Lab 18 Moment of inertia of a frictional torque (lab partners James Okamura and Rodrigo Uribe)

Daniel Guzman

Physics 4A.

Moment of inertia and frictional torque

Purpose: The purpose of this lab is to determine the moment of inertia of a pulley, to then predict the time it would take a car that is attached to the pulley to travel down an incline that is a meter long and inclined at a certain angle.

The theory introduction: the theory for this lab is very interesting due that it uses different important concepts that are all linked to get to the desired  result. The first important part of this experiment is to determine the moment of inertia of the pulley, the moment of inertia of the pulley is determined due that it is going to be used to determine the frictional torque of the pulley as it spins when it has the car attached to it, nevertheless the angular acceleration of the pulley has to be found first in order to determine the rotational torque of the pulley. After the frictional toque is determined using two important concepts such as the moment of inertia and and angular acceleration one will be able to find the time it takes the car to travel down the slope by using the concept of torque. Once the equations for torque are derived correctly one wants to find the tangential acceleration of the system to then use it to find the time it takes the car to travel down the slope.

Description of the apparatus: The apparatus for this experiment consisted of a laptop a big pulley, a car,  a ramp, digital calibers and a phone. The laptop is used in this experiment to analyze the video that is recorded using the phone as the pulley spins in order to determine the angular acceleration of the pulley, the pulley in this experiment is a really important piece of the apparatus due that it plays a big role when determining the tangential acceleration of the system, the digital calibers are used to measure the dimensions of the pulley which are very important to determine the moment of inertia of it and the phone is used to measure the angle at which the car will be descending and the time it will take the car to descend a meter. The experimental set of the apparatus is quite simple due that it only requires one to attached a string to the car and the pullet and set up a ramp at a certain angle at which the string attached to the car is parallel to the ramp.

Apparatus set up for the experiment.

Experimental procedure: the experimental procedure for this lab was very straight forward, the first thing that we did in the lab was to measure the dimensions of the pulley using calibers, to then determine the moment of inertia of the pulley. After the measurements were made using the calibers we proceeded and calculating the moment of inertia of the pulley by adding all the moment of inertia of the pulley. After finding the moment of inertia for the pulley we did a video capture to determine the angular acceleration of the pulley, this was determined by doing a video analysis in which we determine the time it took to stop to then determine the total angle that it rotated, once we had the angular acceleration we proceed and calculated the rotational torque of the pulley, and after having these theoretical results we proceeded and run a trial to measure how long it would take the car to travel the ramp, once we measured the time we proceeded and calculate the time to then compare it to the experimental result.

Using the calibers to measure the dimensions of the pulley to then determine the moment of inertia of it 

Measurement and calculation for the moment of inertia of the pulley

This calculation was made using the dimensions measured from the pulley

Calculation for the angular acceleration and frictional torque used the moment of inertia found in the previous calculation

Calculation for the time it takes the car to go down the incline using the values found in the revious calculations,

The results shown were the theoretical results , which are going to be compared to the actual results. The experimental result in this case found for the time was 8.97 seconds which is very close to the results found theoretically, which lets one infer that the experiment was carried out correctly following the correct assumptions and doing the correct calculations, the percent error for the experiment was very small due that it was less than one percent. The sources of error in this experiment can come from stopping the timer at the correct time, other might be a worong assumption made such as assuming that the string is parallel to the pulley

Lab 17 (moment of inertia of a uniform triangle) Lab partners Rodrigo Uribe and James Okamura

Daniel Guzman
Lab#17 moment of inertia of a uniform triangle)
Physics 4A.


The purpose of this particular experiment is to determine the moment of inertia of a right triangle about its center of mass , for two different orientations of the right triangular thin plate.

The theory in this experiment is very interesting due that it allows one to determine the moment of inertia of an object about a point, which is not necessarily the center of mass of the object, for this experiment we used the parallel axis shift theorem, to find the moment of inertia of a right triangular plate about a center of mass; nevertheless, we found the moment of inertia of the triangle about other point and then used the parallel axis shift theorem to find the moment of inertia of the triangle about the center of mass.
The parallel axis theorem is I parallel axis= I around the center of mass + M(d parallel displacement)^2.
From this theorem one is able to determine the moment of inertia about the center of mass, if one finds the  moment of inertia about the other point, so in this case the parallel axis theorem would be
I around center of mass = I around one vertical end- M(d parallel axis displacement)^2. This theorem is used in this particular experiment to find the moment of inertia of a right triangle about its center of mass when the triangle has two different orientations.

Apparatus description: The apparatus for this experiment consisted of two rotating discs a thin triangular plate, a triangular thin plate, a pulley, a hanging mass, a lab pro and a laptop. The set up for this experiment was quite simple because it only consisted on wrapping the sting around the pulley which rotates between the two rotating steel discs, the apparatus for this experiment uses air which minimizes the friction in the apparatus and allows the pulley to rotate freely, Once the string is wrapped around the pulley and the hanging mass is attached to the string, one can proceed and connect the lab pro to the apparatus and computer which is going to collect the data for the experiment, once this has been done the set up of the experiment is pretty much done, and one can proceed and collect the necessary data for the experiment.

Experimental procedure: The experimental procedure for this experiment was not very difficult due that the variations for the experiment were very simple and minimum. The only two variations for this lab were changing the orientation of the thin triangular plate which made the the procedure quite simple. The first thing we did in the experiment was to measure the mass, this is very important because the equation derived for the moments of inertia depends on this particular value. Once this measurement was made we proceeded and open the valve that lets air flow so the friction of the rotating pulley would be decreased and the friction between the two discs would be decreased as well. Once this was done and when the pulley was rotating already back and forth we started to collect the data for the apparatus itself without the the triangular plate on it. Once the angular accelerations : acceleration up and acceleration down are determined for the apparatus itself, we proceeded and put the triangle with one orientation (small base) on the apparatus and repeated the same process as we did for the apparatus itself, we determined the acceleration up and acceleration down of the apparatus, after this was determined and collected we proceeded and changed the orientation of the triangle again, and repeated the same process , we determined the angular acceleration down and up when the triangular thin plate is on the apparatus.

using the lab pro one can record the position and velocity over time graphs, and from the velocity and and time graph one can take the slope of two different segments, one that is going up and one that is going down and from this one can determine the angular acceleration up and angular acceleration down. ( This graph is just a sample graph of how the angular acceleration was determined).

Data for the experiment:
The data collected in the experiment was the angular acceleration up and down for three different variations: the first variation was for the apparatus itself, the second variation was for the triangular plate over the small base, and the third variation was for the triangular plate over the big base.
The dimension of the triangular plate also were collected and recorded: Base, Height, and the mass of the triangular plate

To calculate the moment of inertia for the three different variations we used the equation derived in class by the professor, equation that was also used for the angular acceleration lab. This equation uses the mass of the hanging mass, the radius of the pulley, and the average between the acceleration up and down, which were collected using the lap pro.

After the moments of inertia are calculated for the three different systems: 1. the apparatus by itself, 2. the apparatus with the triangle over the small base, 3. the apparatus with the triangle over the big base. I proceeded and subtracted the moment of inertia of the apparatus by itself from the variation 2 and variations 3. and in this way one is able to determine the moment of inertia of the triangle: over the small and big base.

Derivation for the moment of inertia of a triangle about its center of mass.

This is very important due that it allows one to compare the moments of inertia obtained experimentally to the moments of inertia obtained theoretically for the experiment. When comparing these two values the percent error would be expected to be very small because the pulley was assumed to be rotating without friction.

When comparing the moments of inertia found experimentally and the moments of inertia found theoretically I found that they were really close to each other, when comparing them and calculating the percent error I obtained the percent error to be less than 5%, which lets me infer that the experiment was carried out correctly.

Conclusion
The moments of inertia about the center of mass of the triangles found theoretically were really close to the experimental values for the moment of inertia, due that when comparing them i obtained percent errors that were relatively small; for the first case the percent error found was 1.67% and for the second moment of inertia the percent error found was 2.91%. These two percent errors are relatively small which lets me infer that the experiment was carried out correctly, the sources of error for this experiment are minimum and might come from making a wrong measurement or from making a wrong assumption such as assuming that the system is friction-less; even though the friction in the system is reduced by the air flow which creates an air cushion on which the disk rotate and the pulley, friction still exists; nevertheless is minimum.

Daniel Guzman. Lab 16 : Angular acceleration (Lab partners-> Rodrigo Uribe and James Okamura)

Daniel Guzman
Physics 4A
Angular Acceleration Part #1


The Purpose for the part one of the experiment was to analyze how the different variations affect the angular acceleration of an object (Disk) that is rotating around its central axis while torque is being exerted, and from the analysis of these different variations determine the moment of inertia of the apparatus or body that is rotating.

Theory introduction: The theory behind this experiment is very interesting because, it will allow one to determine the moment of inertia of a body that is rotating by knowing the angular acceleration and the dimensions of it. Due that is quite hard to determine the moment of inertia of an object just by looking at it. Therefore in this experiment we had to make different measurements of different values to be able to determine the moment of inertia of the apparatus, for example, we measured the dimensions of the apparatus, which were the diameter, and mass of the discs that were rotating, due that is important to know the radius at which the object is rotating and the mass of the object, we also measured the angular acceleration of the rotating bodies; this is key in the experiment due that we are affecting the angular accelerations by different variations, to be able to determine what is the moment of inertia of the apparatus by a derived equation in the class, which heavily  depends on the average acceleration found by making the variations in the experiment.

Description of the apparatus:

The apparatus on this experiment consisted of two rotating discs which rotate about an axis placed at their center of mass, yet the rotation of this discs is generated by a torque which depends on the tension that the hanging mass provides and the pulley on which the string is wrapped around. This particular apparatus is connected to and air valve to reduce the friction between the discs. the lab pro in this experiment is used to collect important data will the discs are in motion, and the laptop is used to analyze the data collected by the lab pro.

Experimental Procedure: The first thing done in the experiment was to  take the different measurements of the apparatus that was going to be used, the measurements taken were the dimensions of the apparatus; therefore the measurements taken were the diameter and mass of the discs in the apparatus, the diameter of the pulleys that were going to be used as well as the masses of the pulleys,the mass of hanging mass was also measured. Once this measurements were taken and recorded we proceeded and connected the lab pro to apparatus and modified it so it would count 200 hundred marks per rotation (the discs on the apparatus have different marks as shown in the picture of the apparatus, important detail in order to obtain the correct result for the moment of inertia). Once this was done the experiment was conducted using different variations, which are going to give a different angular acceleration to the rotating bodies. For every variation we obtained a graph of the angular velocity of the rotating discs, and from this graph we took the slope of the graph, when it was increasing and when it was decreasing, by doing this we obtained the angular acceleration up and the angular acceleration down of the rotating bodies. from these accelerations the average acceleration was determine. After doing the same procedure for the different variations the experimental part for the first experiment is done.

The graphs obtained should look similar to this graph, this is a demonstration graph to show how the graph should look. In this graph is shown where the derivative(linear fit) was taken in order to determine the angular accelerations.

Data Collected

Table of the measured data ( dimensions of the apparatus).

Table of variations with collected data : Data collected after running the experiment with the specified variation.

From the data collected doing the different variations it is clear that when the mass increases the angular acceleration increases as well due that the tension in the string increased, which at the same time increased the the torque, for example for the first trial the average angular acceleration came up to be 1.138, for the second trial 2.374 for the third trial 3.401, which lets one infer that increasing the hanging mass increases the angular acceleration, another pattern found from the data collected is that if the radius of the pulley increases so the angular acceleration due that the torque in in this case depends on the tension in the string and the radius of the pulley the bigger the pulley the bigger the torque, yet when the mass of the rotating discs increase the angular acceleration decreases it can be noticeable in the trial where the aluminum discs was used the angular acceleration is almost as 6 times bigger than when the steel disc was used, which lets one infer that the inertia of the discs depends on a great deal on the material of the disc.

From the data collected and the interpretation of the data we determined the moment of inertia if the apparatus using an equation that was derived in class by the professor.

having these equation and using the data collected the moment of inertia of the apparatus was determined and calculated.

The sources of error in this particular lab were minimum, the measurements of the dimensions of the discs and other pieces of the apparatus were made with electronic calibers, to reduce the error, the other data collected was collected by the lab pro and analyzed using logger pro, which lets one assume that the results are not affected drastically by the error in the experiment and the propagated uncertainty.

Ballistic pendulum lab (lab partners James Okamura and Rodrigo Uribe) Daniel Guzman

Daniel Guzman
Physics 4A.
Ballistic Pendulum Lab


The purpose of this lab was to determine the firing or initial speed of a steel ball from a spring loaded cannon.

Theory Introduction: In this lab we were trying to determine the firing speed of a ball from a spring loaded gun or cannon, due that we do not have an apparatus or equipment that measured the speed of the firing gun, we had to use a different approach to determine the initial speed of the sphere. In this case we used two important concepts: the conservation of mechanical energy and the conservation of momentum in an inelastic collision. Using these two concepts one is able to determine the speed at which the ball was fired. The approach here is to determine the final speed of the system using the conservation of mechanical energy, because if the final speed of the system is known one can use the conservation of momentum to determine the speed at which the sphere was fired.
For the second case the approach is different due that the bullet in this case is going to be fired but it is not going to collide with anything; therefore, we treated just like a projectile. Therefore, to determine the initial speed of the bullet we only had to know a couple things: the distance in the x direction that the bullet traveled, and the height at which the bullet was fired, having these two measurements one can determine the initial speed of the bullet.

 Description of the apparatus: The apparatus for this experiment was quite simple due that it only consisted of a ballistic pendulum, which is composed of a small cannon, a small steel ball, a pendulum and a protractor
that measures the angle at which the pendulum moves. The cannon in this apparatus is used to fire the steel sphere,which gets embedded in the pendulum. Once the sphere gets embedded in the pendulum, the pendulum rises, which makes a lever in the protractor move to a certain angle, depending on how fast the steel sphere was fired.

Experimental Procedure: The experimental procedure was quite simple due that it did not require of putting the apparatus together, which made it quite simple. The first thing done in the lab was to verify if the firing ball would get embedded in the pendulum once it was fired, if the ball would get embedded immediately no adjustments had to be done and one could start making recording the measurements of the angles at which the pendulum would be displaced. In our case the sphere did not get embedded in the pendulum so we had to adjust it by using the two knobs that are right behind the cannon. Once the cannon is adjusted and the ball gets embedded in the pendulum, one can start recording the different measurements of the angles. Once all the measurements of the angles are recorded one can proceed and measure the length of the strings of the pendulum,  the mass of the block and the mass of the sphere, which are important measurements because they are going to be used to determine the initial speed or firing speed of the sphere. Once all the measurements are taken one can proceed and determine the initial speed of the bullet by using the approach described before in the theory introduction.
For The second part of the experiment the measurements are much easier, in these part the only two measurements made were the horizontal distance that the bullet traveled and the height above the ground where the cannon was placed. To measure the horizontal distance that the bullet travels we used a carbon paper that was taped to a blank piece of paper, so every time the bullet would hit the carbon paper it will make a mark on the withe paper, for this part of the experiment we measured a total of 5 horizontal distances, to then take the average that will be used in the projectile motion equations to determine the initial velocity of the bullet.

The data collected in the first experiment was the angle at which the pendulum moved and the final height of the pendulum after the inelastic collision.

The data collected for the second part of the experiment was the horizontal distance that the sphere traveled, the height above the ground and the distance from the table at which the canon was placed.

Table of Calculated Results

Calculated Results part 1

The calculated results for part 1 were made using two important concepts : the conservation of mechanical energy and the conservation of momentum. The calculations made were based on the data collected

Calculated Results part # 2

The calculations made in part two were made using the projectile motion equations, and were based on the data collected


Propagated Uncertainty Calculations

Conclusions
In this experiment the firing speed of the bullet was calculated using two different approaches, one was using a ballistic pendulum. In this approach two important concepts were used to calculate the initial speed of the bullet, one was using the conservation of mechanical energy and the other one was the conservation of momentum. After using these two important concepts the speed found was 4.187 m/sec. In the second part of the experiment a different approach was used in this part we used the the projectile motion equation to found the initial speed of the bullet, we were able to determine the initial speed of the bullet by measuring different distances and taking the average of them, from these distances and some other data collected we found the time it will take the ball to hit the ground, and there we determine the initial speed of the bullet which came out to be 4.98 m/sec, which is slightly higher than the speed found in the first part of the experiment. The propagated uncertainty found for the initial speed or firing speed for the first part was much smaller than in the second part of the experiment. The propagated uncertainty for the first part was = +- 5.76  , while in the second part was +- 8.9, which lets one infer that the first part of the experiment was carried out much better than the second part of the experiment. There are different sources of error for this experiment, which might be; making wrong or not precise measurements, another source of error that could have affected the results can be a miss calculation, which would affect other calculations and increase the error in the calculated results.

Lab#15: PHYS 4A LAB: Collisions in two dimensions. Lab Partners: Rodrigo Uribe and Dylan Valencia. (Daniel Guzman Physics 4A)

Lab #15: Collisions in two dimensions
Daniel Guzman
Physics 4A.

The purpose of this particular laboratory is to analyze a to dimensional collision and from it determine if the momentum and kinetic energy was conserved in the system.

Description of the apparatus: The apparatus for this particular experiment consisted of a glass board or a glass table, two metal marbles, a clamp and a cellphone. In this particular experiment the glad board or glass table was used so the collision between the two marbles could be recorded completely without the marbles falling out of the board or table after they collide. The two marbles in the experiment were the to objects that collide and obviously were the objects analyzed. The phone in this experiment was used as a camera due that some phones can record 120 frames per second, which makes the analysis of the physical phenomena easier.

This apparatus displays the glass board, the clamp and the two metal marbles used in the experiment.

Experimental procedure: The experimental procedure for this particular experiment consisted first on setting up the apparatus correctly, which was very simple, due that apparatus only consisted of the glass board, the two marbles and the clamp. Once the apparatus was set up just as it is displayed on the picture, one can proceed and run a trial to see if the camera has to be adjusted or not, once this is done one run the trials and records them. Once the recordings of the collisions have been made one can go ahead and upload the videos on to logger pro where the analytical part of the lab is carried out. When the videos are uploaded on to logger pro we used a tool that lets one picture the motion of each ball with respect to time from which we analyzed the collision of the balls. From the graphs of position vs time of the marbles we found out which were the initial and final velocities of each marble, and once one has the velocities one can proceed and verify if the momentum and the kinetic energy of the system is conserved.

This graph shows the position of the marbles in the two collisions recorded, also in this graph is shown the initial and final velocities of the marbles (which are going to be used to verify if the kinetic energy and momentum of the system was conserved). Using this particular feature of logger pro is very powerful because it allows one to model the motion of the marbles accurately, which facilitates the extraction of important data such as the velocity before the marbles collide and the final velocity of them.

Video of the collisions

Video of the collision between two marbles of the same mass

Video of the collision between two marbles of different mass

                                         

Data table and calculations 

The data collected for this particular experiment is on red, The data collected was the masses and the velocities, the other values that appear on the table are calculated results

Same Mass (elastic collision)

Different Mass Data collected and calculations.

The data collected for this particular trial was the initial velocity the final velocity and the mass of the marbles, the other values on the table are calculated results for the initial and final momentum.

Calculations

The calculations made were to verify if the initial and final momentum was conserved before and after the collision of the marbles. Also these calculations were made to confirm if the initial kinetic energy would equal the final kinetic energy before and after the collision. The calculation for momentum had to be done for the x and y components due that momentum is a vector, yet for the kinetic energy the calculations were much easier due that one only has to compare the kinetic energy before the collision and the final kinetic energy after the collision.

Conclusion

After doing the calculations and interpreting the data collected in the experiment the kinetic energy and the momentum results before and and after the collision were different, which allows one to infer that there are multiple sources of error and uncertainty, for instance the mass of the marbles was not measured using the right balance, which definitely brings uncertainty and error to the calculations, the mass of the marbles should have been measured using an analytical balance that would read at least to the thousandths place, yet the one used only reads to the tenths place. another source of error in the experiment might come from not editing the video correctly due that if this is not done right the position of the marbles with respect to time before would not be the correct one due to wrong editing. The percent errors in this experiment were really high due to the sources of error mentioned before; for instance in the collision were both masses were the same the percent error was around 30 %. This percent error allow one to say that something in the experimental procedure or how the experiment was carried out was not done correctly.

Lab 14: Physics 4A Impulse-Momentum activity. Lab partner Rodrigo Uribe (Daniel Guzman)

Daniel Guzman
Physics 4A.
Lab 14: Impulse and momentum activity

Purpose of the experiment: The purpose of this particular experiment is to prove that the change on momentum of any body or object is equal to the impulse that the body experiences.

Apparatus description: The apparatus for this particular experiment consisted of a motion sensor, a force sensor, a track, a cart that had to different set ups, one of them with a stopper at one of its ends and the other one had a nail at one of its ends, a piece of clay was also used in this experiment.

First Set up

This is the initial set up the apparatus where the car is placed on a level track on which the motion sensor is at one of its ends and car on which the car on the track will collide.

This particular set up is for the elastic collision, this set up was the same set up as the past picture; however here is shown the position at which the stopper on the car should be placed for t to collide with the spring that is attached to the other cart.

This picture just as the last two pictures show the same exactly apparatus, yet in this picture is represented the collision.

Second set up

As mentioned before the cart will have two variations, one with a stopper at one if its ends and the other one will have a a nail attached to the force sensor. The second variation is used in this experiment due that an inelastic collision has to happen

This picture represents the second set up and the inelastic collision between  the cart and the clay

Experimental Procedure
The experimental procedure of this experiment consisted on setting up first the correct apparatus due that the two collisions are totally different. In our lab group we first did the elastic collision; therefore, we used the cart that had the rubber stopper, so when it collides, the collision will be elastic. Once everything was set up we did  a trial just to see if the stopper placed in the car will hit the target, after this was verified we proceeded and push the cart and hit collect data on the laptop so the initial and final velocity of the cart will be measured as well as the force that is measured by the force sensor that is in top of the cart. We measured the initial and final velocity so that we can calculate the initial and final momentum of the cart, having these two vectors one can find the change in momentum, which will give one the impulse. Also in this first experiment we measured the force using logger pro which give one the graph of the force vs time, from which one can also obtain the impulse that the cart experienced by integrated the area under the graph.

This graph shows the graph of velocity of the car from where we extracted the initial velocity of it, Nevertheless we used the graph of position vs time and determined the slope of it at a certain time, which let us determine the initial velocity of the cart.

This graph shows the graph of force vs time, from which one can get the impulse by integrating the area under the curve, This graph was obtained using the force sensor that was attached to the top of the cart.

Having the initial and final velocity of the cart and having the result of the integration will allow one to compare the impulse obtained using the change of momentum and the impulse obtained by integrating the area under the graph of force vs time.

The procedure for the second part of the lab was somehow simpler due that the collision is inelastic, which implies that the final velocity of the cart will be zero, so we did not have to measure the final velocity of the cart. In this part of the experiment we measured the initial velocity using the same approach as we used for the first part of the lab. Also we measured the force vs time because it will provide one the impulse that the cart experienced when one integrates the graph of position vs time. The approach taken in  first part of the experiment was the same approach taken in the second part of the experiment.

Data Table and Calculations

The data collected is in red: impulse obtained from the integral of the graph, the initial velocity and final velocity obtained using the motion sensor and the mass of the cart which was obtained by using a scale.

Sample calculations

Conclusion

In this experiment one was trying to verify if the change in momentum that an object has would be equal to the impulse that the object would experience. From the data collected and the different calculations made one can conclude that the change in momentum indeed equal to the impulse that the object experiences. Nevertheless, the results obtained in this experiment were not as accurate as one would like them to be due that the percent errors for both the inelastic and elastic collisions were really big. For the elastic collision the percent error was 7.8% and for the inelastic collision the percent error was 20.8 %. When one analyzes these two percent errors one can assume that there are a lot of  sources of error for this experiment. Some of the sources of error come from the apparatus used, for example the stopper in the cart might not collide perfectly with the spring in the cart, which bring uncertainty and error to the experiment and to the calculations. Another source of error for this experiment can come from the method used to measure the initial velocity of the cart due that for the second experiment the percent error calculation, which lets one infer that the measurement of the initial velocity of the cart was not measured correctly.