19-April-2017: Impulse-Momentum

Lab 14: Impulse-Momentum

19-April-2017
Idea: Verify the impulse-momentum theorem by observing collisions in which we observe a cart to simulate an inelastic or elastic collision.

Summary:

EXPT 1:

1. We fasten a cart-track system as shown below. There is a rubber stopper on the cart connected to our force sensor, and once given a pull via a wire, meant to hit and bounce off of an obstacle at the end of the track. In this lab, another cart with a plunger.
   
2. Then, we have a motion sensor on the opposite end of the track, and in Logger Pro make sure that the position towards the motion sensor is the positive direction.
3. After calibrating and zeroing our force sensor, we give the cart a gentle pull towards the stopper multiple times until we get a good set of graphs. These graphs give us Force vs Time and Velocity vs Time, which we need in order to find Momentum, and consequently demonstrate the impulse-momentum theorem.

EXPT 2:

1. With the same setup, we add 500 grams to the cart and repeat the experiment, in order to compare whether the impulse and change in momentum were still equal to each other with a larger cart.
2. Thus we obtain a similar graph to EXPT 1 and compare their results.

EXPT 3:

1. Leaving the extra mass on the cart, we replace the cart at the end of the track with a vertical piece of wood with a lump of clay attached onto it. This experiment is meant to simulate an inelastic collision, wherein the cart stops immediately after collision.
2. I again obtain graphs of Force and Velocity, and calculate the impulse-momentum theorem.

Analysis:

EXPT 1:

Question 1: The net force exerted on the cart just before it starts to collide must be the force exerted by the spring on the extended cart's plunger, which increases as it compresses.

Question 2: The magnitude of the force is at its maximum when the rubber stopper's maximum compression of the plunger 

Question 3: After the collision, it is again the opposing force exerted by the plunger on the rubber stopper.

Question 4: The collision takes approximately a tenth of a second.

Question 5: *Calculations explained below*

Question 6: *Conclusion of EXPT1 explained below*

1) Our Force vs Time graph allows us to find the impulse. We can also calculate the momentum using the data recorded by the motion sensor, using p=mv. This allows me to demonstrate the impulse-momentum theorem by finding the change in momentum, and showing that it is in fact equal to the momentum applied to it.

2) Upon first inspection, on impact, the velocity suddenly increases to zero until it begins travelling in a constant positive direction towards the motion sensor.

3) Our graph gives an Impulse of .5290 N·S. Following my calculation of the change in momentum, I get a value of .560 N·S, relatively accurate to that of the graph, with a percent error of approximately 5%, very accurate. Therefore, I could reasonably determine that this adequately demonstrates the impulse-momentum theorem.

EXPT 2:

Question 7: *Conclusion of EXPT2 explained below*

1) Again on impact, the velocity suddenly increases to zero until it begins travelling in a constant positive direction, towards the motion sensor.

2) In order to demonstrate that regardless of the mass, the change in momentum is equal to the impulse, we again find the impulse and change in momentum of our new graph. Doing so produces an impulse of .841 N·S, and a change in momentum of .892 N·S. This again is accurate to a percent error of approximately 5%, so this indeed demonstrates that regardless of the mass, the impulse-momentum theorem holds.

EXPT 3:

Question 8: *Conclusion of EXPT3 explained below*

1) This time, the velocity does not continue to increase after 0. Because the cart sticks to the clay, the whole system stops. However, I reason that because momentum is still conserved, the impulse still equals the change in momentum.

2) To demonstrate this, I find the impulse and calculate the change in momentum for the last time, and I get values of 0.4314 N·S and 0.455 N·S, respectively. These values are very close to one another once again within 5%, and I can thus reason that even in the inelastic collision the calculated momentum equals the measured impulse applied to it.

Calculate Data

Results for change in momentum for experiments 1, 2, and 3.

Graphs:

EXPT 1
Impulse: .5290 N·S∆Momentum: .560 N·S

EXPT 2Impulse: .841 N·S∆Momentum: .892 N·S

EXPT 3Impulse: 0.4314 N·S∆Momentum: 0.455 N·S

Conclusion:

As demonstrated in experiments 1, 2, and 3, inelastic or elastic, we demonstrated that the impulse is in fact equal to the change in momentum, as for all three cases our calculations showed just how close the two values were. Therefore, we can reasonably conclude that this indeed demonstrates the impulse-momentum theorem. Between the three experiments, my calculations for percent error yielded 5.5%, 5.7%, and 5.2% respectively, showing that indeed this is a very reasonable conclusion.

• As for sources of error: the scale was inaccurate to .1 grams, and although it isn't very significant,  it is important because it does affect our change of momentum calculations to a certain extent.
• Our force sensor is approximately inaccurate to about .001 Newtons, as even in a zeroed state it would randomly read values like .0021 N. This would clearly lead to an inaccurate Force x Time graph, and ultimately an inaccurate value of Impulse. In addition, we had to make sure that the force never exceeded 10N, because apparently the sensors would read inaccurate data.
• In addition, similar inaccuracies to the motion sensor would create an erred velocity x time graph, inevitably leading to a value of a change in momentum with an uncertainty.
• Loss of energy. Clearly, these were not perfect elastic or inelastic collisions, as energy is inevitably lost due to friction, and for all three experiments this means that there is an error in both Impulse and Momentum, Friction would affect the cart's change in velocity.

17-April-2017: Magnetic Potential Energy

Lab 13: Magnetic Potential Energy Lab

17-April-2017
Idea: Model the conservation of energy for a cart-track system with a magnetic potential energy, with an equation we do not yet have. 

Summary:

1. After leveling an air track like the one below, where a cart on an air track opposes a magnet attached to the other end, I tilt the track at various angles (recorded with a smartphone), and I obtain a Force vs Position graph by plotting these test cases. In Logger Pro, I use a power curve to fit this graph, and record the given uncertainties, and reasonably ascertain that this graph gives me the force exerted by the magnet.
   
2. Based on the values given by the curve I determine an approximate function for Force, which refers to the magnet. Its integral would give me the PE of the magnet.
3. Given this function, I now attach an aluminum reflector to the cart, level the track again, weigh my cart, and run the motion detector whilst fixed near to the magnet.
4. From the distance that the motion detector records, I would subtract the distance between the sensor and the magnet in order to obtain the distance between the cart and the track. This is important in order to calculate the Potential Energy. PE = F*x
5. I create a new calculated column given this distance. I then turn on the air track, and give the cart a gentle push.
6. I am then able to obtain a single graph showing KE, PE, and total energy (PE + KE) of the system as functions of time.

Analysis/Explanations:

1) From the first part of the lab wherein we recorded the force at various angles of the incline air track, we obtained a force vs position power fit graph that gave us a value of f(r) as a function of y: 4.281*10^-5 * r^-2.282.

• By graphing Force vs Position, we could obtain it as a function of position along the track. This, we could reasonably assume, was the only force present in the track-system, assuming that friction was zero and loss of energy in general was zero. Therefore, we could further infer that this force must reference the force exerted against the cart by the magnet. This is why it was important that we record record the force using our force sensor.

2) By taking the integral of this function of force, we could obtain the PE of this magnetic force over the distance along the track. Doing so gave me a value for PE of 3.34*10^-1.282 * r^-1.282. From our base assumption, that in this case we could find the PE of the magnet by measuring force and taking its integral, we should see that its change in energy should be equal and opposite to KE. Furthermore, we expect the sums of PE and KE to result in a straight line, because as the KE of the cart decreases, we expect the PE of the magnet to increase. 

3) Therefore, we graph KE, PE, and their sum in order to visually demonstrate this idea.

After graphing the following: KE, PE, and KE+PE vs time, we get a visual representation of the two and if our assumptions were true, then we expect KE to be the opposite of PE, and their sums should result in a straight line.

4) Judging from the graphs that we got, we see that this is accurate: the PE of the magnet peaks just as the cart's KE is at its minimum, and their sum, while not exactly a straight line, demonstrates our base assumption as aforementioned.

Measured Data:

Distance between the magnets at various angles, Theta

Calculated Data:

Calculating the distance(s) between the cart and the magnet for each case.

Integration of function of Force, resulting in a PE of 3.34*10^-5 * r^-1.282

Graphs:

Graph of Force vs Position. The power fit curve as aforementioned gives a function of force as 4.281*10^-5 * r^-2.282.

Graph of KE, PE, and KE+PE.

Conclusion:

We were able to model conservation of energy for a cart-track system and extrapolate from this the equation of PE for our magnet, something we did not know beforehand, because our graphs showed that the change in energy of KE and PE were largely equal and opposite. Therefore, their sum resulted in a graph where we could see this occur. While the graph of their sums isn't a perfect line by any means, it adequately demonstrates this idea with the fact that the graph of KE reaches its absolute minimum just as PE reaches its maximum.

• With sources of uncertainty, there were the usual culprits like the scale, which has an uncertainty to about 0.1 grams. It would damage the graph of KE, as it is the product of a-half, mass, and velocity squared. However, I'd imagine that it didn't hurt the accuracy of my graph very much because this lab was worked entirely in meters, kilograms, etc, units larger upwards to a magnitude of a thousand.
• The other comes from the 'protractor' apps on our phones, as the accuracy of its readings depend on:
• The hardware. For a lot of smartphones the G sensor isn't meant to be fine tuned for accurate scientific measurements.
• The App itself. Various apps had wildly varying levels of accuracy. With the app we used, we determined it had an uncertainty of .1 degrees. Overall, while this would not directly damage the accuracy of our calculations later on, it does create an inaccurate representation of the system.
• The Force sensors and motion tracker. The former was inaccurate to at least.001 N and as a result it would lead to my integration being inaccurate, which means an inaccurate value of PE. The latter, I would not be aware of any errors, but if there were, it would of course lead to an inaccurate KE graph.
• Friction between the cart and the track. Although with the track running, we can reasonably assume friction to be zero, we realize that it isn't a perfect system and there must be, to a certain extent, a loss of energy to such forces. This would affect our calculations for the PE due to force readings that rely on how much the cart accelerates on its way across the track.

10-April-2017: Work-Kinetic Energy Theorem

Lab 11: Work-Kinetic Energy Theorem

10-April-2017
Idea: Model  the relationship between work and kinetic energy using a cart-pulley system.
Summary:

EXPT 1:

1) Set up a cart, track, motion detector, force sensor, and a pulley attached to a hanging mass as shown below. Recording two separate values at the same time serves the purpose of being able to compare W done by the tension and the cart's change in KE via the Work-Energy theorem.

2) After zeroing the force censor, and leveling the track, we verify that the sensor is accurate by hanging a .5 kg mass over the pulley and confirming that the sensors read 4.9N. Then, we removed the mass and added it to our cart, which came out to 1.18 kg.

3) We then hung 50 grams off the pulley, and hit collect as we released the cart. The purpose of this is to demonstrate that the work done on the cart by the tension in the string must equal the change in kinetic energy of the cart. This gave us graphs of Force x Time, and Velocity x Time.

4) To create a comparison graph, we create new calculated columns for KE, which is found using 0.5*mass*"Velocity"^2, and we overlay it on top of our Force x Time graph. We when crop the graph by deleting past the point where Force is again equal to zero.

5) We obtain the area below the graph for Force x Time, which gives us the amount of work done. We then record this value and determine that it is equal to the value of Kinetic Energy at the same point.

6) Highlighting a larger area of the graph starting from the leftmost end, we again obtain the integral, and the value for Kinetic Energy and compare.

EXPT 2:

1) This time, I remove the pulley and string, and attach to the force sensor a spring instead.

2) I then consider a situation in which I set the position of the unstretched spring to be 0, and any subsequent pulling of the cart toward the motion sensor to be in the positive direction.

3) I then hit record, and simply pull the cart toward the motion sensor at a steady slow pace, until the string is stretched about .60 meters.

4) Thus it gives me a Force vs Position graph, which is the key to calculating the spring constant of the spring.

EXPT 3:

1) This time, I start the cart at a stretched position of .60 meters from the origin, and let go. The resulting graphs of Force vs Position, and KE vs Position would allow me to find work done by the spring and the cart's change in energy.

2) To obtain these graphs but with the data points that I'm interested in (where the cart springs back to the origin) I strike through any data points that occur once the cart goes back to its initial position.

EXPT 4:

1) Finally, we watch a video where a professor demonstrates the force of a stretching rubber band over the stretch of the rubber band. In the video, the graph it generates required multiple back-and-forth passes to obtain an average shape. Finding the area beneath this graph gives me the work done to stretch the rubber band.

2) The system is also attached to a cart, and when the rubber band is released, it passes through two photogates a given distance apart in a given period of time. Using this we can determine the kinetic energy.

3) Altogether, I then compare these two values and judge whether the work is truly equal to the change in kinetic energy.

Analysis EXPT1:

In recording the Force x Position graph, we observed that the force is more or less constant - and subsequently that the KE graph appears to peak at what appears to be a cusp of some sorts. Following our integration of the area below F vs t, and comparing it to the value of KE of the cart recorded at the same point, we observe that they are very similar, with about a 2-10% difference whether it be our first set of intervals, or its longer, second interval. Based upon this, we can reasonably ascertain that work done by the string did indeed equal the change in kinetic energy of the cart for both recorded test cases.

Graphs:

First range of Integration for Force vs Time graph overlayed with KE vs Time
Work #1 (area under curve): 0.1036 J
KE #1 (at the point): 0.105 J

Second range of Integration for Force vs Time graph overlayed with KE vs Time over a longer interval than the first.
Work #2 (area under curve): 0.237 J
KE #2 (at the point): 0.225 J

Analysis EXPT2:

Once we obtained our Force Position graph, we obtained a nice, pretty linear slope. Because the only force present accounted by the sensor would be that of the spring, we determine that F = kx. Because it is a linear graph, we can also determine that k, the spring constant, is the slope. Therefore my graph gives a spring constant of 0.7997 N/m!

Following the integration of our graph, we determine that the work done to stretch the spring about 0.6 m is .07010 J.

Graphs:

Graph of Force vs Position, wherein slope m is the spring constant k.

Graph of Force vs Position integrated beneath the line, which gives us the total work done to stretch the spring.

Analysis EXPT3:

This time instead of focusing on the slope of our new Force vs Position graph, we take the integral of the area underneath an arbitrary interval, which gives me the work done by the spring. Comparing it to the value of KE at the same starting point, it lets me see just how close the two are; at most a difference of .5%. Setting up a table let me visualize that for three different positions of the car. Therefore, because the differences in the final values of energy were so minuscule, I could safely conclude that the work done by the spring is equal to the change in kinetic energy of the cart.

Graph/Table:

Comparison of the integral of Force vs Time (work) with Kinetic Energy at the same starting point.

Values of Work and change in Kinetic Energy at various positions of the cart

Analysis EXPT4:

Watching the movie, we notice that the lines drawn by the string vary individually, but with multiple passes of the transfucer, we obtain a graph with a distinct shape. Based upon the shape of this graph we see that once the rubber band pulls the cart to its constant, maximum force, the force decreases steeply and then again stays constant, but not zero. With reference of the scale in intervals 10 Newtons and of .1 meters, we see that force increases from 0 to .26 meters, decreases from .36 to .4 meters, and force is constant at .26 to .36 meters and from .36 on. The area under the graph gives me the work done, which allows me to use the work-energy theorem for the second part of the video.

Using the next part of the video, we record the mass of the cart, the change in position through the photogate, and the time it took to cross the photogates. Because I want to find the final kinetic energy of the cart, I first find the velocity using simple kinematics, and use it to find its final Kinetic Energy.

Once I do, I demonstrate a comparison of my values for work determined using a) the Force vs Position graph and b) KE of the cart. Calculating percent error between the two gives me a surprisingly accurate 1.05%, telling me that 1) Work-Energy is obviously true, and 2) The measurements done to obtain the graph and the speeding cart were accurate to a high degree.

Graph/Calculation/Percent Error

Area under the Force vs Position graph, which gives me approximately 23.65 Joules of work.

Calculation of the cart's final KE. I use kinematics to find the velocity, and subsequently plug it into my equation for KE, giving 23.9 Joules of KE.

Calculation of percent error. As I could see, very accurate.

Conclusions

Uncertainties:

• With regards to clear areas of uncertainty for Experiments 1-3, the most would be that we did not take into account external sources of force like static friction, in assuming that the only force the force sensors read were tension or spring force. When in reality, the energy lost to friction or a more negligible air resistance, affects our integration for work. However, the fact that there is not a very large discrepancy between the results for W or KE suggests that such forces were very very negligible.
• Another potential for error comes from our tools, such as:
• Balancing scale: Introduces an uncertainty in the mass up to .1 grams, affecting our KE calculated column.
• Assuming that the spring used in the experiments was perfect, when in reality it probably had not stretched exactly to .6 m. In fact, I observed that it appeared to be within 0-1 cm, resulting in a value of the spring constant in Experiment 2 that would be slightly inaccurate.
• Inaccuracies in the Force Sensor to about a thousandth of a Newton. Even when zeroed whilst laid flat on the surface, it would read values such as 0.002 rather frequently. Though a very small uncertainty, it nevertheless changes the result of Experiments 1-3 ever so slightly.
• Finally, in regards to the videos in Experiment 4 as previously discussed in my analysis of this part, the graphs drawn by the transfucer were so inaccurate that the professor in the clip ran the device back and forth multiple times to achieve an approximation of what it would actually look like. Therefore, in my calculation of the area under this curve, the final work would inevitably be slightly different from what the system had actually exerted. The same could be said of the speeding cart, as limitations of the tools such as the timer would result in a calculated KE that would not exactly describe reality. However, the fact that the calculated percent error between the two was so tiny demonstrates that this particular uncertainty was not too large of an issue.

05-April-2017: Work and Power

Lab 10: Work and Power
05-April-2017
Idea: Demonstrate the work-energy theorem and power in motion by determining the change in kinetic energy and the work required to move up a certain height, by a) running up a flight of stairs and b) lifting a mass to the same height.
Summary:

1) As seen, the first thing we did was to lift a bag of varying masses in a mass-pulley system, and time how long it took for us to pull it to the top. To obtain the height of the pull, we measured the height of a single step, and multiplied it by the number of individual steps it took to reach the balcony.




























2) Then, we timed ourselves a) walking up the full length of the stairs and b) running up the same length of stairs

3) Using these, we could model the work-energy theorem and subsequently calculate the power it took to achieve these actions.

Data Analysis/Calculations:
In calculating the power it took to lift masses up a height H, I first found the height I was lifting to as previously mentioned, found the force due to gravity on each mass. From that I could find the Work it took to do this, and then by timing the entire process, find the power I'd exerted. 
Calculating this, my data would reflect this: The more work I did in a short time, the more power I output, the more work I do in a far longer time, the less power I output, i.e. as power depends entirely upon the amount of work done over time.

In calculating the work it took to walk and run up the stairs (change in kinetic energy) plus potential energy at the top of the stairs, I needed to obtain my speed, which in order to do, I assumed that the stairs took the form of a triangle with an angle 40 degrees, and second, assumed that my velocity up the stairs was constant. I then also said my starting speed was zero. Using simple kinematics, I obtained a relationship for the vertical component of my velocity using my recorded time and the sine of 40 degrees. I then used that in order to solve for my velocity V along the hypotenuse, again assumed to be the stairs. 
Finally, plugging in these values to my power equation gave me a result that I thought was very reasonable for when I walked up the stairs versus when I ran up the stairs. I expended about 61% less power when I walked up the stairs, even though I had a constant mass of 52kg.

Conclusion:
As I would expect, the faster I was able to get the same mass up the height, the more power I output. Logically, the same was true for the mass: the more mass I had to lift, the more work I did, therefore the more power I had to output. Therefore I could reasonably believe that power depended on both the work done and the time it took to do that work. Thus Power was demonstrated to be equal to the quotient of work and time.
1) In neglecting the kinetic energy in the work I did to lift the masses a height H, based on the fact that I, with a mass 52kg was able to exert 2,213 Joules of work to walk up the stairs in case 2 (WITH a Kinetic Energy and Potential Energy), I can calculate a reasonable percent error by using for the second value only my potential energy up the height H, as shown: 

2) If a microwave oven has a power consumption of 100 Watts, in order to equal this power by climbing the stairs, I would need to climb the following steps per second.

3)If I am cooking in the microwave for a total of 6 minutes, I would have to climb the total flight of steps over 6 minutes to generate enough power to run the microwave.

4) A person can put out 100 watts continuously. A water heater requires 12.5 x  10^6 joules of energy for a 10 minute shower.

This has a power of:

And if I were to gather a group of people to ride generators in order to heat the water in real time, I would require:

Lastly, if I were riding the bike myself, I would have to ride the generator for:

03-April-2017: Centripetal force with a motor

Lab 9: Centripetal force with a motor
03-April-2017
Idea: Model a relationship between angular velocity, and the angle at which an object is at, once a spinning motor rotates a hanging mass attached an R distance away from the center.
Summary:
1) In this lab, we had a set up wherein a hanging mass was attached to a meter stick a distance R away from the rotating center, powered by a motor.

2) By increasing the voltage of the machine, we could get a variety of increasing values of rotational velocity.

    - Before we even turned on the machine, we recorded the height of the machine, from its base to the meterstick of length R.

    - Then we measured the radius of the extended part of the m stick.

    - Once we got the machine spinning, we timed 10 rotations, with which we could calculate angular velocity. Next, by scooting a ring stand and marking the vertical height at which the angled string is now at, we could measure its location.

3) Afterwards, we tested 5 cases at increasing power levels.




Data Analysis:
As shown above, we observe that as the power increased, the time it took for the swinging mass to complete one full rotation naturally decreased. Conversely, when we measure the new heights of the ring stand, we observe that it goes higher and higher. We know that angular velocity is the quotient of a full rotation (in radians) with the time it takes for that full rotation.
Naturally, we discover as a result that angular velocity increases with each test case.
In order to find the angle theta of the swinging mass, we set up a triangle with the length of the string as the hypotenuse, and the height of the machine minus the height given by the ring stand, and use an inverse cosine to find the angle. This shows us again, that like angular velocity, the angle theta also increases as the power is increased as shown by our data below.

Afterwards, we had to accomplish the main purpose of the lab which was to create a model that has a relationship between angle Theta and angular velocity Omega.
To do this, we set up the net forces present in both the x and y direction. In the X direction, we observe that there is a horizontal component of tension in the string, and a force due to centripetal motion. In the y direction, there is a force of the string in the vertical component and force due to gravity. Although I do not know the tension in the string, it is present in both the x and y components of force: therefore I could substitute for it in the x direction, and plug it into my forces in the y direction. Doing so gave me a function for omega in terms of varying angles of theta, which let me see that as theta increased, omega also increased nearly proportionally with it.
Data Table/Calculations:

Data set based on the 5 trials. Note that in this table, Omega was calculated using 2*pi/t

How angle Theta was derived using the height of the stand, big H, and the height of the ring stand, little h.

How Angular Velocity was calculated in terms of angle Theta. We note that the radius of the system is not merely  just the radius of the extended meter stick, but combined with the horizontal component of length of the string!

Final Analysis:
By graphing the above function, I obtain a graph that looks like the following, where Omega is in the y axis, and Theta in the x axis. Looking at the graph we can visually recognize that, in general, as theta increases, so does angular velocity, with clear exceptions to the trend thanks to the asymptotes present in tangent theta.

Comparing this to our values of Omega obtained by using our intervals of Time was a matter of plugging in our values of theta, and before doing any percent error calculations, I observed that they are relatively very close to each other. The fact that my results appear to be within a 5-10% range of each other is a good sign that my measurements were relatively sound.

Conclusion:
Following my percent error calculations, I was pleased to find that none of my percent error exceeded 6%.  From this, I could determine that the time I recorded, AND the measurements of length I took, were both accurate at least to the tenths place, which was very good.

In determining the sources of uncertainty, there were a lot to consider.

1. For one, the spin of the machine was unreliable to a certain extent first due to the unknown fluctuations of power supply, which in turn would directly affect angular velocity, and the second, in a slight wobble in the meter stick holding the hanging mass. This would obviously affect the time it would normally take for a full rotation, and potentially the height recorded by the ring stand: measuring its height at a high point in its wobble would make the recorded height too large, and too short at a low point in a wobble. This would affect our angle Theta.
2. Finally, apart from that, the rest were from the standard limitations of our measuring tools. The first was uncertainty in our meter sticks, to a tenth of a centimeter, which again, would affect our derivation of angle Theta. The second was in the timers we used, primarily our phones. Because we didn't use a photogate, the times we record for a full interval were naturally going to be really inaccurate. We attempted to compensate for this by averaging the time for 10 full rotations to narrow this uncertainity as best as possible, but it would still affect our angular velocity.