Finding the Moment of inertia of a uniform triangle: Jose Rodriguez: Lab Partners: Kevin Tran; Kevin Nguyen: lab completed 5/22/17

The purpose of this lab was to determine the moment of inertia of a right triangular thin plate around its center mass, for two perpendicular orientations of the triangle.

To achieve this experiment we used an apparatus in combination with Logger-pro that allowed us to measure the system's angular acceleration, made some measurements, and used the parallel axis theorem in order to easily calculate the inertia of the triangle around its center of mass.  For instance, we used force equations to derive a formula for the inertia of the system in combination with the given angular acceleration from the graph of Logger-pro and then calculated the inertia with the parallel axis theorem easily.  Then, with the differences in the trials of the calculated inertia per trial we were able to obtain a quantity for the inertia value of the triangle upright and sideways.

First, we set-up the apparatus and measured the triangles dimensions before we hooked up Logger-pro to begin the first trial of just the apparatus spinning.  Next, we made sure to record the data for the angular acceleration during decent and ascent for further calculations and proceeded to complete the next two trials with the change in the direction of the triangle making sure to record the acceleration values as well.  Last, from our calculated inertia quantity per trial we calculated the difference and recorded those values for the triangle's inertia in the two directions.

Image #1) This image illustrates the dimensions of the triangle.

Image #2) This image shows the calculations for the inertia of the triangle around one edge and the inertia around the center of mass.

Image #3) This is the calculation for the center of mass of the triangle.

Image #4) These are the calculations for the inertia of the system starting with just the system, and then the triangle upright followed by sideways.

Image #5) This is the apparatus that we used. Note, this is the set-up for the first trial of the system only.

Image #6) This image shows the setup for trial two when the triangle was upright.

Image #7) This image shows the graph that we got from logger-pro which gives us the angular acceleration on the way up and down.

Image #8) This image shows the graph for trial two when the triangle was upright.  Note that the slope of the graph is the angular acceleration value because it is an omega vs. time graph.

Image #9) This is an image of the final calculations for the moment of inertia for the triangle in the respectively drawn position. Note that the moment of inertia for a horizontal triangle is larger than the vertical one.

In conclusion, we did seem to find the inertia values for the three trials for the system.  The inertia for the triangle itself is seen in image #9 and it shows that one was a larger value than the other.  As for errors in the experiment, there was probably some frictional force involved that interfered and altered the data but regardless I think the horizontal spin would still be larger.  Also, air resistance might have applied some force on the experiment as well.  Lastly, I think the inertia for the horizontal triangle was larger because the radius of the rotation was larger while the mass stayed constant.

Moment of Inertia and Frictional force: Jose Rodriguez: Lab partners: Kevin Tran; Kevin Nguyen: Lab completed: 5/22/17

The purpose of this lab was to find the moment of inertia and the frictional torque of a rotating apparatus so we could predict how much time it would take a metal cart of mass (m) to travel one meter on an angled ramp while being suspended by a string tied to the rotating apparatus.

To complete the experiment we had to use mathematics, physics, and we had to physically measure several variables in order to make needed calculations.  We used a digital balance to measure the carts mass and two types of calipers to measure the diameters of the apparatus.  Next, we used technology to measure the angular acceleration of the system because we needed it to calculate frictional torque of the apparatus since frictional torque is equal to the inertia of the system multiplied by the angular acceleration.  Last we used physics to find the moment of inertia and frictional torque so we could use that data to calculate the time it would take for the cart to travel the previously mentioned destination and route.

For instance, we used a digital caliper and a non-digital caliper to measure the diameters of the apparatus that was composed of three pieces.  As for the cart, we used a digital balance to measure the mass.  Next, we calculated the volumes for each piece of the rotating apparatus in order to calculate the individual mass for each by dividing the individual volume of each by the total volume of the apparatus and then we multiplied by 100 %.  Once we had the individual mass of each piece we calculated the total inertia of the apparatus in order to calculate the frictional torque that the system contained.  But, for the angular acceleration of the system we used an iphone video of the apparatus spinning then we transferred that video to a computer application Logger-pro that has the capacity to calculated the angular acceleration. After that we used Newton's theorems of forces and torque to derive the acceleration of the system with a hypothetical cart of mass 0.500 kg to travel down a ramp angled at 40 degrees while connected to the apparatus.  With all that data, we could know make a calculated predication for how much time it would take a cart of mass (m) to travel one meter on an angled ramp while connected to the apparatus we were using.  Consequently, we used our formulas to predict how long it would take our experimental cart to travel the one meter while tied to our apparatus.  Lastly, we completed three trials for the experiment and recorded the times per each trial in order to take the average value of those three quantities and divide it by our theoretical time then multiply that value by 100% to make sure we were within 4% of a deviation.

Image 1: These are the measurements and calculations for the dimensions of the apparatus.  Note that mass 1 and mass 2 are equal in dimensions.

Image 2: These are the calculations for the moment of inertia and the frictional torque of the apparatus.   

Image 3: This is the work to find the acceleration of the system.  Note our actual cart mass and angle are illustrated on the upper right hand corner of the image.  The angle was measured with an iphone compass application.

Image 4: These are the calculations to find the amount of time it take for a hypothetical cart of mass 0.500 kg to travel the one meter, and the calculations for the acceleration and time of our actual trial cart of mass 0.547 kg and chosen angle of decent to travel one meter as well.  

Image 5: These are the trial times and the % error for the experiment with our actual cart.  

Image 6: This is an image of the apparatus itself.  Note, its mass is 4802 grams.

Image 7: This is a better look of the apparatus and the angled ramp.

Image 8:  This is an image of one of the calipers that we used to measure the diameter of the small cylinder on the apparatus. 

Image 9: This is an image of the apparatus set-up close to actual position that each component was in during the experiment.  Note the larger caliper that we used to measure the diameter of the larger disk on the apparatus.

Image 10: This is an image during the stage when we were using Logger-pro to calculate the angular acceleration.  Note the blue tape on the lower right side of the apparatus, we used the program to plot a dot on the blue tape's center mass while the video ran its length to plot a position vs. time graph that led to a graph of Omega vs. time graph to get angular acceleration. 

Image 11:  This is an image of the final graph that we used to calculate the angular acceleration of the system during our experimental massed car.  Note to get this graph we first used a position vs. time graph, then we applied formulas to those values to get an omega vs. time graph that gave us a slope that is equal to the system's angular acceleration.

In conclusion, it seems like we did the experiment correctly because the accuracy was within 4%. More specifically it was within 1% as seen in image 5.  Furthermore, the moment of inertia that we calculated (seen in image 2 ) was the same value that the professor told us it should of been so that makes me believe the calculations for the volume of the cylinders and main disk were correct.  As for the forces calculations, I think they are correct because the trials that we did resulted in times very close to the calculated time of 6.67 seconds.  Note, it was impressive to see that the car's decent lasted almost exactly the amount of time that we calculated.  Lastly, the errors must have been few because all of the calculations resulted in correct values that matched the experimental expected outcomes like the moment of inertia value of 0.0208 kg*m^2.  Although, I do believe that there was friction against the wheels of the cart during its decent but I don't know how it affected the experiment itself.  Overall, I think the experiment resulted in an accepted moment of inertia value and an accepted frictional torque value because the resulting calculations that depended on those variables seemed to match what we saw in real life. 

Angular acceleration: Jose Rodriguez: Lab Partners: Kevin Tran; Kevin Nguyen: Lab Completed: 5/15/17

The purpose of this lab was to apply a known torque to an object that can rotate, and measure the angular acceleration as well as the inertia of the rotating object to find the relationship between the two.

For this experiment we needed to measure several variables in order to calculate the inertia of the rotating object per each experiment.  In order to achieve these objectives we used an apparatus that has the potential to spin one or two disks simultaneously, had different pulleys, and were tied to a string which was attached to a hanging mass.  The apparatus also had a sensor with the capacity to measure angular velocity, angular position, and angular acceleration vs. time of the system.  We focused on measuring the tension in the string because that value is the value for the torque due to the string at the point of action on the pulley, so we used Newton's second law to derive force and torque equations that we used to solve for the inertia of the disk.    

To begin with, we measured the mass of each lab equipment used except for the main frame of the apparatus with a digital balance.  Then, we connected the sensor to the computer and Logger-pro and began the six experiments that we had to complete.  For the first three experiments we used two steel disks and a small pulley, but the hanging mass (which began with 24.6 grams) increased by one each time.  The fourth experiment changed because the large pulley was used instead of the small pulley and the only hanging mass that was used was the initial hanging mass for experiment one.  As for the fifth experiment, the only change was that we used an aluminum top disk instead of the steel one that we used for the first four experiments.  Lastly, the six experiment was different because we set the apparatus to rotate the two steel disk simultaneously while the initial hanging mass descended and ascended.  Next, we used Newton's second law to speculate a value for the inertia of the spinning disk(s) per each trail and investigate the relationship between angular acceleration and inertia.

This is a table graph that has the values for the initial measurements

Measurements for the different lab equipment that we used for the experiments
equipment used	mass in kilograms	diameter in Meters	radius in Meters
top steel disk	1.357	0.1270	0.0635
bottom steel disk	1.348	0.1260	0.0630
top aluminum disk	0.466	0.1270	0.0635
smaller torque pulley	0.0100	0.0251	0.0126
larger torque pulley	0.0363	0.0489	0.0246
hanging mass	0.0246	N/A	N/A

This table illustrates the effects of the varying experimental variables.  

Effects of various changes in the experiment on the angular acceleration of the system
experiment #		kilograms actually hanging	torque pulley	Disk spinning	acceleration down (rad/2(sec))	acceleration up (rad/2(sec))	average acceleration (rad/2(sec))
1	hanging mass only	0.0246	small	top steel	1.061	-1.235	1.148
2	2x hanging mass	0.0446	small	top steel	2.150	-2.33	2.240
3	3x hanging mass	0.0746	small	top steel	3.238	-3.536	3.387
4	hanging mass only	0.0246	large	top steel	2.074	-2.310	2.192
5	hanging mass only	0.0246	large	top aluminum	5.802	-6.477	6.140
6	hanging mass only	0.0246	Large	top steel + bottom steel	1.176	-1.373	1.275

Image 1 illustrates the apparatus that allowed us to complete the experiments.  Note the sensor box next to the disks.  Also, the larger pulley is on in this picture.

Image 2 illustrates the velocity vs. time graph for the first experiment.  Note slope of the graph is the angular acceleration.

Image 3 illustrates the velocity vs. time graph for the second experiment.  Note slope of the graph is the angular acceleration.

Image 4 illustrates the velocity vs. time graph for the third experiment.  Note slope of the graph is the angular acceleration.

Image 5 illustrates the velocity vs. time graph for the fourth experiment.  Note slope of the graph is the angular acceleration.

Image 6 illustrates the velocity vs. time graph for the fifth experiment.  Note slope of the graph is the angular acceleration.

Image 7 illustrates the velocity vs. time graph for the last experiment.  Note slope of the graph is the angular acceleration.

  Image 8 shows the derivation of the formula for the inertia of disk without friction. 

Image 9 illustrates the formula and calculations for the inertia of the spinning disk(s) for trial 1-4.
Note the inertia is less for experiment number four when the aluminum disk was rotating with the large pulley.

Image 10 illustrates the formula and calculations for the inertia of the spinning disk(s) for trials 5 and 6 as well as the frictional Torque for experiments 1 and 2.
Note that experiment 6 has a greater inertia for the entire lab, as well as the fact that the lowest inertia for the disk was experiment 5.

Image 11 shows the calculations for the frictional torque for the rest of the experiments.  Note that experiment number 6 has the highest value for frictional torque.

 In conclusion, I noticed some relationships for angular acceleration and the inertia for the disk(s) that was or were spinning.  For example, I noticed that for experiments 1-3 the inertia's were similar because all of the values were within 0.0003 kg * (m)^2, but the average angular accelerations differed considerably as seen in image 9.  It seems like the inertia of the disk is not affected greatly by the added massed, even though think it should since the added weight would create a larger value for tension in the string that is the cause of the torque that motivates the disk to move.  Although, the added mass did cause the angular acceleration to increase.  For experiment four, the inertia of the disk was a smaller value than previous trials as seen in image 9.  This may have been due to the added radius of the larger pulley since torque is equal to radius multiplied by force.  Interestingly, experiment 5 had the smallest value for inertia as seen in image 10.  The cause to this is probably because the frictional force is due to the mass of the spinning disk and since the aluminum disk is a lesser mass than the steel disk, it would imply that there was less force to stop the disk from rotating.  As for the last trial, it produced the largest value for inertia of the disks as seen in image 10.  As for the cause, I think it was due to an added force of friction from the lower disk that was spinning as well.  That would imply that there was opposing force towards the inertia of the disk, and image 11 shows that the frictional force for that trial was larger than any other during the entire experiment.  The only errors for the experiment I can think of are mathematical and fundamental physics understanding by my part, as well as possibly have some force of friction from the pulley that detoured the string towards the floor.    

Ballistic Pendulum: Jose Rodriguez: Lab Partners: Kevin Tran; Kevin Nguyen, Lab Completed: 5/1/17

The purpose of this lab was to determine the firing speed of a ball from a spring-loaded gun.

To quantify the speed of ball after it was fired we had measure different variables and apply physics with the help of mathematics.  Just as important, we had to use a spring-loaded gun that had a block tied to four string connected to a pendulum as our experimental apparatus.  As for measurements, we had to measure the height that the ball would take off from, the distance the ball would land after we shoot the ball out of the gun onto the floor, and the length of the strings; the mass was measured with a digital balance.  The physics that helped us accomplish this experiment began when we shot the ball into the block until the end of the experiment when we shot the ball onto the floor because we used conservation of momentum theorem and the conservation of energy theorem to calculate the initial velocity of the ball and the time it takes the ball to hit the ground.  For example, the initial velocity was calculated with the delta momentum technique and the conservation of energy technique and the time it took the ball to hit the ground was calculated with kinematics.  Importantly, mathematics was needed for all of these calculations.  Lastly, physical measurements for the distance that the ball traveled before it initially landed were done by using a carbon paper on top of a white piece of paper so it marked the impact site and a two meter ruler.

To begin with, we had to level the apparatus.  This was done with the use of an iphone and an application named bubble level.  After that, we weighed the block and the ball before we shot the ball into the block five times to calculate the final angle that the pendulum reached before we took an average of those values for our experimental angle value.  Then, we derived a formula for the height the block would reach so we could use that value in our conservation of energy calculations.  That value for the height was used to help find the final velocity of the ball and the block, so we could find the value for the initial velocity of the ball.  Once we had the value for the initial velocity of the ball, we used kinematics to calculate where the ball would land after we shot it towards the floor.  But, to verify that our distance value was reasonable we physically measured where the ball landed with a ruler after we marked the impact with the help of a carbon paper.  Lastly, the physical measurement of the landing distance of the ball and the time calculation was used to calculate the initial velocity that corresponds to those values to compare with the mathematical calculations and physics.


This is an image of the spring loaded firing gun.  Also, notice that I am measuring the height of the initial height of the ball with a two meter stick for further calculations.

This is an image of the physical measurement for the distance of initial impact of the ball when it hit the ground.  Notice the carbon paper that was used to mark the impact site.

This is an image that illustrates the bob technique that we used to mark the starting position.

This image illustrates the given measured variables, the angle trial values, and the derived formula used to calculate the height of the block once it stopped on its rise.

This image illustrates the calculations for the conservation of momentum and energy values that were needed to calculate a theoretical initial and final speed of the ball and the block.

This image shows the calculations for the physical distance and initial velocity value of the ball.

This image illustrates the calculations for the initial speed of the ball, the time it would take the ball hit the ground, and the distance the ball would travel in the air.

 This image illustrates the absolute difference the theoretical and experimental values of the initial speed of the ball and the distance that the ball would travel in the air.

This image illustrates the uncertainty in mass and the angle, as well as the total mass and average measured angle.

In conclusion, we calculated theoretical values for the initial speed and distance traveled of the ball in first by taking some measurements, applying some physics, and some mathematics that resulted in the initial speed of 5.5 m/s and distance traveled of 2.47 m respectively.  Then, we calculated experimental values for these variables with the use of physics and mathematics that resulted in an initial speed of 4.76 m/s and a total traveled distance of 2.14 m.  Importantly, the absolute difference for the initial speed and the distanced traveled was 0.74 m/s and 0.33 m respectively.  Hence, the values for the respective difference were fairly high given that they were 15% off the experimental values for initial speed and distance traveled in the air of the ball.  As for causes of errors, there were probably more than one.  For instance, there was probably some energy lost to friction when the ball went into the block because it seemed like the ball didn't go into the block without friction.  So that could definitely cause an error in the final velocity of the block and the ball as well as the final height of the pendulum swing.  Really friction would affect the whole theoretical and experimental calculations.  Also, another reason for error was that we measured the height and distance traveled by the ball with an old simple ruler that could of easily been misread that would also throw calculations off.  lastly, the balance that we used may of been off a little giving us a mass for the ball that was not correct that would also ruin further needed calculations.  Although, the two values for theoretical and experimental initial velocity and distance traveled were in the same ball park so I think that shows that the physics and mathematics works, but the experiment was not controlled enough to show more accurate and precise values of interest.

Collisions in two dimensions: Jose Rodriguez: Lab Completed: 4/24/17 Lab Partners: Kevin Tran; Kevin Nguyen

The purpose of this laboratory experiment was to look at several two-dimensional collisions and determine if momentum and energy were conserved during each trial.

In order to show that momentum and energy was conserved during the two-dimensional collisions we had to measure and record several variables during the collisions and after the collisions.  But since the theory leading the lab was Newton's third law, it made the needed measurements fairly easy to obtain.  For instance, one of the experiments was a collision between two marbles with nearly equal mass and the other was a collision between two marbles of different masses.  So for the collision between the nearly equally massed marbles we measured the each mass with a digital balance and used a flat glassed leveled surface apparatus for the stage of the collisions.  The apparatus was leveled with the use of an iphone and its application called bubble level; the collision was recorded with an extension of the apparatus that holds a phone right above the glassed surface allowing it to capture the collision with a bird's-eye view of the area of impact.  Consequently, we were able to make a graph of the position vs. time of the two marbles during the collision and shortly after with the use of a computer program called logger pro that can use the video captured with the phone to make the graph.  Just as important, logger pro allowed us to calculate the slopes of the graph which is the velocity of that individual marble during and after the collision.  Then, with the masses and the velocities obtained we did manual calculations to determine if the momentum and energy was conserved during the experiment.  As for the collision with the different masses, the procedure was almost identical except for the difference in masses.

Several steps were needed in order to complete this experiment.  To begin with, we first had to level the apparatus that allowed the surface for the collisions and we had to place the phone in the apparatus the gave the phone a bird's-eye view.  The surface was confirmed to be level with the use of the iphone and its application mentioned earlier, and the iphone was confirmed to be positioned right by looking at the phone during the video capture mode.  Then, we measured the mass for each marble with a digital balance the is available in the lab.  After that, we placed one marble in the middle of the camera view and rolled the other into it while the phone recorded shortly before and after the collision for the same massed marbles as well as the differently massed marbles.  With the videos, we used logger-pro to trim the videos, set the origin on the center of mass of the at rest marble, and plot points in respective to the movement of the center of mass for each marble during the experiment to produce an imposed position vs. time graph for each marble before and after the collisions.  Shortly after, we calculated the slope for each marble before and after the collision that gave us the value of the velocity for each marble at these times during the experiments.  Again, the only difference between the experimental procedures was the mass for one marble so the steps for the necessary values was nearly identical except for measuring the mass of the heavier marble.  Lastly, since we had the masses and the velocities for each marble before and after the crash we manually calculated the momentum before and after each experiment to analyze the experiments.

 This is an image of the apparatus that we used as a stage for the collisions. Note the apparatus extension that can hold a phone in a bird's- eye view position.
  

  This is an image of how the logger pro application looks like when one plots the position vs. time for each marble.  Notice that the origin is placed in the center of the marble initially at rest.

 This is another image that illustrates the intended paths of the marbles.

 This is an image of the experiment with the same masses.  The masses are labeled as x and y with the after collision labels of x2 and y2 that are in different colors respectively.  For example, the blue dots correspond to the velocity of the marble that was rolled towards the other so one can see that it makes it way towards the origin where the at rest marble is with an initial velocity of 418.5 meters per second.  Also, the ball at rest has a plot that corresponds to it being still at first because its dots don't move from the x-axis before the collision.

 This is an image of the different masses trial.  Notice again that the initial slope of the ball at rest is 0.

 This is an image of the calculations for the conservation of momentum and energy with the collective data.  Notice that the calculations in the y-direction are not as close in value as the x-direction values.  For instance, the x-direction absolute difference is 1066.2-560.4=505.8 g cm/s and the y-direction is 875.4 g cm/s.

This is an image for the calculations to check for the conservation of momentum and energy with the same masses.  Notice that these values are much closer in value.  For instance, the x-direction absolute difference is 7.2 units and the y-direction is 101.6 units.

In conclusion, the experiment did result in showing that Newton's 3rd law was reliable because we did show that for every action there is an equal and opposite reaction.  Although our calculations proved a significant difference is values for the different mass trial that was concerning.  The absolute difference in the y-direction was alarming because it was 875.4 units of value, and the x-direction difference was 505.8 units different so that added to the concern that we had done something wrong in the experiment.  Simultaneously, the calculations for the trial with the same masses also had a difference with the highest value being in the y-direction of 101.6 units.  Therefore, there was obviously some propagated errors surely somewhere in the procedure.  I think one error may of resulted because we plotted the movement of marbles before and after the collision wrong so that through off the graphs that we made, which in turn gave us wrong values for the initial and final velocities.  Another possible error may of come because we didn't consider the force of friction between the marbles during contact that may of transferred some energy in the form of heat so we couldn't capture it that resulted in the significant difference in before and after momentum value.  Lastly, there was probably some friction between the surface and the marbles during the experiment that we didn't measure that may of lead to the difference in before and after values.

Impulse-Momentum activity: Jose Rodriguez: Lab partners: Kevin Tran; Kevin Nguyen: Lab Completed, 4/19/17

The purpose of this lab was to try to prove the Impulse-Momentum theorem with the data collected from several experiments.

To accomplish this goal we took a physics and mathematical type approach.  What we did was to set-up a nearly elastic collision between two carts and measured the velocity, force, and time before and after the collision for different masses per experiment; we also did one experiment with a nearly inelastic collision with the heavier of the two masses that was used for the nearly elastic collision so we could try to prove the Impulse-Momentum theorem mentioned earlier is true.  For example, we used the recorded data to calculate the area under the force vs. time graph during each trial with a tool that logger-pro has to obtain an Impulse value and then compared them to calculations for the change in momentum for the respective experiment with handwritten mathematical calculations to get a value for the change in momentum to compare and contrast the data.  The velocity was measured with a motion sensor because we could get a distance vs. time graph and then use the application tool of logger pro to get a value for our velocity which is the slope of that graph.  As for the set-up that we used, it was composed of one metal track, two metal carts, one force sensor, a digital balance, and a motion sensor.  The force sensor was calibrated and the motion sensor was confirmed to be working before we got started with the experiments.

To begin with we set up the track, weighed the two carts, and calibrated the force sensor as well as the motion sensor.  The track was a single metal track about one meter and a half long and about six inches wide; one of the carts had a spring attached to its front side and was attached in a horizontal manner to metal rods and clamps, and the other had the force sensor firmly attached to the roof of it with the actual force sensor facing the collision site to measure the collision force per each experiment.  As for the motion sensor it was facing the rear of the cart with the force sensor attached to it, about thirty centimeters away.  Then we opened up the logger pro computer application, gave the cart with a force sensor on its roof a push towards the one secured, measured the force, distance, and time per respective experiment so we could gather data necessary for the completion of the force vs. time graph as well as the distance vs. time graph for each experiment.  Importantly, the nearly inelastic experiment was done differently then the first two because we took of the cart the was attached with the rods and clamps then placed a clamped piece of wood with clay on its front so the cart with the force sensor could hit the wood and get stuck when they collide.  Also, the tip of the force sensor was changed from a rubber stopper type tip to one that had a sharp point so it can get stuck the impact site to get an inelastic type collision.  For experiment one we used the mass of the cart only, and for experiment two and three we added five-hundred kilograms to the top on the cart with the force sensor on it. Next, for the calculated values of the impulse per each trail we used a tool the logger pro application program has that let us get a value for the area under the graph ( or integral ) that is the value for the impulse for each experiment.  Lastly, the calculations for the change in momentum was done by hand because we had the mass of the cart and the value for the velocities before and after the collision for each experiment.

This is an image of the set-up for the first two experiments of the nearly elastic collisions.  Notice the way that the cart was attached horizontally with the rods and the clamps.  Also, note that the professor was calibrating the force sensor.

This image shows the third experiment for the nearly inelastic collision from the rear.  Note the clay attached to the piece of wood instead of the horizontal cart with the rubber type tip.

This is the image taken of the graphs of the distance vs. time on top and the force vs. time graph on the bottom for experiment one.

This is the image taken of the graphs of the distance vs. time on top and the force vs. time graph on the bottom for experiment two.

 

    This is the image taken of the graphs of the distance vs. time on top and the force vs. time graph on the bottom for experiment three.  Note that the force vs. time graph looks different after the main area under the graph in contrast to experiment one and two.  I think this may be due to the tip of the nail on the cart getting stuck to the wood or the clay on its way back after the impact.

This is an image of the calculations for change in momentum of each experiment and the absolute difference between the Impulse and change in momentum.

In conclusion, the experimental values that we calculated were very similar in value hence the whole lab proved to show evidence that the Impulse and Momentum theorem is true.  Of course, I do believe in the theorem because smarter men discovered it so regardless of the errors that we found in our total experiment does not deflect me from believing the theorem.  Although, our errors were such a low value that I believe we did the experiment adequately well. The absolute difference for experiments one through three: 0.0116 N*s; 0.0055 N*s; 0.0182 N*s.  As for the reason(s) for errors, I think one was that we didn't weigh the cart with the force sensor on top of it so that through the calculations of for change in momentum.  That error would definitely cause a difference in the values of Impulse and Momentum.  Otherwise, the force sensor may be a cause for error because it may of slightly malfunctioned during the trials.  But, that error would be hard to prove.  Interestingly, the force vs. time graph for the last experiment is different I believe because the nail got stuck to the wood or the clay so it pulled back on the force sensor causing a negative force that is seen after the main total force area of the experiment.  Lastly, even though the values for Impulse and change in Momentum were not the same I think the theorem clearly presented itself because of the close precision between values.