of calculation I used the values of the measurements to calculate what would be the density of the cylinder. After having calculated the density of the cylinder I took the partial derivative of the equation of density by taking the ln of both sides of the equation and then I took the derivative of it, which gave me the expression that is boxed in the calculations above.

 Once the for the total partial differential was found I proceeded and used the values that I measured, for the cylinder made out of aluminum and the cylinder made out of zinc, to solve for dp, which represents the uncertainty that exists in the density found experimentally for the cylinders

These final calculations are the calculations for the conclusion part which are the relative error between the accepted values and the experimental values, and the relative uncertainty, which represents the precision of the measurements.

Conclusions
For the aluminum cylinder the calculated value for density and the propagated uncertainty was 2.74 +- 0.04.g/cm^3, The accepted value for the density of aluminum is 2.7 g/cm^3, which lets one infer that the measurements done for the aluminum cylinder were very precise due that we obtain a value relatively close to the accepted value for the density of aluminum. On the other hand when comparing the density for the zinc cylinder I found out that the measurements made for this cylinder were not precise because the value obtained for the density of it is far from the accepted density of the zinc.

The source of error in this experiment might come from the cylinders themselves because they are probably alloids, which is a mixture of different metals and this would definitely affect the results, another source of error in this experiment specially in the final results might come from a miss calculation, which would affect the desired results.

Laboratory #2 Free Fall Lab Determination of Gravity (Daniel Guzman)

Free Fall Lab Determination of Gravity

Daniel Guzman

James Okamura

Alejandro Rodriguez

March 1, 2017

Part #1

In this experiment one was trying to determine the acceleration due to gravity of a falling object that started from rest. In order to determine the acceleration due to gravity that the object was experiencing, one had to measure the position as it was falling down with respect to time. Once the position of the object was recorded one could find the change of position of the object, which is very important when finding the velocity at which the object was moving. Once one knew thee velocity of the falling object one could proceed and graph velocity vs time, and position vs time. When the graphs of position and velocity are graphed one can proceed and determine the equations of the line from, which could derive the acceleration due to gravity that the object was experiencing.

Description of the apparatus and experimental procedure
The apparatus used for this experiment consisted of a sturdy column, a 1.5 meter piece of paper, an electromagnet, a falling object and a spark generator. The object or falling body was held by an electromagnet , which helped the object to start from rest, once the object is falling down a spark generator starts to generate sparks every 1/60 of a second which are recorded on the 1.5 meter paper that is attached to the sturdy column.
Once the position of falling object is recorded on the 1.5 meter paper one would proceed, and make measurements of the dots that are in the paper, once the distance between the dots is measured one could proceed and record that data in excel spread sheets. One had to record in excel : the position with respect to time of the object, the mid time interval between each dot and then calculate the velocity in the time interval. Once this is done one could proceed and graph velocity  with respect to time and position with respect to time. Once this is accomplish one can determine the acceleration due to gravity from the equations of the lines.

Data Tables/ Graphs 

Graph of velocity vs time of the falling object. The equation of the line is displayed in the graph because is very important when finding the acceleration due to gravity of the object

Graph of position vs time of the falling object. The equation of the line is displayed because is very important at the moment of finding the acceleration due to gravity that the object was experiencing.

Questions/Analysis 

1. show that for constant acceleration the velocity in the middle of a time interval is the same as the average velocity for that time interval?

2. Describe ho you can get the acceleration due to gravity from the velocity/ time graph. Compare the result with the accepted values?

Acceleration due to gravity can be obtained from the velocity vs time graph by taking the derivative of the equation of the line, which models the velocity vs time of the falling object.

3. Describe how you can get the acceleration due to gravity from your position vs time graph. Compare your results with the accepted value

As mentioned before the equation of the line is very important when one wants to find the acceleration due to gravity. In this particular case one would take the derivative of the equation of position with respect to time of the object. Once one has the velocity with respect to time one would take the derivative of the equation which will give one the acceleration due to gravity.

Part # 2 ( Analyzing the class data for g )

Data Table 

 Questions and analysis

1. what pattern if any if any is there in the values of our values of g?

The values obtained for gravity in our group were relatively far from the accepted value of gravity which is 9.82m/sec^2 which tells one that the systematic errors really influenced the results obtained. The results obtained for the gravity were in a range from 957 cm/sec^2 to 960 cm/sec^2, which once again tell one that the results obtained are far from the accepted value.

2. How does our average value compare with the accepted value?

The average value obtained for the acceleration due to gravity that our group got was 961.356 cm/sec^2 which, lets one infer that the value was far from the accepted value of gravity, which is 9.8m/ sec^2. Even though the percentage error between the accepted value and the average obtained is only 2.25% it is far from being precise.

3. What pattern if any is there in the class' values of g?

All the values obtained in the class for gravity were relatively far from the accepted value of gravity, which lets one infer that the systematic errors definitely affected the data and the results obtained. The only pattern that I found in the results obtained by the class is that all the results are within a range that goes from 950 cm/sec^2 to 969.7 cm/sec^2

4. What might account with any difference between the average value of your measurements and those of the class? Which are systematic errors? which are random errors?

The difference between of the average values that I obtained for gravity and the class's might come from different factors; for instance, the position of the falling object marked in the paper might have a very little variation, the differences can also come from the measurements made by each group of the distance that there is between each dot, because some groups or even my group rounded the measurements, which will generate a difference between the average values.

Some systematic errors in this particular experiment might come from the measuring devices such as the apparatus itself or the assumptions that one made. for instance, assuming that friction did not play an important role as the object is free falling. Another systematic errors could come from how the data was manipulated. for instance, a miscalculation might change some results, that will definitely alter the expected or final result. 

Some random errors can come from how the measurements between the dots were made because is almost impossible to make each measurement with the same precision as the last one. This random errors are errors that will be propagated or that will be inevitable, even if the best physicist would do the experiment    

5. Summary of the important part of this experiment. What were the key ideas? What were you supposed to get out of it?

The most important part of this experiment was to obtain the acceleration due to gravity that a falling object was experiencing as it is moving. However, to obtain the acceleration due to gravity one had to go through a series of steps that led one to obtain a valuable relatively close to the accepted one. For example, one had to graph velocity vs time and position vs time and from these graphs one had determine the value of the acceleration for the graph. This is a very interesting approach because one can find the acceleration of any object as long as one has the position with respect to time of the object. The interesting feature of these experiment was the numerical approach that one had to have in order to obtain the value for g because it leads one to follow a certain series of steps that eventually will lead one to achieve what one was looking for in the experiment.

Lab 1 : Finding a relationship between mass and period for an inertial balance. ( Daniel Guzman)

Finding a relationship between mass and period for an inertial balance.

Daniel Guzman

James Okamura

Rodrigo

February 27, 2017

       What this experiment is trying to accomplish is to understand the mathematical relationship between mass and the oscillation period of an inertial balance, to then test, the relationship that exist by measuring the period of unknown masses, so the mass could be calculated for the objects.

       In this lab we were trying to measure the different unknown masses of three different objects: an inertial balance, a calculator, and a tape dispenser. In order to calculate the masses for these objects, we measured the period of oscillations of the analytical balance using different cylindrical wights, which ranged from 0 to 800 grams. The period of oscillations was measured for each weight individually.

       The apparatus consisted of a photo gate a clamp and an inertial balance. The inertial balance was clamped into the table and a piece of tape was put on the inertial balance, so the tape will break the light and the photo gate could record the period of oscillations.
     
 The first part for  this experiment was to set up the apparatus correctly and measured the period of oscillations for the cylindrical weights. Once the period of oscillations was measured we had to use the power law equation T=A(m+mtray)^n, which is going to model the data; nevertheless, before modeling the data the ln of the equation has to be taken, so the data can be modeled as a straight, with an equation in the form y=mx+b, which is a straight line. When the ln of the equation is taken the equation would be lnT=nln(m+mtray)+lnA.  After obtaining the equation in this form one can proceed and modeled data as a straight line, on the y axis we used lnT and on the x axis we put (m+Mtray), which gave us a straight line from where we extracted important information, such as the y intercept of the graph and the slope of it. The y intercept of the graph would give us the amplitude of the oscillation and the slope of the graph would give us the exponent of the equation, which are crucial in order to find the unknown masses. However, to find the unknown masses we had first to determine or find the value for the tray. This particular values was found using different numbers when grpahing, until we found the number that will yield the better correlation of the line, which gives us a hint of how the values plotted tend to vary together. Having this Valuable infromation from the graphs we proceed and found the unknown masses of the objects listed previously.

Calculation to see if the period of oscillations is relatively close to the period that one measured using a stopwatch and data table

Graphs of the data

Graphs Analysis  (Graphing process and interpretation of the graphs)

How it was mentioned before the data was modeled using the equation T=A(m+Mtray)^n. However in order to obtain a straight line the log of this graph had to be taken, which will yield the equation of a straight line that looks like this : lnT=n ln(m+Mtray)+lnA, where n represents the slope of the line and lnA represent the y intercept of it (the slope of the line is the exponent for the power law equation and the y intercept represents the amplitude of the oscillation). The key point when analyzing the graph is to obtain almost a perfect correlation, to obtain a good or almost perfect correlation we had to modify the mass of the tray until we found a good correlation, and once this is done we had to come up with a range of values that yielded a very good correlation; these sets of values are going to have an upper bound and a lower bound which will have specific values for the y-intercept and the slope of the line. These specific values found in the upper and lower bounds were used to find the unknown masses of : the calculator, the tape dispenser and the tray.

Calculations using lower and upper bounds to find the unknown masses : The values obtained in the graphs are used here to calculate the unknown values for the objects. A set of different values is used for the upper and lower bound.

Table of results for the unknowns

  The sources of error in this experiment could possible come from the distribution of the mass that was put on the tray;for instance, putting the calculator vertically might yield a different period of oscillation than putting it horizontally. Another source of error is the amplitude that the inertial balance had when recording the period of oscillations for the unknown masses, due that it might record a slightly longer or shorter period of oscillations.

Conclusion

        After having completed the lab one was able to calculate the unknown masses of three different objects using the relationship that exists between the period of oscillations and the mass of any body or object. The calculated unknown masses were relatively close to the actual masses of the different bodies, and the percent errors calculated when using the upper and lower bounds were lower than 10%, which let one infer that the results were relatively accurate.