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.