- Figuring out the spring constant from force vs. position graph
- Determining the relationship between the work done and kinetic energy.
- Verifying that the work done is equal the kinetic energy
Part 1: Work Done by a Non-constant Spring Force.
Objective:
Figuring out the spring constant of our spring and use the integration routine in the software to find the work done in stretching the spring.
Equipment needed:
Ramp, cart, motion detector, force probe, spring, weight, C-clamp, and computer.
Set-Up:
1. Let the ramp lie on the horizontal table.
2. Using the C-clamp to secure the force probe at the edge of the table.
3. Let the motion detector sit on the other side of the ramp, and connect it with the Logger Pro.
4. Connecting the spring to the force probe.
5. Connecting the other side of spring to the cart whose body holds some weight and a square.
Figure 1 shows the apparatus of the experiment.
Figure 1: The apparatus of the experiment
How to start the experiment:
1. Calibrating the force probe with a force of 4.9N applied.
2. Zero the force probe and the motion detector with the spring unstressed or uncompressed.
3. Set the motion detector to "Reverse Direction" so that when the cart moves toward the motion detector, we would obtain the positive direction.
4. Then slowly push the cart move slowly towards the motion detector until the spring is stretched about 1.0m.
Data Collecting:
Since we connect the motion detector with the Logger Pro, we shall get a graph of force vs. position as the cart moves toward the motion detector.
Below is our graph force vs. position (Figure 2)
Figure 2: The graph of force vs. position graph
Back in physics 2AG, we know that as the spring stretches, force is related to position through an equation:
F = k * x
F: force applied to the cart
k: constant spring
x: the stretch of the string.
Since the relationship between force and position of spring is linear, we can figure out the value of the constant spring by taking the value of slope of the graph force vs. position. Thus we linearize the graph of force vs. position and obtain the value of slope like below.
In this graph, we obtain the value of m = 2.855N/m, thus the value of spring constant is 2.855N/m.
Figure 3: Linearizing the graph of force vs. position graph and obtaining the value of slope.
In this graph, we obtain the value of m = 2.855N/m, thus the value of spring constant is 2.855N/m.
After figuring out the value of spring constant, we use the integral to calculate the work done in stretching the spring. The calculation is shown in figure 4.
Figure 4: Integrating the area to find the work done.
W = 0.1541 J
Part 2: Kinetic Energy and The Work-Kinetic Energy Principle
Objective:
Determining the relationship between the work done and the kinetic energy.
Set-up:
We use the same set-up in part 1, which is shown in figure 1.
How to perform the experiment:
1. In this part, we first measure the mass of the cart. (Figure 5)
2. We set up a new column in Logger Pro to calculate the kinetic energy which is this equation:
Figure 5: Measuring the mass of the cart.
m(cart) = 574g.
2. We set up a new column in Logger Pro to calculate the kinetic energy which is this equation:
KE = (1/2) * m * v2
3. Zero the force probe and the motion detector, then pull the cart so that the spring is stretched about 1.0m from the un-stretched position. (Figure 6)
4. Release the cart so that it moves toward the spring.
Figure 6: Pull the cart so that the spring is stretched around 1.0m
4. Release the cart so that it moves toward the spring.
Data Analts:
As the cart moves toward the spring, we would obtain the graph of force vs. position using Logger Pro. Also, on the y-axis, adding the kinetic energy. The graph of force vs. position and kinetic energy vs. position are shown in figure 7.
Figure 7: The graph of force vs. position (Purple) and kinetic energy vs. position (Dark Blue).
Now to figure out the relationship between work done and kinetic energy, we use integral to calculate the work done and also examine the value of kinetic energy, then compare them. Below are three spots we compare work done and kinetic energy.
Figure 8: First position comparing work done and kinetic energy.
| W | = 0.1458N/m,
KE = 0.146N/m
Figure 9: Second position comparing the work done and the kinetic energy
| W | = 0.1232N/m,
KE = 0.123N/m
Figure 10: Third position comparing the work done and kinetic energy
| W | = 0.1076N/m,
KE = 0.107N/m
From three different position comparing the work done and kinetic energy, we notice that the value of the work done and kinetic energy is almost the same. Thus, we can conclude that the work done is equal the kinetic energy
W = KE
Part 3: Work-KE Theorem
Objective:
Comparing the result of the work done and the kinetic energy in order to verify the relationship between two of them one more time.
Set-Up:
In this part, we don't set up any equipment. The professor would turn on a video capturing the moment when a professor did an experiment to figure out the relationship between the work done and kinetic energy in the old time. In the video, the professor uses machine to pull back on a large rubber band, and the force exerted on the rubber band is recorded by an analog force transducer onto a graph. (Figure 11)
Figure 11: Professor is performing the experiment.
The stretched rubber band is attached to a cart of known mass. Then the professor releases the cart to let it pass through two photo gates a given distance apart. Since we know the distance and time that the cart passes through the first and second photo gate, we can use them to figure out the speed in order to calculate the kinetic energy.
Figure 12: The set-up will give us the time and distance as the car passes through two photo-gates.
Data Collecting:
As the cart passes through two photo gates, the professor records the time, the distance, and mass of the cart.
m = 4.3kg
x = 15cm = 0.15m
t = 0.045s
Data Analysis:
Since we have the mass of the cart, the time and distance, we can use this below equation to find out the kinetic energy
KE = (1/2) * m * v2
Calculating the kinetic energy:
Figure 13: Calculating the kinetic energy of the object.
KE = 23.89J
For the work done, we would stop the movie and make a careful stretch the force vs. position graph, then divide the graph into calculable shapes that allow us to use equation to calculate the work done such as triangle, trapezoid, or rectangular.
Calculating the work done:
Figure 14: Calculating the work done.
W = 26.6J
We have W = 26.6J and KE = 23.9J. However, we found out in part 2 that the work done "have to be" equal the kinetic energy of the object. This means that there are some errors while performing this experiment.
Some possible errors in this experiment leading to the unequal value of work done and KE.
1. As professor pull back the machine, she created three different graph of force vs. stretch. Because we randomly use one of three different graph to calculate the work done, we will get the result that a little bit off from the actual value.
2. While we calculate the work done, we also estimate the height or base of triangular or rectangular. Therefore, the result we get for work done have some uncertainty with it.
3. The equipment that professor used in this experiment are too old that may not give us the accurate value like our equipment today. What I mean right here is the photo gate and the time watch that professor used in this experiment.
Conclusion:
In this lab, we manage to find the spring constant in part 1. The result is k = 2.855N/m. In part 2, we determine the relationship between the work done and kinetic energy of the object. After checking three different positions on graph force vs. position, and kinetic energy vs. position, we find out that the work done is equal the kinetic energy. In the last part, we will use our result in part 2 to check an old experiment. We calculate the work done and kinetic energy of the old experiment: W = 26.6J and KE = 23.9J. The results of work done and kinetic energy should be the same, though they are actually not. Therefore, we can conclude that there were some possible errors during this experiment. We come up with three possible errors that explained in part 3.
Some possible errors in this experiment leading to the unequal value of work done and KE.
1. As professor pull back the machine, she created three different graph of force vs. stretch. Because we randomly use one of three different graph to calculate the work done, we will get the result that a little bit off from the actual value.
2. While we calculate the work done, we also estimate the height or base of triangular or rectangular. Therefore, the result we get for work done have some uncertainty with it.
3. The equipment that professor used in this experiment are too old that may not give us the accurate value like our equipment today. What I mean right here is the photo gate and the time watch that professor used in this experiment.
Conclusion:
In this lab, we manage to find the spring constant in part 1. The result is k = 2.855N/m. In part 2, we determine the relationship between the work done and kinetic energy of the object. After checking three different positions on graph force vs. position, and kinetic energy vs. position, we find out that the work done is equal the kinetic energy. In the last part, we will use our result in part 2 to check an old experiment. We calculate the work done and kinetic energy of the old experiment: W = 26.6J and KE = 23.9J. The results of work done and kinetic energy should be the same, though they are actually not. Therefore, we can conclude that there were some possible errors during this experiment. We come up with three possible errors that explained in part 3.
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