Name: Aaron Nguyen
Partners: Daniel and Vanessa
Theory/Introduction:
The Work-Energy Theorem states that the work done on an object by a net force equals the change in kinetic energy of the object:
W = KEf – KEi
The purpose of this lab is to explore the relationship of the Work-Kinetic Theorem with three individual cart experiments.
Experiment #1: We used a system where a constant force is applied to a cart that is initially at rest. According to the theorem the the work done on the cart by the force should equal the kinetic energy acquired by the cart at any point. To help prove the theorem we calculated the left side and right side of the equation separately from our data collected from the experiment.
We calculate the work done on the cart by making a force vs. position graph and taking its area under the curve (Concept: Work=F*d).
We calculated the change in kinetic energy by recording the initial and final velocity of the cart and plugged it into the formula: 
mvf 2 – mvi 2
Experiment #2: We calculated the work done when a non-constant force is applied to the cart. Similarly to experiment #1, we made a force vs. position graph and took the area under the curve.
Experiment #3: We calculated the work done when the cart undergoes a non constant acceleration from rest. Similarly to experiment #1, we are compared the change in kinetic energy and the work done, but this time at three different intervals.
Apparatus/Procedure:
Experiment #1: To find the work done by a constant force we will be using a motion sensor, force sensor, cart, track, pulley, LoggerPro, and hanging mass:
To record the constant force acting on the cart we will be attaching a force sensor onto it. To simulate the constant force acting on the cart we used a cart-pulley system where one end of the string is attached to the force sensor while the other is attached to a hanging mass. The constant tension in the string will pull the hanging mass a set distance.
To record the position of the cart we will be using a motion sensor that will be placed at the end of the track.
To collect the data on LoggerPro we set the cart at around 15 cm away from motion sensor, hit collect, then let the tension in the string pull the cart.
Experiment #2: To calculate the work done by a non-constant force we used a motion sensor, force sensor, cart, track, pole, clamp, spring, and LoggerPro:
To simulate a non constant force we used a spring that was attached to the force sensor on the cart. The other end of the spring was attached to a clamped pole. Since in this experiment the cart was going the opposite way we changed the positive-direction to be towards the sensor instead of the default (away from the sensor). After the system was set up we pressed collect on LoggerPro and slowly pushed the cart from it’s resting position toward the motion sensor.
Experiment #3: We used the same set up as experiment #2, but we changed the motion sensor back to where the positive-direction was away from the motion detector. This time we pulled the cart a set distance, pressed collect, then let go of the cart.
Analysis:
Experiment #1:
After recording our data into logger pro we made a force vs. position graph. We also needed to graph the kinetic energy and since LoggerPro doesn’t give is an option to select kinetic energy as a y-axis we had to implement the equation for kinetic energy into LoggerPro which was mv 2 . We added kinetic energy into the y-axis of the force vs. position graph. By taking the integral of the curve we are able to get the work done on the cart over a set distance (left-side of the equation). By examining the y value of the kinetic energy curve at the same distance we were able to get the acquired kinetic energy at that point (right-side of the equation). After we compared the percent difference between work done and the change in kinetic energy.
*note: Unfortunately I cannot find the picture that shows the integral of the force curve and the y-value of the kinetic energy curve so we could not compare the values.
Experiment #2:
After recording the data into LoggerPro we made a force vs. time graph:
The work done on the cart was the integral of the curve which was 0.321 J. By taking the slope of the line we are able to get the spring constant of the spring, “3.3”. We know the slope is equal to the spring constant from the following derivation:
Experiment #3:
After recording the data into LoggerPro we made a force vs. time graph. To explore the reaches of the work-kinetic energy theorem we took the work done along with its change in kinetic energy over three different intervals:
| Work (Integral) | Kinetic Energy | Percent Difference | |
| Interval #1 | -0.4538 J | 0.522 J | 14.0% |
| Interval #2 | -0.322 J | 0.352 J | 8.9% |
| Interval #3 | -0.1677 J | 0.199 J | 17.1% |
Experiment #4:
After finishing our experiments we ended up watching a video from Professor Wolf. In the video. the professor uses a machine to pull back on a large rubber band. The force on the rubber band was recorded by an analog force transducer onto a graph. The stretched rubber band is then attached to a cart of known mass. The cart, once released, passes through two photo gates a given distance apart.
From the video we were asked to calculate the work done and the kinetic energy of the cart and compare the results just like how we did in the previous experiments.
By making a careful sketch of the force vs. position graph shown in the video and taking the area underneath it by chopping it up into rectangles and trapezoids, we were able to determine the work done by the machine stretching the rubber band.
Adding up the area of all the rectangles and trapezoids gives us a work of 22 J.
To calculate the kinetic energy we need the velocity and mass of the cart to plug into the formula: mvf 2
We were given the mass of the cart: 4.3 kg
We were not given the velocity of the cart, however, we were given the time interval of the cart passing from one photo gate to the other (.0455 sec) and the distance between each photo gate (.15 m). With these two values we are able to calculate the velocity since:
This gives us a kinetic energy of 23.84 J.
The percent difference between these two values is 8%
Conclusion:
Overall our results from our experiments were fairly accurate, but not ideal. Usually for our experiments a percent difference of 5% or less would indicate a well done experiment however ours was unfortunately not in that range. A trend that seemed to occur in the data from experiment #3 was that the calculate kinetic energy was greater than the work done for each experiment. Since only three intervals were taken it could quite easily be a coincidence. Some systematic errors that could have increased the percent difference was the friction between the wheels of the cart and the track. This would decrease the value of kinetic energy.
phys4AF18avnguyen 