Understanding of trajectories: Jose Rodriguez: Lab Partners: Kevin Tran; Kevin Nguyen; Lab Completed: 3/15/17

The focus of this experiment was to use our understanding of trajectories to predict where an object falling from a certain height will land on an angled plane.

This experiment was trying to prove that we have the potential to predict certain variables for an object that has a trajectory path with the knowledge of two direction kinematics.  For instance, what we tried to predict was where a steal ball that traveled on a guided path down a ramp would land on a 26.4 degree angled piece of wood in the path of the ball's trajectory.  To accomplish this we had to make prior calculations and experimental trials to collect necessary data to make an educated prediction of the distance down the angled piece of wood that the marble would land.  For example after we set up the metal-guided path that the marble took we did several trials to calculate the distance where the steal ball would land almost every time; as well, we calculated the height of the starting height of the ball's trajectory.  With this information we then calculated the time it took the ball to land so we could use that information to calculate the ball's initial horizontal velocity.  Lastly, we placed an angled ramp in the path of the ball's trajectory and took the measurement of the angle so we could at last predict where the ball would land on the ramp with the use of kinematics.  Note that the mass of the ball did not matter in this case.

To start this experiment we had to set up a path that the ball could take so we could record necessary data.  So, what we did was use the top of a table to build an Aluminium "v-channel" railway type system so the ball could travel on in a predictable and repeatable manner.  Then, we did two trial runs rolling the ball down the rails so we could see where it would approximately land so we could then place a carbon copy paper on top of white paper to mark the spot where the ball landed about five times.  The ball land in virtually the same spot five times.  Immediately after, we measured the height of the ball with a makeshift plumb device to hold the correct orientation and meter stick to measure the height.  Since the plumb also marked the initial starting position of the ball's trajectory, then we used the initial position to measure the distance to where the ball landed with the use of the carbon print of the ball's impact.  With that data we used two dimensional kinematic equations to calculate the ball's time to hit the floor and the ball's initial horizontal velocity at the point of initial free-fall.  The next step was to place the ramp down the trajectories path, take the angle of the ramp, and use mathematical geometry with kinematics to predict what distance the ball would land down the ramp.  Lastly, we let the ball roll down the ramp and hit the ramp on a carbon paper to mark the landing site so we could measure the distance.

This image shows the guided railway type system that we built so the ball could travel while on its path to the floor.

This image displays the string and paperclip style plumb that we made to measure the starting distance of the ball's trajectory.

 This is the carbon paper on top of the white paper ready to mark the ball's impact.

This image is where the ball marked the impact and it also displays the way we measured the distance of air travel.

This is a picture of the data and calculations of the ball's initial height, distance traveled, time in air, and initial velocity as well as the actual distance that the ball landed on the ramp.

This image illustrates the expression we used to calculate the value of "d" distance that the ball would land on the ramp and the actual calculations of our prediction of .444m.

This image shows the ramp that we used and the device that we used to measure the angle of the ramp to the horizontal.

In conclusion, our prediction of the distance down the ramp the ball would travel was .444-meters and the actual measured distance was .481 meters.  Considering that we did this experiment in a college class room, I fell like it was fairly accurate.  The difference was .037 meters or 37 centimeters.  I think that the inaccuracy may come because we didn't consider the friction force that the ball had while it was rolling down the ramp, or because of the simple tools that we used to measure the distance, height, and angle of the ramp used during the experiment.  For example, the friction force may change the initial velocity which in turn can affect the calculations for initial velocity of the ball, and the distance measured may be inaccurate because we used a worn down meter stick that has distorted cm units so it makes it hard to see exactly the correct values.  Although, I feel like the main point of this experiment was proven which is that we can use two dimensional kinematics to predict some characteristics of an object during trajectories.
   

Air Resistance and Mass: Jose Rodriguez; Lab Partners: Kevin Tran; Kevin Nguyen: Date Lab Completed: 3/13/17

The focus to this is experiment was split into two parts that were equally important.  The first part was to find a relationship between the force of air resistance towards an object during free-fall and the terminal velocity of the object. The other part was focused on using the data obtained through the experiment to model a formula for the relationship between both variables.

What we tried to measure was the terminal velocity of an object at different speeds so we could use that data to predict an exponential formula to model the relationship between velocity and air resistance.  To accomplish this task, we recorded the professor of our class drop several coffee filters from a fixed height with varying masses so we could record data to create an Excel sheet.  The Excel sheet was composed of velocity, force of air resistance, acceleration, velocity, and position during small time intervals to create several graphs to evaluate the experimental results.  

To begin with, this experiment required the class to walk to the auditorium of the school because we needed a tall enough height to complete it.  We needed the height because we needed to drop different masses of coffee filters from the same height so we could calculate the terminal velocity at different speeds.  We assumed that more mass would result in a faster terminal velocity so that's why we used different masses during the experiment trials.  Fortunately, the professor did the dropping of the coffee filters while one of our lap partners filmed the process so we could later use that data to create the graphs.  Also, the professor luckily did the work of weighing a collection of 50 coffee filters to obtain a mass of 43.9-g so we could calculate the mass per one coffee filter.  As well he obtained a black cloth long enough to measure more than two meter sticks and marked the length of one meter stick with blue tape so we could use that length to calculate the distance within the needed frame of reference.  Once we were done with that part we walked back to class and began to export the data from the videos into logger pro by using the distance reference and a set time interval to begin plotting points on the center mass of the coffee filter(s) during its decent to create a position vs time graph.  With those first variables velocity, acceleration, force net, and displacement were easily obtained through the Excel application.  Lastly, we used the final velocities of each trial per different masses and created a graph of the force of air resistance (which was equal to the product of mass and gravity per coffee filter) vs terminal velocities of each trial, created a parabola curve and obtained an equation for our model formula for the proportionality of air resistance and terminal velocity.  

This is an image of the professor dropping a coffee filter. Note the two pieces of blue tape, this distance is the one he indicated measured one meter.

This is an image that shows the data from the recorded videos of the drop trials.  The time starts at t=0 and the time intervals are 1/30th of a second, but we plotted points for every third frame of the video.

This image is of the position vs time graph and show the section of the plotted graph that we focused into to measure the objects terminal velocity.  That is the slope of the graph towards the end of the fall. 

Yet again, another picture focused into terminal velocity.

This portrays the calculations for velocity, acceleration, net force of air resistance, and distance with the help of Microsoft Excel.  Notice that the highlighted areas indicate terminal velocity of the object because the velocity doesn't increase with time.  Also, the fact that velocity doesn't increase indicates that we choose a small enough time interval. 

          This image indicates how the excel graph was formulated and also the initial mass calculations for each coffee filter.

Our assumed formula was Force of air resistance= Kv^n and as seen in this image we got a value for k of .0032 and n of 2.5114 with the use of the air resistance vs terminal velocity per trial graph with a parabolic graph.  

In conclusion, we did get a final result for our assumed formula for air resistance vs terminal velocity of a free falling object.  But more interesting is the fact that that formula helps prove the hypothesis that the force of air resistance at terminal velocity is equal to the force of the object falling.  I think this is why there is a terminal velocity even possible.  It seems like if not for the force of air against a falling object, then the object might increase it fall speed indefinitely.  Never the less, our experiment did result in a value for our velocity constant of K= .0032 and of the exponent n= 2.5114 that was very accurate in calculating the force for objects of different masses.  Our assumption was correct about the existence of a formula for air resistance even though there was many areas where our calculations were not exact.  For example, the fact that we recorded the objects free-falling with a lab top computer instead of a high powered effective camera and that the distance reference was makeshift could easily add errors of data to our calculations.  In fact, the graph above of the air resistance vs terminal velocities doesn't have all the plots exactly on the curve so this proves that the results are not as accurate as they can possibly get.  Also, the fact that we were inside a building during the drop trial might influence the results slightly because there was less air movement than if we were outside?  Never the less, the results do make some sense because it did prove to me that during an objects free-fall the encountered air resistance becomes equal to the force of the object to point of causing the object to have a constant velocity without acceleration.

Calculating Non-Constant Acceleration problems. Jose Rodriguez; Lab partners: Kevin Nguyen; Kevin Tran: Lab Date: 3/8/17

This experiment was intended to teach the students how to solve a non-constant acceleration physics problem with the help of Microsoft Excel in a numerical style approach in contrast to an integration analytical style approach.

What we tried to measure was what distance a 5000-kg elephant on roller skates (with an initial speed of 25-m/s rolling down a hill) and with a 1500-kg rocket strapped to its back would have after the rocket finishes a constant 8000-N thrust in the opposite direction with a fuel-burn rate of 20 kg/s.  To solve this problem we calculated the solution with two approaches.  The first approach was to use Newton's second law of gravity (F=ma) to find a function for acceleration (a=F/m) of the elephant with the rocket in respect to time, and then integrate it twice to get an equation for the position of the elephant for at any time (t).  The other approach was to use Microsoft Excel to calculate the distance at any time by entering specific formulas into the needed Excel cells.  Both approaches are based on physics and require mathematics to solve.

This experiment was completed by first looking at the analytical approach that was completed for us in the lab handout.  The integration began with the function of the elephants acceleration, and then the integration was done twice to get an equation for the position of the elephant at any time (t).  The fact that this was already done for us made this part very easy.  Then, the next part required us to use Microsoft Excel so that the program would do automatic calculations with the given parameters and provide us almost instantly with the distance when the elephant's velocity was equal to zero.  Therefore, we entered time, avg acceleration, acceleration, delta velocity, velocity, displacement, and finally distance. Lastly, we entered different delta time values of 0.05 s and 0.1 s to see how it impacted the results.

This is an image of the lab manual that illustrates the integration and the mathematical process that was used to solve for the position equation of the elephant for any time (t).

This image illustrates the final calculation for the elephant of 248.7-m.

This is an image of the Excel graph that proves in a numerical sense that when velocity equals zero, the distance is 248.6 meters.

This image illustrates the values at every 1/10 of a second.  Notice that the distance at 1.5 seconds is the same compared for every second.

This image is for every .5/100 th of a second.  Notice again, that the distance at 1.5 seconds also approximately matches the other time intervals of 36.1 meters.

This image of the work done by my lab partner Kevin Tran illustrates the calculation when some coefficients for the variables changed.  The changes: Initial mass= 5500kg; Fuel burn rate=40kg/s; Thrust force=13000N.  Note: the distance was less.

In conclusion the distance that the elephant traveled due to the rocket's thrust was only 1/10th of a meter in difference.  The analytical approach shows a distance of 248.7m in comparison to the 248.6m that the numerical approach yielded as seen in the prior images.  This is mainly due because the picture I captured for the time interval of every 5/100th of a second was not of the sections where the velocity was equal to zero.  If I would of done that, then I think the distances would of been more precise to one another.  Although, there was a sense that the numerical approach will be faster to calculate when the given variables that were given change, of possibly even if new variables are introduced.  Note that the last image shows a result for a change in magnitudes of some variables and that the distance is less.  Sadly, we did not capture an image of the distance with the new changes so we don't know if that answer matched an answer using the numerical approach.  Also, I feel like there is an equal change to make a mistake in both calculations.  For example, the analytical approach makes a mathematical error easy, and the numerical approach lends to errors during the entering formulas part of the lab.  Consequently, if the numerical approach is entered correctly I feel like it would be a much easier and accurate technique to use.      

Determining the density of two cylinders and calculating the propagated uncertainty of the experimental value. Jose Rodriguez; Lab partners: Kevin Tran; Kevin Nguyen. Lab Date: 3/6/17

The purpose of this experiment was to calculate a density value for each of two unknown metal cylinders and to calculate the propagated uncertainty of each experimental density value with the help of a mathematical derivative of the formula for Density=Mass/Volume.


There were several measurements that we needed in order to complete this lab.  To begin with, we had to record the mass, height, and diameter of each cylinder so we could use those values to calculate the density of each cylinder.  Therefore, the density formula we choose to use was density=mass/volume since we had the needed components for each of the cylinders.  In order to measure the height and diameter of each cylinder we used a vernier caliper that has a micrometer.  This instrument allowed us to record a more precise measurement than a meter stick.  As for the mass, we used a digital balance.  The technique we used to calculate the propagated uncertainty for our experimental values of density required use to take the natural log of our density formula, take the derivative of each term, calculate a result, and then multiply the result with our respective experimental density values to obtain a propagated error value for each cylinder.

To start, we recorded the mass, diameter, and height for each cylinder.  A digital balance was used to record the masses, and a vernier caliper was used to record the heights as well as the diameters.  A vernier caliper micrometer allows one to record a more precise reading to the 1/100cm.  Once we recorded those values, we plugged the data of each cylinder into our respective formulas and produced experimental density values.  For the density values we used Density=Mass/Volume.  Then, to calculate the propagated uncertainty we had to first calculate the propagated error for each experimental density value.  So, first we had to derive a formula from the density formula by taking the natural log and by taking the derivative of the natural log to finally fill in the respective values to calculate the propagated error of each density value.  Although each propagated error value only gave us the percentage of uncertainty for each respective experimental density value, so we had to multiply each propagated error value to its' respective experimental density value to find the propagated uncertainty values of each cylinder.  All of these calculation were performed by hand.

This image was taken when we were measuring the diameter for one of the cylinders.  Note that the vernier caliper is the instrument being used to measure the diameter. Also note that the micrometer section of the caliper is the smaller measuring scale that appears like the light is reflecting off of it in the picture.

This is an image when we were measuring the height of one of the unknown cylinders.

This is an image of the calculated experimental density value for cylinder 1. Note the formula for volume and density.

This is an image of the calculated experimental density value for cylinder 2.

This is an image that illustrates the derived formula of density and the calculations for the propagated uncertainty of cylinder 1 as well as the experimental density value for cylinder 1.

 This is an image that illustrates the derived formula and calculations for the propagated uncertainty of cylinder 2.


In conclusion, I had some assumptions before the experiment.  One was that I thought the final calculation would not be completely accurate because the vernier caliper's measurement units on the micrometer section were very small, so I think that allowed room for error.  Another, was that the balance we used is so sensitive that I think mere wind movement may have affected the mass value that in return may fault the experimental density values for each cylinder.  And, my last assumption was that we might make a mathematical mistake and therefore get an inaccurate result.  Although, the final experimental density values were not verified because we didn't find out exactly what kind of metals the cylinders were to compare the accepted values to the calculated experimental values of their densities.  Never the less, the experimental density result for cylinder 1 was 7.33+-.08g/cm^3 and the result for cylinder 2 was 2.90+-.03g/cm^3.  It's important to note on the cylinders' density calculations that even though they had approximately the same diameters and different heights, the shorter cylinder was calculated to have a higher density value.  As for my assumptions, they could not be confirmed or denied because we didn't find out what kinds of metals the cylinders really were.  Also, The propagated uncertainty for each cylinder density result is the number after the plus & minus sign following the larger value of each respective experimental density value.

    

Determining Gravity With The Help of Technology and Physics; Jose Rodriguez; Lap partners: Kevin Nguyen and Kevin Tran; Lab Date: 3/1/2017

This experiment was composed of two parts and the students worked in groups of three to complete it.  The first part of this experiment was to try to prove that the accepted value for acceleration on Earth is 9.81m/s/s with the help of a simple apparatus, physics, and technology.  The second part honed into the errors and uncertainty section of an experiment; it required us to calculate each group's deviation of their experimental acceleration value from the class's mean average of experimental acceleration values, the class's standard deviation of the mean average, and the normal distribution of the class's experimental values of acceleration.

What we tried to measure was how much distance a free falling object (with no force but gravity imposed onto it) travels in consistent time intervals, and how much standard deviation each group's experimental acceleration values had from one another.  To accomplish this experiment, the use of an apparatus that is designed with a 1.5m falling distance, a free-falling object, a spark-generator, and a spark-sensitive tape provided the data that allowed us to generate computer graphs for distance vs time and velocity vs time that aided us in our pursuit to obtain a value for Earth's acceleration.  The logic behind this experiment was based on calculating the distance over a consistent time period of the falling object.  For instance, the spark-sensitive tape provided that information because we measured each gap and we knew that the spark-generator provided data for every 1/60th of a second.  Then, we will transfer that data onto the computer application Microsoft Excel to find the velocity of the object during its' fall at any time because the graph will display the slope-intercept equation of the velocity so we could obtain an experimental value of acceleration if we take the derivative of that equation.  Then, the second part of the experiment required us to calculate each group's deviation from the class's mean experimental acceleration value that lead to the standard deviation value of the class in whole to each group's experimental acceleration value to on another.

This experiment began with the professor's help because he assembled the apparatus so it was ready for use.  The apparatus was assembled by placing a (white-colored) spark-sensitive tape vertically on the main support beam, by placing a free-falling object in front of that tape at the top that was held in place by an electromagnet, and by connecting the power source to an outlet so that the spark-generator had power to produce an electrical charge.  Then, the professor flipped a switch so that the electromagnetic magnet disengaged and the object fell while a spark-generator marked its position every one-sixtieth of a second till it reached the bottom.  Next, we measured the distance between the marked sections with a meter stick (using the centimeter units) and recorded the data. Simultaneously, we set formulas on Excel to calculate time, distance, displacement, mid-interval time, and mid-interval speed to better aid in the graphing process and entered the distance we had measured as well as the time for every 1/60th of a second.  For the mid-interval time we used every 1/120th of a second and entered that value into Excel; the mid-interval speed was calculated as well.  Immediately after that, we created a distance over time graph and a mid-interval speed over mid-interval time graph which displayed the slope-intercept formulas that handed us our experimental acceleration value in the units of cm/s/s. Lastly, for the second part the professor asked all the groups for their experimental value and found the mean average, deviations of the class's mean average, and the standard deviation of each group's experimental value in respect to one another so we can debate whether or not the values were precise and therefore accurate.

This is a picture of the apparatus that was used to capture the distance vs time descent of the free-falling object.  Note that the professor is lifting the free-falling object so the electromagnet holds it at the top.
Also, the spark-generator power interface can be seen to the right of the apparatus; it is blue.

This is an image of the spark-sensitive tape with the meter stick laying along its side. The black dots on the white tape are where the spark-generator made its mark to capture the position of the free-falling object.  The distance between the black dots after the first mark is the distance we measured for the displacement of the object every 1/60th of a second.

Here one can see the data entry of the measured distance between the time interval of 1/60th of a second and mid-interval time of 1/120th of a second.  The mid-speed was calculated by entering a formula into Excel the divides the displacement over 1/60th of a second.

This is an image of the distance vs time graph and the equation for velocity in respect to time is seen in the middle of the image;the experimental acceleration value can be obtained by the derivative of this equation's slope.  Note also that the slope of the parabola is the velocity of the free-falling object.

This graph is composed of the speed on the y-axis vs the time on the x-axis and its' slope is our experimental value for acceleration.  The equation's slope that is visible in this slope-intercept format style equation is the free-falling object's acceleration at any time.

In conclusion, I assumed that we would automatically get the experimental value for Earth's accepted value for acceleration of 9.81m/s/s but that was not the reality because our experimental acceleration value was 9.58m/s/s. Also, I assumed that the speed between time intervals would be constant, but as one can see from a previous data picture it changed throughout the fall of the object so it was inconsistent. On the other hand, the pattern of the speed may be inconsistent because of other forces that were active on the free-falling object during the fall.  For example, it may be possible that there was some kind of friction applied on the object as it fell that we didn't anticipate before we began the experiment.  Or, possibly there was some kind of electrical resistance force applied onto the object when the spark-generator produced a spark to mark the object's position.  Although, a more logical reason for the discrepancy of the experimental value is that the meter stick that was used is not precise enough to make accurate measurements, or my eye sight may have failed when we measured the displacement value so the following calculations carried the error throughout the entire calculations. The meter stick has a standard deviation equal to +- 1/100th of a meter so it's not the most accurate tool that we could have used for the measurement for the displacement.  For example, if we had access to make some kind of laser measurement device I think we could have more accurate results.  The absolute difference of our calculated result to the accepted value of Earth's acceleration was 9.58m/s/s - 9.81m/s/s=-0.23m/s/s; the relative difference was (9.58m/s/s-9.81m/s/s / 9.81m/s/s) x 100% = -2.3%. Consequently, the graphs did give us equations for velocity and acceleration for the object at any given time and the velocity equation can be derived to obtain the acceleration value; as for the acceleration equation, its given slope is the experimental acceleration value.  As for the standard deviation of the mean average of the class, it was recorded as +- 6.1186m/s/s as seen in two pictures below.  The standard deviation value means that the class's group's average experimental values for acceleration were about 6.1186 units far from the average value in either a positive of negative direction.

This image proves that the magnitude of the speed for mid-interval is the same for the average speed for any displacement value for constant acceleration.  Note that this time interval is in between 2.3 - 3.8 seconds and that the average speed value matches the value for mid-interval speed at any time.  That can be seen in this picture.

This image illustrates the classes standard deviation value of 6.11860732m/s/s that can be calculated by taking the square root of 37.43735556 which is the average deviation value squared.  Also note that the class's mean average for our experimental acceleration value was 961.5m/s/s.

27-Feb-2017: A power law for an inertial pendulum. Lab Partners: Kevin Tran; Kevin Nguyen.

This experiment was focused primarily on teaching students how to use an inertial pendulum, how to find correlations between masses and periods of oscillations, and on how to interpret the data to obtain a graph in the form of a mathematical point-slope formula.

This is an image of the inertial pendulum and how it was set up with the monitoring sensor of the Logger Pro application. NOTE: This is also a picture taken during the part of the lab that we had to find an unknown mass.

This experiment began with the set-up of the inertial pendulum and a monitoring system application known as Logger Pro that has the ability to sense periods of oscillations, make graphs, and derive a mathematical formula with the ability to derive one accurate enough to obtain unknown masses of different objects.  Then, known masses (in the form of molded brass) were placed on an inertial pendulum and the periods of oscillations were recorded in order to create several graphs.  Although, the data in respect to the mass of the tray of the balance used had to be manipulated so that we could derive fairly accurate formulas for different ranges of precision.  Next, an iphone and a wallet were placed on the balance and the periods of oscillation were recorded so that we could use our formula to conclude the masses of the two items used.  Lastly, the data that we gathered from the experiment proved helpful in the cause to find the unknown masses of the iphone and the wallet.

This is an image of the natural log graph recorded with the use of the data that was obtained during the experiment.  Note that the formula (in the form of point-slope) is visible to the right of the image. The time axis is in respect to the mass on the balance and of the tray of the balance; the speed axis is in respect to the period of oscillation.    

                                    Another image of the data recorded during the experiment.

This is an image of the calculations used to determine the masses of the iphone and the wallet.  On the top of the image one can see the mathematical formula used for this experiment.

In summary, we used an inertial pendulum and an application named Logger Pro to create data that was later used to create a formula that can facilitate the approximation of an unknown mass without the need of gravity.  The data was manipulated and the results concluded to be fairly accurate.  As for the possibility of errors, they are very likely since the chance of human error and technological error feels like a high possibility given the experiment was not strictly controlled.