Lab 11: Work-Kinetic Energy Theorem Activity
Chris Ceron
Goal of Lab
Prove the work-kinetic energy theorem by comparing the calculated value of 
KE and comaparing it to force times distance
Theory/Introduction
We define work to be a force multiplied by a distance:
The work-kinetic energy theorem states:
In this lab, we tested different experiments to compare the calculated value of work and comparing it to the value returned through LoggerPro.
Experimental Procedure
Experiment 1: Work Done by a Constant Force
For experiment one, we used a level track for a frictionless cart that was attached to a hanging mass. The hanging mass also went through a frictionless pulley system.
Our apparatus also includes a force and motion sensor, and a stopper for the cart.
Once the sensors were calibrated, we attached the force sensor to the cart, and attached the hanging mass to the force sensor using a lightweight string. We placed the motion sensor behind the cart to give us the displacement vs time of the cart while it is being pulled by the hanging mass. Through a free body diagram we found that the tension on the string was equal to the hanging mass times gravitational acceleration. We used this tension force and multiplied it by the displacement to calculate the work done.
We also found the change in kinetic energy by using LoggerPro to find the velocity of the cart. From there we used our kinetic energy formula:
Experiment 2: Work Done by a Nonconstant Spring Force
For experiment 2, the work done changes based on the displacement of the cart. The cart is attached our force sensor using a spring.
For experiment 2, the work done changes based on the displacement of the cart. The cart is attached our force sensor using a spring.
The force of a spring is:
To find the value of k, we recorded the cart when the spring was unstretched to stretched to a known distance. LoggerPro produces a Force vs. Distance graph, and this is used to calculate both the value of k by using a proportional fit and the work done on the cart. To calculate work, we used the change of the spring's potential energy:
The spring's initial potential energy is 0, so the final equation used was:
The spring's initial potential energy is 0, so the final equation used was:
Experiment 3:
For experiment three. we stretched the cart to the same known distance as experiment two. We recorded the changing values of force vs. distance as the spring pulls the cart and the changing values of kinetic energy. These values are predicted to be the same. Similarly to experiment two, we use LoggerPro to find the area under the graph of force vs. distance and compare the two values of work.
For experiment three. we stretched the cart to the same known distance as experiment two. We recorded the changing values of force vs. distance as the spring pulls the cart and the changing values of kinetic energy. These values are predicted to be the same. Similarly to experiment two, we use LoggerPro to find the area under the graph of force vs. distance and compare the two values of work.
Experiment 4:
For experiment four, we watched a video in which a professor uses a machine attached to a rubber band. The set up in the video included a force reading, photogates to measure the displacement, and time intervals. We used the results in the video to calculate work by multiplying the force times the distance.
For experiment four, we watched a video in which a professor uses a machine attached to a rubber band. The set up in the video included a force reading, photogates to measure the displacement, and time intervals. We used the results in the video to calculate work by multiplying the force times the distance.
Data:
Experiment 1:
The hanging mass in experiment one was .050kg
The total mass of the cart was 1.19kg.
Experiment 1:
The hanging mass in experiment one was .050kg
The total mass of the cart was 1.19kg.
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| Our initial readings for experiment one. Shows a constant force. |
We manipulated the force graph above to also display kinetic energy by creating a new column that multiplied the mass (constant) and velocity of the cart based on the sensor's recording.
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| Force graph also displays kinetic energy |
We changed the time axis from three to one and found the area under the force graph using LoggerPro's integral feature. We then displayed the value of kinetic energy for our highlighted section and compared the value to the area under our force graph.
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| Displays the values for kinetic energy and the area under the graph for the highlighted section |
The value we received for the area under the graph was 0.1659 N*m
The value we received for the change in kinetic energy for the same section was 0.156 J (J = N*m)
The two values are relatively close.
Experiment 2:
As mentioned in the Theory/Introduction, we first found the value of k by recording the cart when it was pulled to a known distance. The distance we stretched our cart to was 0.6 m. Through LoggerPro, we could find the slope of the graph to give us our value of k.
The value of k we obtained was 3.460 N/m.
Using this graph, we could also find the area under the graph to find work.
Experiment 4:
Using this graph, we could also find the area under the graph to find work.
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| Work done by stretching the spring |
The value we obtained for work was 0.5146 J
We also calculated our work using the elastic potential energy formula
Experiment 3:
The following graph shows a specific point where we compared the area under the Force graph vs. position to kinetic energy. The value for the integral of the force graph is negative due to the positioning of our motion sensor, so we compare the magnitudes.
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| Area under the graph = -.6583 N*m. Kinetic Energy of the same section = 0.675 J |
For experiment four, are calculations were based on splitting up sections of the graph into simpler calculations of triangle, trapezoids, and rectangles.
For triangles, we used:
For trapezoids, we used:
For rectangles, we used:
Calculation:
Using the displacement and time provided in the video, we can calculate our final velocity, and final kinetic energy:
The two values are relatively close.
Conclusion:
The work done on the cart by the spring are relatively similar to its change in kinetic energy. As the spring exerts a force on the cart, the cart continues to increase in velocity. Similar to the experiments, you can find any two points while the spring exerts a force on the cart, and use the velocities at those points to find work. Work is the change in kinetic energy according to the work-kinetic energy theorem, and the change in velocities come from the spring's force accelerating the cart.
Overall, the data followed our expectations as the area under the graph of force vs. distance was very similar to the kinetic energy. Some uncertainties that caused our numbers to slightly differ was the calibration of our force sensor. We calibrated the sensor more than once because it was recording incorrect values when tested. Another uncertainty could have also been the motion sensor. The sensor will not accurately record the carts distance if the cart begins its displacement too close to the sensor. This will cause a fluctuation in our velocity, which results in it affecting our kinetic energy.
Although these uncertainties existed, they were only slightly present. Our numbers in each experiment were very close and we could easily conclude that the area under the graph of force vs distance and kinetic energy are the same value.
For rectangles, we used:
Calculation:
Using the displacement and time provided in the video, we can calculate our final velocity, and final kinetic energy:
The two values are relatively close.
Conclusion:
The work done on the cart by the spring are relatively similar to its change in kinetic energy. As the spring exerts a force on the cart, the cart continues to increase in velocity. Similar to the experiments, you can find any two points while the spring exerts a force on the cart, and use the velocities at those points to find work. Work is the change in kinetic energy according to the work-kinetic energy theorem, and the change in velocities come from the spring's force accelerating the cart.
Overall, the data followed our expectations as the area under the graph of force vs. distance was very similar to the kinetic energy. Some uncertainties that caused our numbers to slightly differ was the calibration of our force sensor. We calibrated the sensor more than once because it was recording incorrect values when tested. Another uncertainty could have also been the motion sensor. The sensor will not accurately record the carts distance if the cart begins its displacement too close to the sensor. This will cause a fluctuation in our velocity, which results in it affecting our kinetic energy.
Although these uncertainties existed, they were only slightly present. Our numbers in each experiment were very close and we could easily conclude that the area under the graph of force vs distance and kinetic energy are the same value.
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