Lab 11: Work-Kinetic Energy Theorem Activity
Joel Cook
Eric Chong
Max Zhang
Eric Chong
Max Zhang
Lab performed April 10, 2017
The goal of the lab performed was to confirm the work-kinetic energy theorem that states that the work done on an object is equal to the change in kinetic energy of the object.
Intro: In this set of experiments, we measure force, position and velocity in different conditions to calculate the work done on an object. The area under the force versus position graph is the work done on the object. By calculating kinetic energy from mass and velocity, we can compare the change in kinetic energy over some portion of the graph to the area under the graph for the same portion (the work done). By the work-kinetic energy theorem, these two values should be equal.
Apparatus/Procedure:
Experiment 1:
In the first experiment we set up a track, cart, motion sensor, force sensor and hanging mass as shown.
The force sensor was calibrated and both sensors were zeroed. We then pulled the cart toward the motion sensor and released the cart. Logger pro was used to record the force vs time, position vs time, and velocity vs time. Work equals the product of the tension force in the string and the displacement of the cart. This work should be equal to the kinetic energy of the cart during the same displacement. By plotting kinetic energy (in joules) on the graph of force (newtons) vs time (seconds), we can take the integral of the force graph during some displacement and compare this value to the change in kinetic energy during the same displacement. The two values should be equal.
Experiment 2:
For the second experiment, we set up the apparatus as before but instead attached a ring stand to the table and placed the force sensor on the ring stand. A spring was attached between the force sensor and the cart.
The graph below shows force vs position for a constant force of 3 Newtons over 7 meters. Since work is equal to F*X, the work would be 21 J. Since work is force times displacement, work is equal to the sum of force at each position, multiplied by the position, which is the area under the curve.
For a non-constant force, such as below, we expect the work to equal the area under the curve as it did above. Using calculus, if we partition the graph below vertically, we can find the sum of these partitions as the width of each gets infinitely thin. We find that the work is equal to the integral of the force function times the infinitely thin partitions. For the function below, f(x)=x^2, the integral of the force function from position 0 to 7 meters equals the area under the curve, or approximately 114.3 joules.
The force and motion sensor were zeroed with the spring completely slack. Data for force vs position was collected as the cart was pulled toward the motion sensor. As the cart was pulled closer to the motion sensor, the spring was stretched further and the force increased as the position increased. From this graph we could determine the spring constant for the spring. Since force equals the product of spring constant and displacement, a graph should have a constant slope, which would be the spring constant. Finally, integration was used to calculate the area under the curve, which would be the work done.
Experiment 3:
For the third experiment, the same setup as above was used with the spring. For this portion of the lab, we want to verify that the work done is equal to the change in kinetic energy over the same distance. We pulled the cart toward the motion sensor and collected data once the cart was released and the spring pulled the cart down the track. We expect that the force will decrease as the cart moves away from the motion sensor and the kinetic energy will increase as the cart speeds up. KE=0.5*mass*velocity^2
Integration was performed on the graph of force vs position to determine the work done and kinetic energy was calculated and plotted vs position. We expect the work done over some interval is equal to the change in kinetic energy over the same interval.
Experiment 4:
For the last portion of the lab, we watched a video in which a person pulls back on a machine attached to a large rubber band. The machine manually graphs force vs position while the rubber band is stretched. The machine is attached to a cart of know mass and released. The time for the cart to pass between two photogates and the distance between the photogates is known. We sketched the graph of force vs position from the video and calculated the work by adding together the areas under the curve. We then compared this value to the change in kinetic energy calculated from the velocity of the cart.
Data:
Experiment 1:
The photo below shows the graphs generated for position vs time, velocity vs time and force vs time. The velocity was used to generate a kinetic energy graph from KE=1/2*m*v^2 to plot kinetic energy vs position.
As the two photos below show, we integrated the force vs position graph over an interval and compared the value to the kinetic energy at the end of the interval. The following table shows the data:
Position (m)
|
Integral Value (N*m)
|
Kinetic Energy (J)
|
0.192
|
0.09598
|
0.132
|
0.765
|
0.2017
|
0.269
|
Experiment 2:
The first photo below shows the graph generated from part two of the lab in which the cart attached to a spring was pulled slowly toward the motion sensor. Force vs position was graphed as shown below. In the second photo, the integral of the function was calculated to find the work done and the slope of the line was calculated using a proportional fit. The slope of the line should be the spring constant of the spring. A proportional fit was chosen because the force at position zero was zero and we expect a linear graph.
The work done (integral) was 0.3130 N*m. The slope of the force vs position graph was 2.206, which is approximately the spring constant for the spring used.
Experiment 3:
For experiment three, we generated the following graph from recording force and position after releasing the cart. We struck through the data from after the cart impacted the stop on the track.
As shown below, we plotted kinetic energy vs position and force vs position on the same graph to analyze and compare. We expect that the area under the graph of force vs position will equal the change in kinetic energy. We performed three different integrations on the graph to make three comparisons of work vs change in kinetic energy.
The following table shows the results for work done vs change in kinetic energy.
Change in Position (m)
|
Work Done (J)
|
Change in KE (J)
|
0.009-0.000
|
-0.06206
|
0.544-0.592 = -0.048
|
0.155-0.009
|
-0.1947
|
0.155-0.352 = -0.176
|
0.352-0.155
|
-0.2888
|
0.352-0.531 = -0.274
|
Experiment 4:
The graph below is the graph generated from the rubber band machine in part 4 of the lab. We calculated the work by adding the areas under the graph of force vs position. Next we took the information from the video to calculate change in kinetic energy and compare the value to the work calculated.
The area under the curve (work) was calculated as:
W= 60(0.2)/2 + 60(0.1) + (60+32)(0.05)/2 + 32(0.25) = 22.3 J
The change in kinetic energy was calculated as:
Displacement = 0.15m
Time = 0.045s
Mass = 4.3 Kg
Velocity = Displacement/Time = 0.15/0.045 = 3.33 m/s
KE=1/2*m*v^2
KE = 1/2*4.3*3.33^2 = 23.9 J
Analysis:
Experiment 1:
For the first experiment, we expected the integral of force vs position (work) to be equal to the kinetic energy. As the table shows below, the first value was approximately 27% off and the second value was approximately 25% off.
Position (m)
|
Integral Value (N*m)
|
Kinetic Energy (J)
|
0.192
|
0.09598
|
0.132
|
0.765
|
0.2017
|
0.269
|
Experiment 2:
For the second experiment, we calculated the spring constant of the small spring we were using as the slope of the graph of force vs position. Our value was 2.206 N/m. The integral of the function was taken to find the work done, which was 0.3130 J.
Experiment 3:
The mass of the cart was 0.548 kg. Again, we expect the work done to be equal to the change in KE. As the chart below shows, our values were between 6 and 23% off from each other. The values were closer to being equal when the magnitude of the values were larger.
W=ΔKE
W=KEf-KEo
W=1/2*m*vf^2-1/2*m*vo^2
Change in Position (m)
|
Work Done (J)
|
Change in KE (J)
|
0.009-0.000
|
-0.06206
|
0.544-0.592 = -0.048
|
0.155-0.009
|
-0.1947
|
0.155-0.352 = -0.176
|
0.352-0.155
|
-0.2888
|
0.352-0.531 = -0.274
|
As shown previously, we calculated the work done from the graph in the video as 22.3 J and the change in kinetic energy from the velocity of the cart as 23.9 J. The values are within 7% of each other which again proves that change in kinetic energy is approximately equal to work done.
W= 60(0.2)/2 + 60(0.1) + (60+32)(0.05)/2 + 32(0.25) = 22.3 J
KE = 1/2*4.3*3.33^2 = 23.9 J
Conclusions:
After conducting multiple experiments with different conditions and sensors, we can confidently state that the work done on an object is equal to the change in kinetic energy of that object. None of the values came out exactly equal in the experiments performed but we do not realistically expect that the values would be exactly the same. For the experiments involving a cart on a track it is possible that our track was not perfectly level, which would affect force in a way we have not accounted for in our calculations. Additionally, even though the sensors were calibrated and zeroed, it is always possible that the sensors are not reading perfectly accurate data. Through experimentation, it has been determined that the force sensor, which uses magnetism to measure force, may have been affected by the magnets in the stop we were using on the track. This interference would have adversely affected our data and therefore our calculations. For the final experiment, we were watching an older video in which the experiment was performed with manual, analog methods and outdated equipment. This would affect the accuracy of the measurements in the video. These inaccuracies would be exacerbated when we may not have accurately interpreted, or transferred, data from the video for our calculations. Regardless of inaccuracies that may have existed, the results are sufficiently accurate for our purposes to conclude that the lab was a success and worthy of a future physics student's time and attention.
Conclusions:
After conducting multiple experiments with different conditions and sensors, we can confidently state that the work done on an object is equal to the change in kinetic energy of that object. None of the values came out exactly equal in the experiments performed but we do not realistically expect that the values would be exactly the same. For the experiments involving a cart on a track it is possible that our track was not perfectly level, which would affect force in a way we have not accounted for in our calculations. Additionally, even though the sensors were calibrated and zeroed, it is always possible that the sensors are not reading perfectly accurate data. Through experimentation, it has been determined that the force sensor, which uses magnetism to measure force, may have been affected by the magnets in the stop we were using on the track. This interference would have adversely affected our data and therefore our calculations. For the final experiment, we were watching an older video in which the experiment was performed with manual, analog methods and outdated equipment. This would affect the accuracy of the measurements in the video. These inaccuracies would be exacerbated when we may not have accurately interpreted, or transferred, data from the video for our calculations. Regardless of inaccuracies that may have existed, the results are sufficiently accurate for our purposes to conclude that the lab was a success and worthy of a future physics student's time and attention.
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