Lab 22 Physical Pendulum Lab

Partners: Matthew Ibarra, Billy Justin
Date of Lab: 28 November 2016 

Lab 22 Physical Pendulum 

Mission Statement:To first derive expressions for the period of  various physical pendulums, then to verif your predicted periods by experiment. 

Theory and Experimental Procedure: For the sake of expediting the duration of the lab, most of the calculations involved had already been completed during previous exercises. We were expected to begin the lab equipped with our completed derivations for the respective center of mass and moment of inertia of a ring and triangle so that we merely need collect the dimensions of our pendulums and promptly plug them into our formulas to obtain our anticipated, theoretical values.

The derivations for the ring and the associated theoretical values are shown below, together with our collected experimental values and the associated percent error.

Above, we see that our theoretical value for the period was expected to be 1.067 seconds, and the experimental  value resulted in being  1.071 seconds, which yielded a percent error of 0.37%, well within our permissible margin of error. Below are photographs of the experiment proper.

Our apparatus consisted of a photo-gate which detected a blue tape strip as the ring oscillated back and forth. The sensor would then input the detected interruptions into LoggerPro, which in turn produced the graph shown below.

From the graph shown above, the mean value for the duration of time between two interruptions (the period) was 1.071 seconds, considerably close to our expected value.
Next we were tasked with determining the period of an isoseles triangle pivoting around an apex and base, respectively. Plugging into our derived equations and then proceeding to conduct the experiment ourselves, we obtained the following data.

For the trial during which the triangle pivoted about the apex, we obtained a theoretical value of 0.72 seconds and an experimental value of 0.72445 seconds, yielding a percent error of 0.6%, again well within the margin of error. Below are photographs documenting the apparatus setup and the graph from which we obtained our experimental value.

Above is the graph displaying the experimental value of the period for our triangular pendulum when pivoting about its apex. Below is the same triangle, except now pivoting about the midpoint of its base.

For the trial during which the triangle pivoted about the base midpoint, we obtained a theoretical value of 0.6238 seconds and an experimental value of 0.6237 seconds, identical to the thousandths place, yielding a percent error of 0.016%, our smallest ever percent error.
Conclusion: 
It should be noted that the data that I decided to include into this lab for the triangle-pivoting-about-apex pendulum is actually from our second attempt. Our first attempt yielded a percent error of 9%, which is not tolerable, and later attributed to the added weight and aerodynamic drag of the excess masking tape and a previously-thought-to-be-negligible paperclip, which we promptly removed, shown below. Upon removing these protrusions, our percent error fell from 9% to far more reasonable parameters. Even so, we knew the paperclip located at the pivot point shouldn't be contributing to this discrepancy, since there the net torque is zero. After resolving the detrimental effects of the other paperclip and excess tape, our experimental and theoretical values corresponded very well, giving credence to the physical models which we employed.

Lab 21 K-constants/ Harmonic Motion Lab

Student: Xavier Lomeli
Partners: Matthew Ibarra, Billy Justin
Date of Lab: 12 October 2016 

Lab 13 Magnetic Potential Energy

Mission Statement:To demonstrate how simple harmonic motion and oscillations work.

Theory: Every kind of calculation is in truth an approximation, and the tolerance of uncertainty depends on the application the calculation will be for. When multiple calculations are involved, then the uncertainty ripples, or propagates, through to the final result. Learning the method by which this kind of propagated uncertainty is determined is the essence of this lab.

Experimental Procedure:

 We were first given a single spring with a certain k and tasked with identifying said k. We concluded that we should use the effective mass of 115 grams ( 1/3 mass of spring [13.8 g] + mass of hook [3.7 g] + hanging mass [107 g]) to aid in this calculation of k before comparing with the results of other groups.
The methodology of our calculation went as follows: in order to obtain the k constant, we first found the displacement without the hanging mass and then displacement with the hanging mass. Since k = W/x we used the delta in weight / delta of displacement from absent hanging mass to present hanging mass, shown below.

next we required the oscillation time for our spring. To this end, we counted the time for our spring to oscillate 10 times after being displaced 2 cm, conducting 5 trials and taking the average time, shown below.

 Dividing our average oscillation time by the number of oscillations, we determined our period of oscillation to be 0.550 seconds.
With our contribution completed, we collected the data amassed by the other groups, shown below.

With this data now in hand, we proceeded to plot our collective k vs T graph, making sure to employ the power fit option so as to make sure our period took the desired form of 1/k^1/2, shown below to be the B value.

Following this exercise, we continued on to the second part of the lab where we varied the effective mass and found the accompanying period. After inputting the pertinent data, we made the graph of mass vs period to see if the period is indeed m^1/2 as expected. To measure  the period of oscillation, we used the same method as for the previous part.

our effective mass variation is noticeably small, possibly affecting our graph, shown below.

This time, we got the b value to be 0.7323, which is not quite the value we are looking for, which is 0.5, resulting in a percent error of 46.46%.
Conclusion: 
 The predominate source of error in our opinion is most likely attributable to other groups, for we feel very confident in our own procedure and results. Furthermore, besides human error, other error could be from the mass variation not being big enough.
Despite not getting the value we are hoping for, we still developed a model nonetheless, and understand the relationship between period , k and mass.

Conservation of ang momentum

Student: Xavier Lomeli
Partners: Matthew Ibarra, Billy Justin
Date of Lab: 21 November 2016 

conservation of ang mome

Mission Statement:To investigate the conservation of angular momentum about a point that is external to a rolling ball. Furthermore, to identify the angular velocity of the system after the ball is in the ball catcher by using the theorem of conservation of angular momentum.
Procedure and Analysis:
First, the apparatus, involving a track, metal ball , and spinning metal disk system, is shown below.

For this lab we conducted 2 trials, each with different radii, but not before collecting the required information, namely the diameter and mass of the steel ball.

Upon collecting this information, we needed to find the velocity of the ball when it left the track.
To do this, we employed kinematic equations using the measurements shown below. 

 Now equipped with the knowledge that the incoming velocity was 1.35 m/s,

next we set up the disk system and used logger pro to find the angular acceleration up and down.

 -we obtained two different places where the ball hit along with the omega that we would later need to compare, shown below.

 We also calculated the alpha, shown below.

armed with both the desired omega and alpha, we could now find the moment of inertia of the system, shown below.

Now equipped with the desired moment, we plugged into the conservation of angular momentum formula in order to find our theoretical omega, shown below.

 Conclusion:

As shown above, we found the percent error between our theoretical and experimental omegas to be 7.7% and 9.9%, respectively. Sources of this error likely involve the possibility that the collision between the ball and the arm was not, in fact, inelastic. Otherwise, we showed that angular momentum is conserved. 

Lab 19 Meter-Stick/Clay Exercise

Student: Xavier Lomeli
Partners: Matthew Ibarra, Billy Justin
Date of Lab: 21 November 2016 

Centripetal Force

Mission Statement:To release a meter stick, pivoted at or near one end, from a horizontal position. Exactly when the meterstick reaches the bottom of its swing it collides in-elastically with a blob of clay. The meter stick, with clay attached now, continues to rotate to some final position. After measuring the appropriate masses, we are to come up with a prediction for how high the clay-meterstick combination should rise. Then are then to capture the experiment on video and compare the actual results to our prediction.
Procedure and Analysis:
For the theoretical portion of this experiment, we first needed to devise a method of predicting just how high the clay-stick combination would rise under the aforementioned circumstances. To begin with, we first identified that this could be done by segmenting the experiment into three distinct sections.
The first segment would examine the rotational motion of the meterstick from its initial flat horizontal position down until just before the moment when it impacts the clay, the point being to determine the angular velocity (omega 1) of the meterstick just before colliding with the clay.
The second segment would examine the inelastic collision between the meterstick and clay, where we were to employ the principle of conservation of angular momentum in order to determine the initial angular velocity (omega 2) of the combined system.
Finally, the third segment would use the principle of conservation of energy to predict the final height of the clay.
Below are our calculations.

Our calculations first involved finding omega 1 (5.48 rads/second) by establishing an energy relationship in which the initial gravitational potential energy transformed into rotational kinetic energy. Next we employed conservation of angular momentum, in conjunction with the parallel axis theorem (since the pivot point is not located at the center of mass of the meterstick). Doing so yielded us omega 2 (2.88 rads/second). Now equipped with omega 2, we set up another energy relationship, this time to calculate the theoretical height which would be reached by the clay-meterstick combo. This is when we were presented with a computational dilemma, the dilemma being that the vertical displacement of the meterstick (if the point of collision is considered the origin) is not equal to the vertical displacement of the clay. In actuality, the clay rises some distance more than the meterstick due to its center of mass being farther from the pivot position. Hence, we needed to devise a ratio between these heights with respect to the pivot point. Using this ratio, we were able to complete our calculation, shown above. We obtained a theoretical value a height of 32 centimeters, or .32 meters.

Next, in order to examine the veracity of our computation and our theoretical value, we conducted the experiment for ourselves, setting it up as shown below.

Capturing video of our experiment, we obtained the following graph.

From the graph above, we see that the maximum height reached by the clay is shown to be 20.71 centimeters, or 0.2071 meters. This value is considerably lower than our predicted height.
Conclusion:
We identified our percent error of 35.4% and then brainstormed reasons for this discrepancy. We first reviewed our calculations thrice, the third time being under the direct supervision of the professor, in order to identify computational errors.When we found no fault in our calculations, we considered experimental factors which we did not consider initially. To begin with, we neglected to consider the friction at the pivot position and at the bottom between the clay and the paperclip stand on which it rested. Furthermore, another potential source of error likely involved unintended oscillations perpendicular to the arc of motion, aka "wabbling."

Lab 17 Moment of Inertia of Uniform Triangle about Center of Mass

Partners:
Date of Lab: 2 November 2016 

Lab 16 Angular Acceleration

Mission Statement:To determine the moment of inertia of a uniform right triangular thin plate around its center of mass for two perpendicular orientations of the triangle. Then to compare the experimental and theoretical results for the moment of inertia of the triangle for each of the two orientations. 

Theory and Experimental Procedure: 
 The apparatus was composed of two spinning disks next to a Pasco rotational sensor. We connected a torque pulley to the top of the two disks with a string strung around it. The opposite end of the string slides over another pulley and dangles off the edge of the table with a hanging mass. The aforementioned triangle was then mounted to the top of the torque pulley by slotting into a protruding holder rod. Activating the compressed air permits the disks to rotate without friction, in turn prompting the hanging mass to oscillate up and down.

In essence, this experiment tested our understanding of the parallel axis theorem, which states the following:

I(parallel axis) = I(axis through center of mass) + M*d^2, where d is the displacement from the original cm and M is the mass of the object in question.

Since the limits of integration are more straightforward if we first find the moment of inertia about a vertical edge of the triangle, we can calculate said moment of inertia and then obtain the moment of inertia around the center of mass by rearranging the relationship above into the relationship below.

I(around cm) = I(around one vertical end of the triangle) + M*d^2

With this in mind, we needed to derive the moment of inertia of the triangle around its center of mass and then utilize the parallel axis theorem to identify the moment of inertia around its new axis. My  derivation for this is shown below.

 The desired moment of inertia of the triangle resulted in being I = 1/18*M*R^2. Worthy of note, in order to solve for the moment of inertia of the right triangular thin plate, we first had to measure the the triangle's mass, base length, and height. These came out to be 0.455 kg, 0.098 m, and 0.14950 m, respectively. The base and height length would depend on the  orientation of the triangle.

Now commenced the experimental portion of the lab. The moment of inertia of the whole system was solved for by knowing the torque exerted by the tension from the hanging mass together with the angular acceleration of the system which was produced by said torque. The connection between these qualities is shown below. 

Torque = Tension*radius = I*alpha  

Hence, by obtaining the total moment of inertia of the system and then proceeding to subtract from it  the moment of inertia of the disk/holder subsystem, we can yield the moment of inertia of just the triangle. This moment of inertia was henceforth our experimental value of I.

However, since there was some unknown amount of  frictional torque in the system, namely due to the the disk not being completely frictionless or the presumed-to-be-frictionless pulley nevertheless not being massless, the angular acceleration of the system during the descent of the hanging mass would not be equal to the alpha of ascension. Hence we were obliged to derive an equation which identified this mysterious frictional torque. The derivation is shown below

 Armed with this information, we proceeded to document the data of the angular acceleration for only the disk/holder system. Much like in prior labs, the angular acceleration was equivalent to the the slope of the angular velocity vs. time graph, shown below.

 We first finding the average upward angular acceleration, then the corresponding downward counterpart, and then the average of the two. Putting our derived equation into practice, making sure to use the proper mass and base/height values, I was thus able to calculate the experimental moment of inertia of the disk/holder system, shown below.

 Next, I positioned the triangle while oriented vertically and commenced the experiment anew, with the respective omega v time graph shown below.

Getting the average acceleration, I was able to calculate the moment of inertia for the vertical system, shown below.

Following the professor's suggestion, by subtracting my disk/holder/triangle system inertia from the disk/holder system inertia, I was able to obtain the experimental moment of inertia of only the triangle which resulted in being 2.09x10^-4 kg*m^2. After using the theoretical equation we derived earlier, we were able to say that the theoretical moment of inertia should be 2.44x10^-4 kg*m^2. Our percent error came out to 11.3%

Finally, I flipped the triangle so that the longer side was horizontal and rinsed and repeated.

Using the above average angular acceleration, I was able to calculate the triangle's experimental moment of inertia when horizontal, shown below.

Contrasting the experimental value for moment of inertia of the triangle (horizontal) I =5.32x10^-4 with the theoretical value I=5.65x10^-4 yielded a percent error of 6.84%.

Conclusion: Equipped with the Parallel Axis Theorem, I was able to derive an equation for the triangle's moment of inertia around choice axes of rotation. Using my hand-measured (and thus susceptible to human error) data of the disks' and triangle's dimensions, I was able to calculate the experimental value for the triangle's moment of inertia. Then, by juxtaposing my experimental results for moment of inertia with the theoretical value obtained through my derived equation, I was able to determine how precise we were able to get using our understanding of newton's second law.. My results, even though the percent error was larger than desired, were still close enough to verify our experiment.

Errors occurred during our experiment beginning with the fact that our vertical triangle was not Truly vertical.  Other things such as the friction from not having a perfectly clean disk, or non-uniform disk, and slight error in measurements of the triangle may have skewed our results a bit.

 

Lab 18 Moment of Inertia and Frictional Torque

Partners: Matthew Ibarra, Billy Justin
Date of Lab: 7 November 2016 

Lab 18 Moment of Inertia and Frictional Torque

Mission Statement:To predict the time of descent (the distance being 1 meter) of a cart down an angled, frictionless (presumed) slope while attached (via string) to a metal disk pulley itself subject to friction. 

Theory and Experimental Procedure for Part 1 of Lab 16: In this lab, our end goal was to determine, with as best precision as we could muster, the time it would require a cart to roll down a distance of one meter down an incline while simultaneously strung to a metal pulley. In order to carry out this calculation, we first needed to determine the moment of inertia of the pulley system when spun without the added torque applied by the cart. Since the pulley system consisted of a large disk together with cylindrical protrusions, we needed to find the respective moments and then sum them together to obtain the desired moment.

However, besides determining the moment of inertia of the system, we likewise needed to consider the opposing torque being exerted by the friction between the pulley and the supporting armature. Hence, we needed to somehow determine the angular deceleration of the pulley when spun (again, without being connected to the cart). To do this, we set up a photogate such that we could graph the theta-time graph from which we could obtain the desired angular deceleration. We taped a white strip onto the rim of the disk so that the motion could be sensed. The photogate setup is shown below. 

Using this technique, we obtained the graph shown below.

 From the graph above, we identified the angular deceleration to be twice the A value. Now equipped with this important information, we were able to proceed with our calculations.
Calculations: 
Below is our calculation for the moment of inertia of the pulley system. As aforementioned, we first found the respective moment of inertia for the disk and then the cylinder, which was broken into two slices since the disk bisected the cylinder. Once we found the component moments, we added them together and discovered our desired moment of inertia for the pulley system, which resulted in being 0.019 kg*m^2.  

Now equipped with the knowledge of the systemic moment of inertia, it was key to understand that this moment of inertia of the system, in theory, is equal to the frictional torque which eventually stops the spinning. Hence we could carry out our second calculation, shown below, in order to find the acceleration of the cart, and thus the time taken to cover a given distance. That theoretical time ended up being 7.388 seconds. 

Performing the experiment three times, we obtained three different times of 7.35, 7.32, and 7.36 seconds, all of which were approximately close to the theoretical time, and well within 4% off. Nonetheless, sources of error likely dealt with the fact that our entire calculations were based on the presumption that the pulley was uniform in terms of mass-distribution. In actuality, the pulley is not uniform, and hence the moment of inertia which we calculated is likely not the actual moment of inertia.

Lab 16 Angular Acceleration

Partners: Matthew Ibarra, Billy Justin
Date of Lab: 2 November 2016 

Lab 16 Angular Acceleration

Mission Statement:To determine the angular acceleration and moment of inertia of an object that can rotate when a known torque is applied. 

Theory and Experimental Procedure for Part 1 of Lab 16: In this lab, we set up the following apparatus, shown below:

- As we can see, the apparatus involved several stacked disks (the bottom disk is kept from ricocheting by a drop pin) which would rotate independently when spun due to a cushion of air between them, with the top disk (either steel or aluminum, depending on the trial) floating upon this cushion of air when the blower was activated.

-In order for the top disk to rotate, we installed a hanging mass, shown above, such that the hanging mass would exert a force, translated through the pulley, onto the floating top disk, hence producing a torque, which in turn would rotate the disk. Diagrams of this interaction are shown below.

-

-As can be discerned from the diagram immediately above, the entire point of this setup is to determine the various angular accelerations of the top disk. To do this, we utilized a rotational sensor which, when synchronized with LoggerPro, could yield graphs of angular position velocity, and acceleration verses time. Due to systemic deficiencies, we discarded the angular acceleration graph outright and were forced to manually calculate said accelerations. With everything good to go, we activated the air and released the hanging mass so that the mass would move up and down while the top disk would rotate clockwise and counterclockwise. As aforementioned, we were to use the omega verses time graphs obtained via the sensor to obtain the associated alpha values.
-In total, we conducted six different trials, each beginning with different experimental conditions designed to document the effects said conditions would produce. Below is the empty table specifying those conditions.

Below is now our completed table specifying those conditions along with documenting our collected and calculated data.

Conclusions for Part 1 of Lab 16

Now equipped with the necessary information,we were able to draw numerically specific conclusions about the effect on the acceleration of the system by increasing the hanging mass, radius of the torque pulley, and/or the rotating mass, all shown below.

Theory and Experimental Procedure for Part 2 of Lab 16::
Next, we needed to determine the relevant experimental and theoretical moments of inertia. First, we utilized the experimental formula derived for us in the instructions to calculate the experimental moments of inertia, shown below. Also, armed with the knowledge of the standard formula for the moment of inertia of a disk, we found the respective moments for top steel, top + bottom steel, and aluminum plates, shown at the bottom of the page shown below.

As can be seen from the above calculations, the correspondence between our experimental and theoretical values proved to be exact to three significant figures. Hence we verified the validity of the standard formula. Given the accuracy of our results, sources of error were likely negligible save for the hanging mass possibly swaying side to side rather than be kept straight as it rose and fell.