Lab 11: April 10, 2017: Work-Kinetic Energy Theorem
Lab 11: Work-Kinetic Energy Theorem
Amy, Alex, and Cristian
April 10, 2017
Today's lab was to prove the work energy theorem by comparing the change on Kinetic energy and using the formula for work which is the force times distance.
Theory: By comparing the two formulas for work we can calculate the Kinetic Energy in the system of a cart with a hanging mass. In theory both results should equal each other. We are also measuring the work done on a cart by a constant force as well as a non constant one.
Apparatus and Procedure:
For this lab we had to measure the work done on a cart using a motion sensor, a force probe, a track, and a laptop. The motion sensor had to be constantly calibrated in order to get accurate data.
The first experiment had a level track with a friction-less cart riding on the track. On the cart was a 500 gram mass and a force sensor that had a string tied to it that ran over a pulley on the edge of the track tied to a hanging mass off the side.
On the side of the track where the cart starts was a motion sensor that would detect the motion of the cart (this gave us position and velocity readings).
Since the force of tension in the string pulling the cart is known (equal to the gravitational force from the hanging mass) we could calculate work by using this Tension force times the displacement reading from the motion sensor.
We can also find the kinetic energy gained by the cart at any point the during the movement of the cart by plugging the value of velocity at that point into our KE formula:
KE=.5mv^2 by graphing the values we calculated from the changing work of the cart we came up with similar results for our Kinetic Energy.
The second part of the experiment we had to use a non constant force where we used a spring connected to the cart. Due to the spring not being a constant force the work of the cart changes during the motion of pulling the cart and letting go. The force of the spring can be calculated with Hooke's Law where k is the spring constant. If we have the value of spring force from the force sensor, and we know the displacement of the cart from the starting from the point where the spring is unstretched, we can record this data and do a proportional fit to find a slope of the graph which is our spring constant. Since the graph is a plot of Force vs distance, we can also find the work done by finding the area under the curve of this graph. LoggerPro has a function that finds this value for us (integration), but we can also integrate to derive a familiar expression:
The third part of the experiment we use the same setup as experiment 2 but now we start from the point where the spring is stretched and release the cart. As the cart moves it records changing values of force vs position as well as changing values of kinetic energy vs position. We compare these two values again just as we did in experiment 1 and theoretically they should be equal. From initial release point, a couple different points are tested to compare the values at that given position. Similarly to experiment 2, we utilize the integration function within LoggerPro to find the area under the curve of of the force vs position graph to find the work. We also use our kinetic energy formula to find the value at the given speed at that position.
The last part of the experiment involved watching a video and sketching the graph and measuring the work done on a cart that was hooked on a rubber band.
Data:
Mass of our cart was 0.499 kg
Original hanging mass was 0.5 kg but then we went lower to 0.05 kg
Overall the data follows our expected trend that work done by a force over a given distance is equal to the change in kinetic energy. Some sources of uncertainties within our experiment gives us values that are slightly higher or lower than expected. One main reason is the calibration of our force sensor. Through multiple trials of experimentation we found that the force readings were going slightly off. With a hanging mass of 0.050 kg, our expected force reading in Newtons should be 0.050 kg * 9,8 m/s^2 = 0.49 N. But we see from our graphs the the readings for force that we are getting are slightly lower, hovering around 0.380 to 0.410 N. This would actually give us higher readings for our area under the force position graph. In any derivation of values we did in previous labs, we found whenever a value with uncertainty is taken to some higher power (squared in this particular calculation) the margin of error it produced increased greatly. It could be that our readings for velocity weren't quite as accurate from the motion sensor. This would explain some of the dips in kinetic energy readings. While we would expect the KE graph of an object with constant acceleration to continue speeding up, there are small dips and irregularities in our graph. The readings that we took from our motion sensor were erratic and the length of time we collected data was also rather short. Squaring this value introduces even greater uncertainty in our calculations.
Similarly for our calculations for non-constant spring force, our values were very close but slightly off in the same trend as the previous, our calculated value for work from spring force was slightly higher than change in KE. Main source of error is most likely the same as previous experiments where our equipment and readings obtained for velocity had a lot of fluctuation in data. We initially also had a lot of trouble setting up our equipment because of faulty cables and the force probe and sensor would not connect.
Theory: By comparing the two formulas for work we can calculate the Kinetic Energy in the system of a cart with a hanging mass. In theory both results should equal each other. We are also measuring the work done on a cart by a constant force as well as a non constant one.
Apparatus and Procedure:
The first experiment had a level track with a friction-less cart riding on the track. On the cart was a 500 gram mass and a force sensor that had a string tied to it that ran over a pulley on the edge of the track tied to a hanging mass off the side.
On the side of the track where the cart starts was a motion sensor that would detect the motion of the cart (this gave us position and velocity readings).
Since the force of tension in the string pulling the cart is known (equal to the gravitational force from the hanging mass) we could calculate work by using this Tension force times the displacement reading from the motion sensor.
We can also find the kinetic energy gained by the cart at any point the during the movement of the cart by plugging the value of velocity at that point into our KE formula:
KE=.5mv^2 by graphing the values we calculated from the changing work of the cart we came up with similar results for our Kinetic Energy.
The second part of the experiment we had to use a non constant force where we used a spring connected to the cart. Due to the spring not being a constant force the work of the cart changes during the motion of pulling the cart and letting go. The force of the spring can be calculated with Hooke's Law where k is the spring constant. If we have the value of spring force from the force sensor, and we know the displacement of the cart from the starting from the point where the spring is unstretched, we can record this data and do a proportional fit to find a slope of the graph which is our spring constant. Since the graph is a plot of Force vs distance, we can also find the work done by finding the area under the curve of this graph. LoggerPro has a function that finds this value for us (integration), but we can also integrate to derive a familiar expression:
The third part of the experiment we use the same setup as experiment 2 but now we start from the point where the spring is stretched and release the cart. As the cart moves it records changing values of force vs position as well as changing values of kinetic energy vs position. We compare these two values again just as we did in experiment 1 and theoretically they should be equal. From initial release point, a couple different points are tested to compare the values at that given position. Similarly to experiment 2, we utilize the integration function within LoggerPro to find the area under the curve of of the force vs position graph to find the work. We also use our kinetic energy formula to find the value at the given speed at that position.
The last part of the experiment involved watching a video and sketching the graph and measuring the work done on a cart that was hooked on a rubber band.
Data:
Mass of our cart was 0.499 kg
Original hanging mass was 0.5 kg but then we went lower to 0.05 kg
We manipulate the first graph from above and change the x axis to a function of time. We also add a new column for finding kinetic energy (using the velocity readings from the motion sensor) and plot it on this same graph. We use the integration function on LoggerPro to find the area under the curve of the Force vs distance graph and compare it to the KE value at a specific point
Conclusion: The work done on cart by spring is always slightly higher than change in KE but the values are similar. As the force of the spring accelerates the cart, it gains speed. Starting from a position at rest, at the points on the graph we chose to find values of work by spring compared to the KE at that point, the cart has gained speed up to that point giving us the change in kinetic energy. We know from the work-kinetic energy theorem that work is equal to change in kinetic energy and the change in speed comes from the force of the spring accelerating the cart.
Overall the data follows our expected trend that work done by a force over a given distance is equal to the change in kinetic energy. Some sources of uncertainties within our experiment gives us values that are slightly higher or lower than expected. One main reason is the calibration of our force sensor. Through multiple trials of experimentation we found that the force readings were going slightly off. With a hanging mass of 0.050 kg, our expected force reading in Newtons should be 0.050 kg * 9,8 m/s^2 = 0.49 N. But we see from our graphs the the readings for force that we are getting are slightly lower, hovering around 0.380 to 0.410 N. This would actually give us higher readings for our area under the force position graph. In any derivation of values we did in previous labs, we found whenever a value with uncertainty is taken to some higher power (squared in this particular calculation) the margin of error it produced increased greatly. It could be that our readings for velocity weren't quite as accurate from the motion sensor. This would explain some of the dips in kinetic energy readings. While we would expect the KE graph of an object with constant acceleration to continue speeding up, there are small dips and irregularities in our graph. The readings that we took from our motion sensor were erratic and the length of time we collected data was also rather short. Squaring this value introduces even greater uncertainty in our calculations.
Similarly for our calculations for non-constant spring force, our values were very close but slightly off in the same trend as the previous, our calculated value for work from spring force was slightly higher than change in KE. Main source of error is most likely the same as previous experiments where our equipment and readings obtained for velocity had a lot of fluctuation in data. We initially also had a lot of trouble setting up our equipment because of faulty cables and the force probe and sensor would not connect.
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