The purpose of this experiment was to prove the work-kinetic energy theorem holds to be true. The work-kinetic energy theorem states that the total amount of work done on a system is equal to the change in kinetic energy of the system.
Apparatus:
Figure 1: Apparatus of this experiment
Figure 2: Experimental setup used to obtain data
The apparatus in this experiment consisted of a computer with the LoggerPro software, which was used to obtain data needed for this experiment. The data needed was the position of the cart, which was obtained with a motion detector, and the force applied by the spring, which was obtained with a force sensor. As just stated, a cart and a spring were also used. Some PostIt notes were also pasted to the cart to allow the motion detector to more effectively sense the presence of the cart. Two 250 g weights and a "weight hook" (used to hold the weights) were also used to calibrate the force sensor. Although weights are shown on the cart, they were actually removed off the cart later on in the experiment since they affected the data obtained in a slightly negative way. A calculator was also used to hold the spring up and keep it in its "natural" position (unstressed). Lastly, a C-clamp was used to hold a metal stick, which was used to hold the force sensor in place.
Procedure (Part 1):
First, the motion detector and force sensor were set up to the LoggerPro software. The force sensor was then zeroed, and calibrated with a 500 g mass on LoggerPro. The rest of the apparatus was then set up, as shown in Figures 1 and 2. One end of the spring was attached to the force sensor, and the other end was attached to the cart. A force (N) vs position (m) graph was then opened on LoggerPro. The force sensor was again zeroed with the unstressed spring attached. The cart was then pulled toward the motion detector so that the spring was stretched about 0.4 m, with the cart at rest (the force applied by the hand equals the force applied by the spring). Collecting data with the motion detector and force probe, the cart was then let go so that the only force acting on it was the force of the spring until it reached the zero position (where the force sensor is located. The force vs position graph was then formed with the collected data.
Discussion (Part 1):
Figure 3: Force vs position graph of the cart using raw data
As seen in the graph above, the farther the position of the cart, the larger the force of the spring (consistent with Hooke's Law: F=kx). The rest of the graph after the origin (in the negative x axis) is just bad useless data resulting from the collision of the cart with the force sensor. Removing this data and forcing the graph to go through the origin (since the spring does not apply a force when its in its natural state) by subtracting each point by the amount the point on the y axis is above the origin, the actual graph of the force applied by the spring and the amount the spring is stretched (the position of the cart) for this experiment was obtained.
Figure 4: Actual force vs position graph of the spring
Since the graph goes through the origin, a proportional fit was created. Relating the slope-intercept form equation of a line (y = mx + b, where b = 0 since the graph's y-intercept is the origin) and Hooke's law (F = kx), it can be seen that the spring constant k can be found from this graph by taking the slope of the graph. The spring constant k of the sued spring was found to be 1.758 +/- 0.01568 N/m.
Conclusion (Part 1):
The uncertainty in the slope of the graph is very small, proving that the method to obtain the spring constant is effective. The uncertainty in "k" results from the fact that an ideal spring was not used, since an ideal spring is unattainable and not realistic, while Hooke's law assumes that an ideal spring is used.
Procedure (Part 2):
The mass of the cart was measured to be 0.505 kg. Using the same data from part 1, a new column for kinetic energy was created. The kinetic energy at each point was calculated using the kinetic energy formula: KE = (1/2)mv^2, where the velocity was obtained by the motion detector and the mass of the cart measured was used. The force vs position curve and the kinetic energy vs position curve were both displayed on the same graph, and, using the integration feature in LoggerPro, the work done by the spring on the cart and the kinetic energy of the cart were compared by taking the area under the curves.
Discussion (Part 2):
The work done was found from the area under the force vs. position curve and the kinetic energy of the cart was found from the area under the kinetic energy vs position curve. They were both compared at three different positions to prove the validity of the work-kinetic energy theorem. These positions are shown below:
Figure 5: Comparing the work done by the spring and the KE of the cart at the first position
Figure 6: Comparing the work done by the spring and the KE of the cart at the 2nd position
Figure 7: Comparing the work done by the spring and the KE of the cart at the 3rd position
As seen in Figures 5 through 7, the work done by the spring and the kinetic energy of the cart at the three different positions were very close (only off by an average of around 0.001), proving that the work-kinetic energy theorem is applicable and holds to be true.
Conclusion (Part 2):
The small error (difference) between the work and kinetic energy is mostly caused by the fact that an ideal spring was not used, which Hooke's Law assumes. It is also caused by the uncertainty in the measurement of the velocity of the cart by the motion detector.
The work-kinetic energy principle for this system can be stated as follows: The work done by the spring on the cart is equal to the change in kinetic energy of the cart; therefore, energy is conserved in the spring and cart system.
Procedure (Part 3):
Using LoggerPro, a previously made video that also proves the work-kinetic energy theorem was investigated. The professor in the video uses a machine to stretch a large rubber band as much as possible. The force exerted is recorded by a force transducer onto a graph. The stretched rubber band is then placed onto a cart, and the cart is released. It travels through two photogates of a known distance apart, and the time it takes to travel through the two photogates is measured.
The graph created from the force transducer was replicated onto the graph provided, with the y axis containing the force applied by the rubber band (in N) and the x axis consisting of the amount of the stretch of the rubber band (in m). The amount of work done by the rubber band can then be calculated by finding the area under the curve of the graph. The curve of the graph was simplified so that different shapes can be constructed under the curve, making it easier to calculate the area.
The kinetic energy of the cart was then found by first finding the velocity of the cart, which was done by dividing the total distance between the photogates by the total time it took for the cart to travel through the photogates. Then, the kinetic energy of the cart was found by plugging in the known mass and velocity into the kinetic energy fomula.
Discussion (Part 3):
The graph of the force of the rubber band vs its stretch is shown below:
Figure 8: The graph of force exerted by the rubber band vs its stretch
The area under the curve was split up into 6 usual shapes whose areas are easy to find. They include a large triangle (A1), two rectangles (A2 and A3), a thin rectangle (A5), a small thin triangle (A4), and a trapezoid (A6). Summing the areas of these shapes, it was found that the work exerted by the rubber band was 26.6 J.
To compare the work exerted by the rubber band to the kinetic energy of the cart, the previously proven work-kinetic energy theorem was used. The kinetic energy of the cart was found by first finding the velocity of the cart in between the two photogates. Using the distance it traveled (0.15 m) and the time it took to travel that distance (0.045 s), the velocity of the cart was found to be 3.33 m/s. Then, using the velocity and the given mass of the cart (4.3 kg), the kinetic energy of the cart was found to be 23.9 J.
As can be seen, there is a relatively small difference between the work exerted by the rubber band and the kinetic energy of the cart (2.7 J).
Conclusion (Part 3):
The error (difference between the KE of the cart and work done by the rubber band) is caused by many things. First, the uncertainty (simplification) in the replicated curve of the graph is one of the major contributions to the difference between the two. This simplification resulted in the addition of extra area under the curve, therefore causing extra calculated work of the spring that the spring actually did not do. In addition, other sources of error include the fact that the professor in the video tried to move the graph at constant speed, which causes some uncertainty in the curve of the graph. Friction between the graph and the marker as well as between the graph and the sliding holders of the graph are some sources that resulted in the uncertainty of the graph moving at constant speed. Lastly, the uncertainty in the force transducer and the timer that measured the time it took for the cart to go through the two photogates also caused the difference between the kinetic energy of the cart and the work done by the rubber band. However, this difference is very small, also proving that the work-kinetic energy theorem is legitimate and is an excellent model that is applicable to almost all systems.
No comments:
Post a Comment