Eric Chong
Lab 11: PHYSICS 4A Work-Kinetic Energy Theorem Activity
Lab Partners: Joel Cook and Max Zhang
Purpose: The goal of this activity is to observe the work done by a constant force, the work done by a non-constant spring force, and overall how the work-kinetic energy theorem works.
Introduction/Theory: The work-kinetic energy theorem states that work is equal to the change in kinetic energy. Through this definition, it is possible to calculate the work done on an object based on its change in kinetic energy. Energy is also transferred from one form to another. Energy cannot be destroyed nor created. Through this lab, we can highlight some of the mechanics that goes into the work-kinetic energy theorem in order to help us understand how energy in a system works. It is also noteworthy to mention that kinetic energy KE=0.5mv^2, where m is mass and v is velocity.
EXPT 1: Work done by a Constant force
Procedure: For this activity, we will be observing the work done by a constant force. We will be using a cart that is connected to a hanging mass by a string over a pulley on a leveled track. Here is a picture of the setup:
The first step is to set up a motion sensor at the back of the track in order to track the cart's distance. A force sensor with a string attached to the hanging mass is also set up on top of the cart. This way, the force and distance can be recorded simultaneously. We first zeroed the force sensor so it could accurately record the force of the hanging mass. We also made sure the track was leveled as best we could so the cart could move at constant speed. To make sure the cart doesn't fall off, we set up a bumper at the end of the track. After setting up Logger Pro and the sensors, we were ready for the next step.
We used a 500 g mass and added it on top of the cart. We also used a 50 g hanging mass at the end of the string and pulled the cart back on the track. From here, we started to collect the data and let the car roll. Here are the graphs for the data:
As seen, the graph for the velocity vs. time is very unstable, and this could be due to the leveling of the track issues we faced. The force is constant throughout, which is expected, and is promising for our calculations.
From here, we made a kinetic energy axis on the force vs time graph. This allows us to compare the kinetic energy with the force vs time graph. Here is what it looks like (the graph at the very bottom):
As expected, the kinetic energy is very unstable due to the velocity being not as linear as predicted.
Afterwards, we changed the x-axis to position, creating a force vs. position graph. We took the integral of a part of the force vs. time plot in order to give us the kinetic energy of the cart. We also compared the actual kinetic energy based on the data calculations to the integral. The result looks like this:
We first took the integral of a small section of the force vs. position graph and found the value to be 0.09598 J, and the actual kinetic energy to be 0.132 J. We then took the integral of a larger area of the graph and found the value to be 0.2017 J, and the actual kinetic energy of the cart to be 0.269 J. The results were not as close as expected, and this may be due to the problems with leveling the track. It was expected that the kinetic energy of the cart is the same as the integral of the force vs. distance graph. Had we done better the leveling of the track, the results would have probably been even closer.
EXPT 2: Work Done by a Nonconstant Spring Force
Procedure: For this activity, we will be measuring the work done when we stretch a spring through a measured distance. Before we start, we first sketched an F vs. x graph constant force and an F vs. x graph with nonconstant force. Here are the sketches:
For both graphs, the area under the graph should be equal to the work done by the applied force because work by definition is force multiplied by time. Therefore, it makes sense for the work done on an object to be the area under the curve of a force vs. distance graph.
At the start of the experiment, we first calibrated the force sensor. Then, we set up the cart connected to a spring that is connected to a force sensor attached to a rod. We also set up the motion sensor at the back of the track. The set up looks something like this:
We set up Logger Pro by opening up a file with a preset setup of a force vs. position graph. We then zeroed the force probe and the motion detector with the cart next to the spring in an unstretched position. We also reversed the direction of the motion detector so that towards the detector is the positive direction. We also set up a formula within Logger Pro so that it would calculate the kinetic energy for us. We then began to graph the force vs position graph as the cart is moved slowly towards the motion detector until the spring is stretched to about 0.6 m. Here is a sketch of the graph:
We found the spring constant of the spring by looking at the slope of the graph, which turns out to be 2.206 +/- 0.01720. The spring constant is the slope of the graph because the equation of the spring force is F=kx, which resembles an equation of a line. Thus, the spring constant is the slope of the graph. We then used the integration routine in the software to find the work done in stretching the spring. Here is the result:
The work done in stretching the spring turns out to be 0.3130 J, according to the graph.
EXPT 3: Kinetic Energy And The Work-Kinetic Energy Principle
Procedure: For this part of the lab, we used the same set up as experiment 2. We zeroed out the force sensor and motion sensor at starting position, again, and then we stretched the cart with the spring attached to 0.6 m. Then we began graphing the graph once we let go of the cart and succumbed it to the spring force. Here is the resulting graph:
We used the integration routine in Logger Pro to find the area under the graph for three separate parts of the force vs. position plot. Here are the other two parts:
From this, we were able to calculate the change in kinetic energy and the work one on the cart. We summarized the data in a table:
Note that the sign of the work and change in kinetic energy is negative, and this is due to our "reversing" the direction when setting up the motion sensor.
Though our data does not show, due to the leveling of the track, the work done on the cart by the spring is supposed to be similar to its change in kinetic energy. And this goes according to the definition of the work-kinetic energy theorem, where work is equal to the change in kinetic energy. For this experiment, since the spring did some work on the cart, the spring changed its kinetic energy and increased its speed.
EXPT 4: Work-KE theorem
Procedure: For the last part of the lab, we watched a movie file that showed a professor using a machine to pull back on a large rubber band. While the machine pulled, the professor was also recording the force while the rubber band stretched a certain distance. Here is a sketch of the data:
Since the work done is the area under the graph, we found the work by splitting up the graph into its shapes and adding up all of the areas together. The shapes we used are a triangle, two rectangles, and a trapezoid. Here is the calculation:
According to the calculation, the work done pulling the rubber band is approximately 22.3 J. Afterwards, we calculated the final kinetic energy of the cart attached to the machine with the information we were given at the last few frames of the video. Here is the calculation for it:
According to this calculation, the final energy of the cart is approximately 23.8 J, slightly higher than what we calculated from the sketch.
Some sources of uncertainty for this part of the lab is the fact that we sketched the graph, rather than examined the raw data. Since the video depicted the graph as uneven and has many peaks and dives, the total work calculated might be slightly less than the actual work done. There was also a slight increase at the last part of the graph, but we sketched it as a straight horizontal line. This could have contributed to the slightly reduced number we calculated from the graph. Furthermore, there is also the method in which the professor recorded the data. She herself has to pull the paper across the surface, and this may have caused a bit of error to the data. There is also the device that stretched the rubber band. Perhaps the cart was in contact with friction and slightly reduced its kinetic energy. Overall, there were many sources of uncertainty, but the numbers are still relatively close, supporting the work-kinetic energy theorem.
Conclusion: The lab holistically showcases the work-kinetic energy in action . We understand that the change in kinetic energy can be used to calculate the work done on an object. Some areas of improvement could be replacing the track used, or leveling it better. Though the uncertainties exist on the experiments with the track, due to the leveling of the track, and our velocity is very skewed from its actual value (with an error of up to 70% at times), we learned through the last experiment that the change in kinetic energy is the same as the work done and how we can derive the spring constant from the graph of the spring force vs. position. We also learned that the area under the force vs. position graph is used to calculate the work done on an object, and by extension the change in kinetic energy.
No comments:
Post a Comment