Mechanical energy lab - Lab Report PDF

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Group members: Morgan Nguyen, Jacob Berna, Joseph Eisold, Micheal Rukab Investigation/Objective: The objective of this lab was to investigate the claim that Mechanical Energy is conserved in a closed system. In the case of our lab we will be testing this claim with a cart on an inclined track with a hoop spring at the bottom of the track. To figure out the Mechanical Energy conserved we will have to take the data for the Kinetic Energy, Potential Elastic Energy, and Gravitational Potential Energy and then show graphically that they equal to the mechanical energy at any point along the track. The first objective we had to do was to determine the spring constant, this was done by applying a force to the spring and then measuring the distance it compressed. We could then graph the distance vs. force to get a slope that would give us the spring constant. We then just put the cart on the track at a certain angle and let it hit against the spring and measured the position, time, and velocity. With the K constant, parameters (measured and given), and the position and time graph we could then calculate using logger pro the mechanical energy, potential gravitational energy, and the potential elastic energy. From there we then graphed them and showed that they equal to the mechanical energy of the cart at any point along the track. We hypothesized that for each time the cart hit the spring it would have a lower mechanical energy even though it was supposed to be in a closed system with no energy loss. This energy loss happens because of the friction on the track and the spring which releases energy in the form of thermal energy. We predict that for the graph it will show the mechanical energy decreasing in a step like fashion for each compression with the spring as energy is lost. We also hypothesize that when the cart moves down the ramp it will transfer its potential energy first to kinetic energy and then to potential elastic energy and then back to the kinetic energy and then finally potential gravitational when it is back at the top of the ramp. Overall the goal of this experiment is to graphically display that the cart's kinetic energy, potential gravitational, and potential elastic equals to the mechanical energy of the cart at any point on the cart's trip (displaying conserved energy). We are supposed to show that energy is conserved in the cart's closed system but in reality we will have some energy loss making our graphs not perfect. Assumptions: Some assumptions can be drawn for this lab. The first assumption that can be made is that friction is to be ignored for the cart on the track. Even though friction still plays a role in the motion of the cart up and down the track, it will not be included in any calculations. We will also assume that there is no friction in the spring. Since this cart on the track is a closed system we will assume that there will be no other non-conservative forces and energy in the cart is conserved. We will ignore the energy lost as thermal energy in the system from the spring and the friction. We assume that on the cart the gravity is always pulling straight down and that there is a normal force that is perpendicular to the track. We assume that the cart on the ramp is started at zero velocity and is only pulled down by the gravity until it hits the spring. Another assumption that can be made for this lab is that all the sensors and provided materials are in working order. This being the cart rolls up and down the track with no issues, the sensors are calibrated and damage free, and the spring is not warped and produces consistent reactions.

We assumed that the spring was in perfect working order when in reality it was a little bent which would have impacted its elastic performance. Since this experiment is not being done in a controlled environment we can assume that there may be outside elements also acting on the system, however, for this lab we will be ignoring these elements and moving forward as if they were not there. Uncertainty: The first type of uncertainty that we could come across would be personal (error) uncertainties, these could be simple mistakes like putting in the wrong parameter when calculating the energys. Some other personal uncertainties could be not dropping the cart from the same position on the track every time making some data runs have different time frames, in addition to the parameters having to be entered into the sensors, another error that could be made is forgetting to change the direction the motion encoder reads at. If the encoder is left on the normal setting it will read in negative measurements which will mess up the data collected. We could have also not started the sensors at the right time when the cart was released. Another issue could have been having the loop force sensor on the +/-50N insead of the +/-10N Along with the personal uncertainties there are also systematic uncertainties, issues that may arise due to the sensors and the readings that they give us. Since this is not the first time they have been used they may have damage or hidden issues in the force encoder and the motion sensor. The environment that the lab is performed in also plays a role into the uncertainty of the investigation. Since it is not being done in a controlled environment, the entire system is subject to outside elements that may not be accounted for. These being dust, A/C, another person bumping the table, etc. There were also statistical uncertainties where there were random changes during the experiment. One of these was when we were applying the force to get the K value. When the force was applied it was hard to push on the spring at a constant rate. The force when pushing could be slightly larger or smaller because pushing with a hand is not consistent. There could have also been random uncertainties when the cart was moving down the ramp and bouncing off the spring. Experimental Design: Set up - For this experiment we set up a track on a piece of wood with a loop force sensor on the bottom of the ramp and a force motion sensor at the top of the track. The angle of the ramp was calculated measuring the length and height of the track.The mass of the cart was calculated using a scale. Part 1. We first needed to calculate the K constant of the force loop spring. We did this by putting the cart near the loop force sensor and zeroing the sensors and then applied a force to the cart with a hand. The logger pro would record the force applied and the distance the spring compressed. With this information we could graph it as displacement vs. the force applied and the slope of the graph gave us the K constant of the spring. This process was done three times and the average of the three spring constants were taken.

Part 2. The second part of the experiment we needed to calculate the potential elastic energy, the potential gravitational energy, and the kinetic energy. With this data we could then calculate the mechanical energy. To do this the cart was first positioned at the top of the track and then released and allowed to hit and bounce off the loop force sensor for 20 seconds. This records the position, time , and velocity of the cart as it moved down the ramp and bounced on the spring. Before the energies could be calculated some more information/parameters were needed. The mass of the cart was measured and the angle of the ramp was calculated by measuring the height of the block and the length of the track and then doing the inverse of sin (opposite/hypotenuse length). This information could then be entered into logger pro as parameters. With the parameters and the time vs. the position graph the energies could then be calculated by entering their formulas into the calculated columns in the logger pro. The energies were then compared graphically at every point on the carts positions to show that the potential energy, potential elastic energy, and kinetic energy equals the mechanical energy. Lastly uncertainties for these values were calculated in logger pro. Reducing Uncertainty: Some ways that personal uncertainties can be limited/reduced in this investigation could include double checking calculations, making sure the sensors are always zeroed, and starting the sensors at the same time the cart begins.. Another way that we can reduce uncertainties is by making sure the sensors have not reverted back to their previous settings, but that they have stayed in the parameters we set and are recording the correct units, signs, and the correct amount of samples. To keep track of the position the cart starts on the track a tape line could be placed as a start line so that the cart travels the same distance across the different runs. To reduce the uncertainties created by the environment, we could move the investigation into a more controlled room, clean the track and carts, and limit the amount of people around the system. To reduce statistical uncertainty we can run multiple trials. When calculating the K constant of the spring we ran multiple trials so the uncertainty of the hand pushing the cart against the spring would be reduced. Also by doing the position vs time trials more than once we can reduce uncertainty in the cart and the spring interacting. To reduce systematic uncertainty we can look up the manufacturer's specifications of the uncertainty on the loop force sensor and the force motion sensor to reduce issues with these pieces of equipment. For the calculations to reduce uncertainty it would be best to do them with more digits after the decimal place to be more accurate with the numbers. Calculating Uncertainty: Based on the raw numbers the logger pro gave us we can estimate the input position uncertainty of the cart to be +/-0.001 m, the input time +/-0.001s or +/- 1/35 hertz, the force reading to be around +/-0.01N and the input velocity at +/- 0.03 m/s. The uncertainty of the massed cart was +/- 0.001g. The output uncertainties which were averaged was +/- 0.0008 m/s for velocity, +/- 0.001084 kg/m/s for kinetic energy, +/- 0.000112 kg/m/s for potential elastic energy, +/- 0.000125 kg/m/s for potential gravitational energy, and +/- 0.001763 kg/m/s for the mechanical energy.

Table 1. Chart showing the uncertainties of the outputs of the velocity, potential gravitational energy, potential elastic energy, kinetic energy, and mechanical energy. The uncertainty averages are presented at the bottom of the chart since the whole chart was too big to display the whole thing.

0.0008 m/s

0.001084 kg/m/s

0.000112 kg/m/s

0.000125 kg/m/s

0.001763 kg/m/s

Averages of 1,000 rows of data for

uncertainties

Presentation of Data: Part 1: Calculating the spring constant

Figure 1. Graph displaying the displacement vs. the force of a cart as it is pushed against the force loop sensor. The slope represents the spring constant. Table 1. Chart displaying the K constants of the spring calculated from three trials. Average spring constant of the three trials is provided.

Figure 2. Graph of the time vs. the position of the cart as it moved down the ramp and bumped against the spring. Table 2. Parameters used for the energy calculations

Table 3. Chart showing the position, time, velocity, potential elastic energy, potential gravitational energy, and the kinetic energy along with all their uncertainties. This is not the whole chart, it is about 1,000 rows long.

Figure 3. Graph displaying the potential gravitational energy, potential elastic energy, kinetic energy, and the mechanical energy as the cart moves up and down the ramp and hits the spring multiple times.

Observations: When the cart went down the ramp it increased in speed as the potential gravitational energy was transferred into the kinetic energy. When the cart hit the spring the kinetic energy was transferred into potential elastic energy. When the spring was fully compressed then the potential elastic energy was transferred back into the kinetic energy. On

the journey back up the ramp the kinetic potential energy was eventually transferred back to the potential gravitational energy. During each trip duration the cart would lose some energy to thermal energy from friction which resulted in lower mechanical energy after each trip up and down from the spring.

Results: After completing part one of the lab it was determined that the average spring constant for the force loop meter was 70.496 N/m. The parameters for the lab that were calculated were the mass of the cart which was 0.3197kg. The angle of the ramp using the hypotenuse (ramp length) and opposite (height of block) was determined to be 2.28 degrees. For part 2 the position and time were recorded of the cart and the results of that is visible in figure 2. As time increased the position of the cart decreased. From there then the potential gravitational energy, potential elastic energy, kinetic energy, and the mechanical energy were calculated. The results of those are visible in (figure 3.) where the green line is mechanical energy, the blue line is potential elastic, the pink line is potential gravitational, and the yellow line is kinetic energy. As seen on the graph after each collision event with the cart and the spring the total mechanical energy decreased. Since the mechanical energy decreased it caused the potential elastic, potential gravitational, and kinetic energy to also decrease. Discussion: For this lab we were able to successfully get the spring constant first. The spring constant value had some variation so we thought it would be best for three trials and then take the average of the spring constant for a more accurate value. After we got the spring constant value we then calculated the angle of the ramp and acquired some other parameters. We then successfully ran three trials of the cart moving down the ramp and hitting the spring to record the time vs. position. We did three trials of this and then chose the best one to use for the calculations. All three trials came out nice so we just chose the last one. With the parameters, k constant, and the position vs. time coordinates we were then able to calculate the energies of the cart successfully. When we got these energy variables we then graphed them to see if the results made sense. Upon looking at the graph in (figure 3.) the different line slopes make sense. The mechanical energy line is the highest because it equals to all the energies (potential gravitational, potential elastic, and kinetic energy. As seen on the graph as the potential gravitational energy decreases the kinetic energy increases. The kinetic energy keeps increasing until it hits the spring where the kinetic energy is transferred to the potential elastic energy. When the spring decompresses as seen on the graph the energy is then transferred back into kinetic energy as the cart is pushed back up the ramp. As the cart nears the top of the ramp the kinetic energy slows as the energy is transferred back into the potential gravitational energy. Comparing parts of the lab: When looking at the graph for each time the cart hits the spring

the mechanical energy decreases. It looks to decrease in a stepwise fashion decreasing faster at first but then slowing down as the time progresses. Since the mechanical energy equals all the other energies we also see a stepwise decline in those energies. The likely reason for this decline is that friction is getting involved when the car is moving on the track and hitting against the spring. This causes energy to be lost as thermal energy which results in the total mechanical energy decreasing. In principle this lab was supposed to be a closed system but the friction still had a significant impact. When comparing all the impacts of the cart and the spring it was interesting that at each interaction when all the energy was transferred to the potential elastic energy the mechanical energy went down to zero and then increased back up when the potential elastic energy was transferred back to the kinetic energy. When comparing the time vs. position graph and the mechanical energy graph there were some similarities whereas the time progressed they both decreased because of the thermal energy released from friction. The graphs having similar outcomes makes sense because as the position of the cart after each collision decreases the mechanical energy should decrease because there will be less (potential gravity, potential elastic, and kinetic energies). After performing this experiment and looking at the results most of are hypotheses were supported. We predicted that after each collision with the spring and the cart that the mechanical energy would decrease because of losses with thermal energy from friction. This is what happened after each collision: the cart's mechanical energy did decrease. We also predicted that the potential gravitational energy of the cart would be transferred to kinetic energy and then at the bottom the kinetic energy would be turned into potential elastic energy. We said the same was true going back up the ramp. This was exactly what happened as seen on the graph whereas the cart moved down the ramp the potential energy line sloped down while the kinetic energy line sloped up. Conclusion: For this experiment our group was able to successfully graph the energies of the cart as it moved up and down ramp as it was hitting against a spring. The hypotheses that we formulated were correct and were supported by this experiment. In theory the cart on the ramp coming into contact with the spring at the bottom of the ramp should have been a closed system. We didn’t observe this though because we did see the effects of friction on the cart because the thermal energy caused our mechanical energy to decline after each time it hit the spring and moved up the track. It would be best to remove this friction to see the real conservation of energy in the closed system but this would be difficult. If the track was frictionless the cart would not be able to make it back up the ramp. The best strategy seems to just ignore the friction and accept the decline in the mechanical energy. In the future it would be interesting to try this experiment at different angle heights and then see how mechanical energy changes due to the nonconservative forces. Overall this lab was a success where we were able to display how a cart's mechanical energy stays constant in a closed system (cart, track, and spring) and that all the energies equal to the same mechanical energy at any point along the track (in theory). Ways to improve the Lab: In the future it may be best to get new force loop sensors because

they were a little worn and bent. The sensor despite its age did still work for this experiment but it could have led to higher uncertainties in our end results. Other than that the lab did not have any big issues and was easy and straightforward to complete. In the future it would be interesting to see the graphs for the mechanical energy of the cart at different angle heights. By doing this we could see how the non-conservative forces slow the carts at different ramp angles. It would be expected that the smaller the angle of the ramp the quicker the cart would slow down from friction. Extension Questions: 1.) If the friction between the cart and the track is increased, what would be the effect on your plots? If the friction increases then the thermal energy would increase. This would result in a lower amount of mechanical energy after each run (going down the track hitting the spring and bouncing off). The mechanical energy would decrease faster. 2.) In addition to a friction force that is parallel to the track, the force that the track exerts on the cart also has a normal component (a component perpendicular to the track). This component is in fact relatively large. Why were we able to ignore it in our analysis of Mechanical Energy? For our experiment the cart was in a closed system meaning that nonconservative forces had no effect and were ignored in calculating the cart mechanical energy. Since normal force is a non-conservative force we did not have to include it. We only considered the action-reaction forces. (ex cart and t...