Work-Kinetic Energy Theorem Activity
Annemarie Branks
Professor Wolf
Objective: Using Hooke's Law, find the spring constant for a spring attached to a cart on a track.
Part 1: Work Done by a Non-constant Spring Force
Procedure:
1. Set up your cart on a track with a motion detector, force probe, and spring as shown in the picture below. Using LoggerPro, calibrate the force probe. You may want something underneath the spring to support it.
2. Using the file L11E2-2 (Stretching Spring) in LoggerPro, you will create a Force (N) vs. Position (m) graph. Make sure the motion detector is set to the Reverse Direction so when the cart goes toward the motion detector, it is travelling in the positive direction. Zero the motion detector and the force probe.
3. Stretch out the spring by pulling on the cart and collect data for when you release it. This is the graph we got from our cart and spring:
You can see by the extreme variation in the line, where the chart had its collision at the end of the track.
4. To find the spring constant for our experiment, highlight the part of the graph that is usable data and find the slope. We are doing this because of Hooke's law which tells us that the stretch of the spring is proportional to the force or, F=-kx. In our experiment we found that k= 1.759 +/- 0.01568.
5. Using the integration routine in LoggerPro, you should be able to find the area under the curve. This is how you find the work done in stretching the spring.
Objective: Prove the Work-Energy Theorem by graphing and comparing Force vs. Position and KE vs. Position graphs.
Part 2: Kinetic Energy and the Work-Kinetic Principle
Procedure:
1. Using the same set up as in part 1, we are going to compare Force vs. Position to Kinetic Energy vs. Position. In order to graph for Kinetic Energy we need to know the cart's velocity and its mass. So we measured the cart's mass, which was 0.505 kg, and we can find the changing velocity in various ways like (F/m)*t. We make a new column, for KE, in our data table with the equation 1/2*m*v^2, and graph it.
2. Click on the y-axis title to include both Kinetic Energy and Force versus Position on the same graph.
3. Using the Analysis Feature in LoggerPro you can find and compare the graph's position, force, kinetic energy, and integral (or work). Do this for multiple points.
For this highlighted area on the Force vs. Position graph, the area is 0.02664 N*m and the KE is 0.027 Joules (or N*m).
For this highlighted area on the Force vs. Position graph, the area is 0.07105 N*m and the KE is 0.069 Joules (or N*m).
For this highlighted area on the Force vs. Position graph, the area is 0.1100 N*m and the KE is 0.108 Joules (or N*m).
Though we could continue this for more areas, we can already see that the values for the areas under the curve are almost the same to the Kinetic Energy at that position. These values are not exactly the same due to errors in our apparatus. The graph should be a straight line with a y-intercept at zero. Considering the imperfection in this experiment, these graphs still prove that 1/2*m*v^2 = Work.
Objective: Using an alternative method as shown in the video, prove the Work-Energy Theorem. Discuss discrepancies in this method.
Part 3: Work-KE Theorem
Procedure:
1. In LoggerPro there is a file called Work KE theorem cart and machine for Phys 1.mp4 with a video demonstrating how Force vs. Stretch graphs of a rubber band used to be made before programs like LoggerPro existed. The rubber band is attached to a cart which eventually passes through two photogates. This give the final speed of the cart which was can use to find the final Kinetic energy. As the rubber band is stretched, a marker is drawing on a large graph that the professor in the video is trying to move as a constant speed. You will see many variations within the lines due the human error, so the professor repeats the process until they get a good estimate of what the graph really should look like.
2. There are two methods to finding the area under the curve. The first method is weighing the paper, measuring the area of the paper, cutting out the area under the curve, weighing the area under the curve, and determining the area under the curve using a conversion factor. The second method is counting how many squares are under the curve and multiplying that number by the area of one square.
Just to get an idea of the second method, we approximated the area under the curve by dividing it up into 4 different sections and making either triangles or trapezoids out of those areas, and finding the areas that way. By doing this, this is how our graph turned out:
We found the area for each section, added them up and found that the total area under the curve was 26.595 N*m. This is how much work is being done in the system. Let's compare this value to the change in Kinetic energy, because according to the Work-Energy Theorem they should be equal. The mass of the cart was 4.3kg, the change in distance was 0.15m, and the time it took to travel from one photogate to the other was 0.045 seconds. With 1/2*(4.3)*(0.15/0.045)^2, the kinetic energy is 23.9 Joules.
These answers are of course not completely accurate but they are relatively close to each other. Using calculus is the best method but it would be difficult. If we divided the graph up into smaller sections and used better, more expensive equipment we would find an answer with fewer errors. There are errors in the professor pulling on the graph paper and errors due to friction in the system. This is not a very accurate representation of the desired experiment but it was the best the professor could do with her limited resources at the time of the film's production. This experiment is not the best way to prove that the Work-Energy Theorem is correct.
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