07-June-2017: Physical Pendulum Lab

Physical Pendulum Lab

07-June-2017
Purpose: Predict the period of two pendulums by deriving an expression for the inertia of a semicircle and an isosceles triangle. Then, verify the results experimentally.

Summary:

1) Before class, we went and derived the various expressions of the moment of inertia of an isosceles triangle and a semicircle at various pivots parallel to the lamina by first finding their center mass, and then utilizing the parallel axis theorem. In order to get an expression for the period, it was a matter of finding their various angular frequencies.

2) Next we set up a stand with a clamp fixed perpendicular, from which we hung a triangle and semicircle at a pivot fixed about the peak of the triangle, and the base-midpoint of the semicircle, after measuring their dimensions. We hung them from the rod by taping two paper clips to the tops. In addition, taking their mass was not necessary, as it is was variable that cancels out in our calculations.

3) Because we wanted to record the period of its harmonic motion once we gave it a push, we taped a piece of paper running down the middle of our two objects, to trigger a photogate as the paper swung through its motion.

4) Once we got our data from Logger Pro, we compared our experimental results with our calculated theoretical results.

Analysis

To ultimately predict the period of our two pendulums, we would have to find the moment of inertia about our fixed pivot points for the triangle, with base B height H, and the semicircle with radius R.

To do this in the first place, we have to know how far these pivots are from the objects' center mass. For both of our objects, I placed them base-down, with their x-midpoint at the origin, so that our shapes were symmetrical about the x-axis. This meant that we only cared about their Y center of mass, which I did by deriving an expression for the triangle and the circle, and plugging them into our known expression for the Y_cm.

Then, I derived expressions for its inertia along and edge.

• For the semicircle, I found the inertia at the midpoint of its base - and using the parallel axis theorem, solved for the object's inertia about its center of mass.
• For the triangle, I found the inertia along the pivot on its peak, and again, using the parallel axis theorem, solved for the object's inertia about its center of mass.

In addition, I went ahead and derived expressions for the angular frequency, which we would later need to calculate for our period.

Then, I solved for the inertia of our objects at our last pivot points.

• For our semicircle, we found its inertia about the point directly opposite the midpoint of the base, at the circle's outer rim using the parallel axis theorem.
• For our triangle, we found its inertia about the midpoint of its base, again using the parallel axis theorem.

Again, I went ahead and derived expressions for the angular frequency, which we would later need to calculate for our period.

After running our experiment and gathering our data from LoggerPro for our triangle rotated about its peak and the semicircle about the midpoint of its base, we plugged in our calculations for the shapes' various dimensions into our expressions for angular frequency, and subsequently for period.

Doing so, we got values with excellent accuracy, and when compared to our experimental results from LoggerPro, for the triangle, 1.3%, and for the semicircle, 2%.

Measured/Data:

Base of triangle: .205 m

Height of triangle: .227 m

Radius of circle: .152 m

Period of the triangle

Period of the semicircle

Calculated Data:

T_triangle: 0.856 s

T_circle: 0.849 s

|%Error Triangle| = 1.3%

|%Error Circle| = 2.0%

Conclusion: 

Based on how small our percent errors were, I can conclude that the predicted periods I got through my derivations for the period of our two pendulums were very reasonable. However, there were basic assumptions we made that certainly contributed to our uncertainty, no matter how small they ultimately turned out to be.

Aside from the usual: possible inaccuracies in logger pro, uncertainty in our measuring instruments, etc, I observed two that were most significant to this lab:

First, when attaching the object through the hoops (made via the loops in a paperclip), we assumed that the pivot was perfect - without friction, attached perfectly about a pivot. However, this was not true, as the contact between the paper clip and the perpendicular hinge introduces friction about this point, which would lead to our period increasing over time (T = 2t, more time it takes to make a full swing results in a longer period). It's safe to assume this is a reasonable observation, as our experimental data is indeed larger than our predicted.

Finally, the cutout shapes weren't perfect by any means: the dimensions themselves were not perfect, with margins of error up to 1 cm. The assumption that we made by deriving a formula for inertia for a perfect triangle and semicircle would no doubt mean the actual inertia of the cutouts (whatever that may be) would not be exactly the same, albeit they still would be very similar.

31-May-2017: Conservation of Energy/Conservation of angular mom

Lab 19: Conservation of Energy/Conservation of angular mom

31-May-2017
Purpose: Demonstrate the conservation of angular momentum, by first using conservation of energy with a rod swiveling around a pivot, picking up a clay object at its midpoint.

Summary:

1) For this lab, we set up a clamp at the edge of the table, with a metal rod pointing out that would let us insert a meter stick through one of its pivots.

2) We put double sided tape at the end of the meter stick, so that when it starts to swing up again after released from rest, it would grab a clay target placed at the bottom of its arc. 

3) Directly facing the apparatus, we stationed a mobile phone to capture the process at 60 fps.

4) Once the video was captured, we took the footage into LoggerPro, plotted the points, and obtained the experimental value for the final height of the clay blob by using its "scale" operation to determine the height of a point placed directly over the clay.

This was our set up as seen in LoggerPro:

5) After that, it became a matter of deriving my various expressions for energy and angular momentum in order to calculate the theoretical value of the clay's height!

Analysis

1) LoggerPro gave us a value for the blob's actual height, which we will use as the experimental value with which to compare our calculated, our theoretical, result. In order to do the energy calculations that would give me this value for height, we first needed to know three things:

1. When released from rest, what angular velocity does the meter stick have?
2. After it makes contact with the blob, what is its inertia?
3. Once it begins to rise again, what is its new angular velocity?

Number 1 allows us to solve for number 3 by letting us set up an expression for angular momentum. For number 2, because the meter stick is by itself for now, we use the expression for the parallel axis theorem to solve for the inertia of this stick when its pivot is 10 cm from the end. Plugging this into our expression for the stick's angular momentum, we arrive at a value for the stick's initial angular velocity for Number 1.

Then, we needed to know what angular velocity our system had following the collision, which we set up exactly the same as Number 1, but with the expression for inertia we derived using the parallel axis theorem, PLUS the inertia of the blob that is a distance away from the pivot. We got a value that was significantly less than the angular velocity before the collision, which is expected.

Finally, we could set up our expression for conservation of energy to solve for the height of the clay. The system starts with a kinetic angular velocity, and ends with a potential energy at some height. However, we took note that it was important not to forget about the potential energy of the stick; so we had two variables for h on the right: the height of the stick's center mass, and the height of the blob. Because the stick is rotating, we used the energy expression for the stick-blob system. Then it became a matter of isolating my h_clay variable, and solving for it.

Having arrived at my theoretical value, percent error reveals that my calculations were reasonable, but had some important uncertainties, at 10.9%.

Measured Data

M_stick =  0.141 kg

m_blob = .021 kg

H_exp = .4003 m

Calculated Data

W_0 = 5.67 rad/s

W_f =  3.79 rad/s

H_clay_theoretical = 0.449 m
|% Error| = 10.9%

Conclusions

Based on how high my percent error was, the sources of uncertainty here clearly had a significant impact on my final results. The first of which was no doubt the fact that in real life, no collisions are perfect, and the stick+clay collision results in a loss of energy that leads it to the stick+blob not making it up as high as its theoretical value shows it should be.

Our set up was not ideal either, as energy is realistically lost due to friction where the meter stick's hole meets the pivot, again, resulting again in a loss of energy that leads to the stick+blob not making it up as high as it should. 

Another important assumption we glossed over was how we treated the blob as a point mass at the end of the stick, when in reality, it was an amorphous object likely with a moment of inertia in accordance with its own shape, an error that likely makes our prediction for the theoretical height of the blob too big.

In our calculations, the propagated uncertainties with this lab were minimal, such as the mass of our objects. What could possibly be more significant however, is LoggerPro. As previously explained, we used LoggerPro's "scale" tool to let the program know that the length of the stick was 1 meter. However, the camera shot the footage from the ground, looking up some angle toward the stick. Therefore LoggerPro has an inaccurate understanding of the actual scale of our system, which could explain that our experimental value of height was simply the fault of scaling.

However, based on how my calculations ultimately did compute, and return reasonable numbers, it is safe to assume that the calculations above reflect what is going on in the apparatus in terms of conservation of energy and conservation of momentum.

22-May-2017: Finding the moment of inertia of a uniform triangle about its center of mass

Lab 17: Finding the moment of inertia of a uniform triangle about its center of mass

22-May-2017
Purpose: Determine the moment of inertia of a right triangular plate around its center mass, upright, and on its side.

Summary:

Our set up here is identical to that of the angular acceleration lab's disk-hanging mass system, where we have a sensor at 200 counts per rotation. Thus we obtained graphs of angular position and angular velocity vs time, because with our angular velocity vs time graph, getting the upward and downward slopes of this graph  gives us our values of angular acceleration. The hanging mass exerts a torque on this system. Getting the aforementioned data would therefore let us find the inertia of the various systems.

1) Subsequently, we first record the upward, downward, and calculate the average of, the angular acceleration for the steel+aluminum disks+holder.

2) Then, we weigh, and measure the dimensions of our triangle plate, then place the triangle in an upright position with the triangle's center-mass placed directly into the holder. Then we record the upward, downward, and calculate the average of, the angular acceleration.

3) I repeat step 2, but rotate the triangle 90 degrees.

Knowing the angular acceleration of these systems lets us calculate for the inertia of each individual total systems (holder, upright triangle+holder, sideways triangle+holder), and because we can also derive the moment of inertia for the machine with just the holder, we can also derive the inertias of both triangle orientations by subtracting the inertia of the holder-system from our last two tests.

Analysis

The goal of this lab is to calculate the inertia of the triangles at upright and sideways orientations rotating about their center mass. To that effect, using the spinning disk-hanging mass system required us to find the inertia of the disks with the triangle-holder, the upright triangle+holder, and the sideways triangle+holder. Subtracting the inertia of the disks+holder from the last two cases lets us do this.

As aforementioned, our setup for finding the inertia is identical to our angular acceleration lab. In order to calculate the inertia, we measured the mass of the various parts of our apparatus, and used logger pro to find the angular acceleration, which was a matter of finding the slopes of logger pro's angular velocity vs time graphs as the hanging mass goes up and down. For our calculations, we obtain the average of these values. 

Afterwards, it's a matter of plugging in our values into our given equation of inertia, once again ignoring the friction that may be present in the disk, and in the pulley itself. This is so, for all three of my trials. These values represent the experimental part of the lab, because we want to be able to compare this to calculations for the same inertias, using derivations for the moment of inertia of a right triangle.

Angular accleration and dimensions of our triangle

Derivation of inertia for the three systems.

For our theoretical values for the moment of inertia, we use the equation for the moment of inertia of a right triangle that we calculated in the lecture videos. We first derived an expression for the center mass of the triangle, then the moment of inertia about the leftmost edge, and using the parallel axis theorem, solved for the moment of inertia about the triangle's center mass, and again, plug in our values from the experiment.

Doing that, we performed percent error calculations between the theoretical and experimental results for our triangles' center mass, and found them to be rather similar. Between our upright triangle, I got a percent error of 7%, and for the second, 4.7%, values, demonstrating a reasonable accuracy between my calculations and experiments. This is how we determined the moment of inertia of a right triangular plate around its center mass, upright, and on its side.

Calculating moment of inertia about the center mass of our triangle

Calculating our theoretical values for Icm using our derived expression.
+
Final percent error calculations

Measured Data:

Graphs:

Angular acceleration of the holder system

Angular acceleration of the holder+upright triangle system

Angular acceleration of the holder+sideways triangle system

Calculated Data:

Conclusion:

Because of the small calculated percent errors for inertia, I conclude that my values for the inertia of the right triangles were reasonably valid.
Regardless, there were uncertainties in this lab that explains the percent errors that I got.
As usual, uncertainties like the scale wouldn't affect our calculations that much -- our most important source of error carries over from our angular acceleration lab - the friction present in the disk and the pulley.

Though the air pressure reduces the friction between our disks significantly, it is still present, both between the disks, and the string-pulley. In our angular acceleration lab, we ignored this friction in calculating inertia -- and we do so again, for our experimental values of the triangle's inertia. This means a value for our angular accelerations that are inaccurate, which creates uncertainty for all of our experimental inertia calculations.

Our base assumption that we can ignore friction is what most likely lead to the modest percent errors - as 7% is not insignificant - therefore, this assumption was clearly flawed, and if we had taken it into account, no doubt our final result for the theoretical would've been much smaller, and much closer to that of our experimental results.

22-May-2017: Moment of Inertia and Frictional Torque

Lab 18: Moment of Inertia and Frictional Torque

22-May-2017
Purpose: Demonstrate and calculate the moment of inertia, and frictional torque, of a large metal disk on a central shaft.

Summary:

1) We first took measurements of the various parts of the apparatus above. We took the radius of the metal disk and the metal shaft, and recorded the mass printed on the disk, and from this we calculated the mass of the individual parts of the apparatus, which allows us to calculate the moment of inertia of the disk.

2) Using a stopwatch (when the video capture failed), we timed how long the blue tape on our disk would take to come to rest once I gave it an initial spin, and then recorded where it stopped. This would allow us to find the angular velocity, and subsequently the angular acceleration, which would let us solve for the frictional torque exerted by the shaft. We did this multiple times until we got values for angular velocity that was reasonable.

Down the slope

3) Then, we propped up a wooden board in front of the disk at an angle, and connected a dynamics cart to the shaft of our apparatus with a string.

4) Using our calculated value of frictional torque in step 2, we set up equations for the cart and the disks' torque, which lets us solve for the acceleration of the cart, and subsequently for the time it might take for the cart to travel down the ramp 1 meter. This calculation of time would be our theoretical value.

5) To get the experimental value of time for comparison, we simply let the cart roll down the ramp, and our group recorded the time it took for it to travel 1 meter down the ramp. Our experimental value of time was the average of all our measurements.

Analysis:

In order to calculate the moment of inertia for the disk, and subsequently the frictional torque, following my measurements of the disks' radius and mass, solving for its inertia was a matter of using the form of inertia for a circular disk of mass M and radius R. We took the radius of the metal disk and the metal shaft, and recorded the mass printed on the disk, and from this we calculated the mass of the individual parts of the apparatus - the large disk, and the shaft to calculate the moment of inertia of the disk.

Calculating m_1/m_3, mass of the left//right hand shaft, and m_2, mass of the disk using the total mass M printed on the disk.

Calculating the inertia of the disk using m_2.

Then, I had to somehow find the frictional torque of the disk and shaft. To do this, I needed the moment of inertia of the disk, and its angular acceleration, which would be equal to whatever torque the friction was causing. Because I just calculated inertia, I had to find an angular acceleration. To do this, I first thought about using video capture to plot the position of the blue tape over time, which in theory should give me an angular acceleration, presumably less than 1 rad/s^2, because I was not spinning the disk nowhere near that fast. However, the video capture gave me values for alpha that were ridiculously wrong (about 5.1 rad/s^2) - so I turned to using a stopwatch and calculating omega and alpha by hand.

To calculate this, I first found how long it took the disk to stop after its initial spin, how many times the blue tap crossed the top of the disk, and lastly, where the blue tape was stopped relative to the vertical. Determining how many radians the disk spun divided by our average time^2 would give us the angular acceleration, which came out to a much more reasonable 0.5874 rad/s^2.

Calculating for the frictional torque, I stated that the product of the disks' inertia and its angular acceleration is the frictional torque, because once I gave the disk a constant spin, no other force but friction would have been acting perpendicular to the disk. Using my calculation for Omega, solving for the frictional torque gave me a value of .012278 kg*m^2/s^2.

frictional torque

In the next part of the lab, in order to verify just how accurate our calculations for inertia and frictional torque have been so far, we needed to set up an equation that would allow us to predict how long it would take for the dynamic cart to roll 1 meter down the ramp.

It is important to solve for the cart's linear acceleration (which is also equal to the disks'), because we can then use kinematics to solve for time t:

I started by writing my axis in the x and y directions, and using newton's second law I came up with an expression for the dynamic cart and the tension in the string as they both moved down the ramp:

mgsin(0) - T = ma

Then writing out an expression for I*Alpha using torque:

T*r - τ = Iα

Because I want to solve for linear acceleration, 'a,' I convert Alpha to 'a' in my torque expression, and solve for 'a' by first solving for T in my second law expression, and substituting this into my torque expression gives me a value of the cart's linear acceleration to be 0.03233 m/s^2, and a short kinematics calculation gives me an expected time of 7.865 seconds on the ramp as our theoretical value.

Calculating the theoretical time

Lastly, our experimental value for time was determined with three of us timing how long it takes for the dynamics cart to roll down the ramp inclined at 47°, and finding the average of our three values (provided they weren't too different from each other, which they weren't). Our average recorded time came out to be a remarkably close 7.68 seconds, with a percent error of only 2% compared to our  theoretical value. Clearly, our calculations had been very accurate, and demonstrated that the calculations of inertia and torque simply worked.

Experimental results for time

Measured Data

Time measurements

Calculated Data:

Conclusions:

Based on the accuracy of my calculated result of time compared with the experimental result of the dynamics cart rolling down the slope, I conclude that the moment of inertia, and frictional torque, of a large metal disk on a central shaft can be calculated with a high degree of accuracy, and this does indeed work.
Regardless, there were a lot of uncertainties in this lab.
Apart from the usual, trivial uncertainties (like the scales having an uncertainty of a tenth of a gram, or the calipers a tenth of a millimeter) that would hardly affect all of our calculations, the first one to note was the mass printed on the metal disk. While we assumed and trusted what it said, the fact that we were not able to verify its accuracy leads me to assume that our subsequent calculations for each of the individual components of the apparatus and our frictional torque - also have an uncertainty as a consequence.
Next, when calculating omega, because we did not use logger pro's video capture -- we timed it instead -- our measurement of the period and the radians it travels in that time have significant uncertainties of their own. Obviously, this means that my results for the angular acceleration, and subsequently the inertia of my disk, also have an uncertainty.
Lastly, when rolling the dynamics cart down the ramp, we recognize that some energy is lost whilst the wheels make contact with the ramp, and that this would mean our recorded values of time would be inaccurate as well, because we treated the cart as if it were just a sliding mass at some angle in our calculations.
Ultimately, however, our percent error demonstrates that these were not significant enough to detract from our conclusion.