The purpose of this experiment was to prove that the conservation of energy principle applies to a vertically oscillating spring that does not have a negligible mass.
Apparatus:
Figure 1: Mass-Spring Apparatus Setup
The apparatus for this experiment consisted of a long vertical rod held pointing upwards by a C-clamp. A horizontal rod was also attached to the vertical rod, which was used for holding up the spring in the vertical direction, as shown in figure 1 above. Some masses and a mass "holder" were also attached to the spring such that it allows the spring to oscillate continuously. An index card was also applied to the bottom side of the mass "holder" to allow the motion detector under the mass-spring system to detect the presence of the oscillating spring and mass. A meter stick was also used to measure the dimensions of the system. Lastly, a computer with Logger Pro software was also needed to allow the collection and analysis of data obtained from the motion detector.
Abstract:
The three types of energies involved in this system include kinetic energy, gravitational potential energy, and elastic energy. The hanging mass contains kinetic and gravitational potential energy while the spring has elastic potential energy. However, since the spring in this experiment does not have negligible mass compared to the rest of the system, it must be taken into account that the spring also has kinetic and gravitational potential, however not as much as the mass does. In addition, because the potential energy and kinetic energy of the spring varies throughout its mass, calculus must be used to determine functions that represent it gravitational potential energy and kinetic energy. To do so, a part of the mass (dm) along with its corresponding length (dy) must be observed, and the ratio between the two (dm/dy) must equal the ratio of the mass and length of the spring as a whole. Then, dm must be solved for in terms of y and must be substituted into the energy equation for that piece (in this case, dGPE and dKE). Then, through integration the sum of the energies of all the pieces of the spring must be found. Finding these functions of energy for the spring allows the use of the spring as a point particle, making it easier to measure its energies as it oscillates. The derivations of these two energy functions are shown below:
Abstract:
The three types of energies involved in this system include kinetic energy, gravitational potential energy, and elastic energy. The hanging mass contains kinetic and gravitational potential energy while the spring has elastic potential energy. However, since the spring in this experiment does not have negligible mass compared to the rest of the system, it must be taken into account that the spring also has kinetic and gravitational potential, however not as much as the mass does. In addition, because the potential energy and kinetic energy of the spring varies throughout its mass, calculus must be used to determine functions that represent it gravitational potential energy and kinetic energy. To do so, a part of the mass (dm) along with its corresponding length (dy) must be observed, and the ratio between the two (dm/dy) must equal the ratio of the mass and length of the spring as a whole. Then, dm must be solved for in terms of y and must be substituted into the energy equation for that piece (in this case, dGPE and dKE). Then, through integration the sum of the energies of all the pieces of the spring must be found. Finding these functions of energy for the spring allows the use of the spring as a point particle, making it easier to measure its energies as it oscillates. The derivations of these two energy functions are shown below:
Figure 2: Derivation of the gravitational potential energy (GPE) of the spring
In this derivation, H is the height from
the ground to the top of the mass-spring system and y "naught" is the height from the ground to the bottom of
the mass-spring system. Therefore, the length of the spring can be found at any point during the oscillation by subtracting the height H from the ground to the top of the spring (which is constant) by the height y "naught" from the ground to the bottom of the mass (which constantly changes throughout the oscillation. Because H is constant, MgH/2 is a constant in the equation for GPE of the spring. The value of y "naught" at each point was obtained from the position data of the oscillating mass gathered by the motion detector. The derived formula allows us to treat the spring as a single mass located at the center of the spring, but only when using this formula.
The same derivation, shown below, is done for the kinetic energy of the spring, with the exception of different data values and a different base formula (0.5mv^2).
Figure 3: Derivation of the kinetic energy (KE) of the spring
In this derivation, the length y of the spring is the length L of the relaxed spring. The length of the relaxed spring is the needed length because it is the length at which the mass of all the pieces of the spring (dm) are proportional. In addition, the velocity of the piece of spring is equal to the velocity of the end times y/L. This is because the velocity of the spring at its different points varies, and the only velocity that can be collected and recorded is the velocity of the end of the spring.The velocity of the piece is proportional to the velocity of the end times y/L because, for example, at the top of the spring y=0, so the velocity of the top of the spring is zero, which is seen. At the bottom of the spring, the length L is equal to y, which results in the velocity of the bottom to be equal to the velocity of the end, which also makes sense. Therefore, based off of this, the velocity of the center of the spring must be half the velocity of the end, and so on. This derived formula allows us to treat the spring as a mass at the end of the spring, but only when using this formula.
Therefore, there are now five different energies found in this mass-spring system: the kinetic energy of the mass and the spring, the elastic potential energy of the spring, and the gravitational potential energy of the mass and the spring,
Procedure/Discussion:
First, the dimensions of the mass-spring system were measured. The length L of the relaxed spring was measured as 47.3 cm. In addition, the mass of the spring was measured to be 85 g. Also, the height H from the ground to the top of the sprig was recorded as 145.7 cm. The distance from the top of the motion detector to the bottom of the unstretched spring was measured and recorded to be 85.7 cm. Lastly, the height y "naught" from the ground to the bottom of the mass holder when the spring was relaxed was recorded to be 98.5 cm.
Then, the spring constant of the spring was calculated by hanging various masses onto the spring and recording the change in the length of the spring. Then, using Hooke's Law (F = kx), where F is the weight of the added mass and x is the change in the length of the spring, the spring constant k was calculated. Using a 200 g (F = 1.96 N) mass, the spring stretched a distance 0.216 m. Then, the spring constant k was calculated to be 9.07 N/m (k = F/x = 1.96 N/0.216 m). The same procedure was done with 300 g and 400 g masses, whose k values were calculated as 8.80 N/m and 8.87 N/m, respectively. Taking the average of the k values, the actual experimental spring constant k was found to be 8.92 N/m.
Next, a 250 g mass was attached to the spring and the spring was allowed to oscillate vertically by pulling down on the mass slightly and letting go. Then, using the motion detector, the position and velocity data of the system was collected. The experimental position and velocity graphs are shown below.
Figure 4: Position and velocity graphs of the oscillating mass spring system
These graphs correspond with the predicted because, theoretically, the velocity of the mass is greatest when the spring is unstressed and is zero when the string is at its maximum stretch and compression, which is seen in the graphs above. As can be seen, the maxima and minima of the position graph translate to zero velocity in the velocity graph and the maxima and minima of the velocity graph are caused from the center between the maxima and minima of the position graph, as was expected.
Next, the graphs of KE, EPE, and GPE were predicted based on the position and velocity graphs. After the prediction, the experimental graphs of the same energies were found with respect to time. New columns of EPE, GPE (including both mass and spring), and KE (including both mass and spring as well) were created using the derived formulas as well as 0.5kx^2 for EPE, 0,5mv^2 for KE of the mass, and mgh for the GPE of the mass. the value of h for the GPE of the mass was calculated by subtracting the unstretched position by the position data gathered. These graphs are shown below:
Figure 5: KE, EPE, GPE and Total E vs time graphs of the mass-spring system
These graphs also corresponded with the predicted graphs. As predicted, the kinetic energy graph must correspond to the velocity graph; the maxima and minima of the velocity graph should result in peaks in the KE graph, as seen. This is because the velocity is greatest at those time intervals, resulting in the most kinetic energy. In addition, as predicted and seen, the kinetic energy graph must read zero at the maxima and minima of the position graph. In addition, as predicted, the maxima and minima of the GPE graph must correspond directly with the maxima and minima of the position graph, as seen. In addition, as predicted, the GPE graph is never zero since the mass never reaches the motion detector (the defined zero GPE point). Also, the maxima of the elastic potential energy graph are proportional to the minima of the gravitational potential energy graph, as predicted since the elastic potential energy is greatest when the spring is stretched the greatest (lowest GPE). Lastly, the total energy is nearly a straight horizontal line, as predicted. This proves that the conservation of energy principle is true and applies to situations that involve energy, such as the oscillation of a spring system seen in this experiment.
Then, the following experimental plots were found using Logger Pro: KE vs. position, KE vs. velocity, EPE vs. position, EPE vs. velocity, GPE vs. position, and GPE vs. velocity, The KE graphs are shown below:
The KE graphs shown in Figure 6 are as expected. For the position graph, as the position of the mass-spring system reaches the unstretched position of the spring, the KE is the greatest (max velocity), and as the position of the system reaches the maximum stretch or compression of the spring, the KE is zero (zero velocity), which was expected and witnessed in the oscillation of the graph. Looking at the velocity graph, the kinetic energy is proportional to the velocity squared, as expected and seen from the result of a parabolic graph. When the velocity is zero, the KE is zero, and when the velocity is highest (in both directions), the KE is the highest, as expected. Next, the GPE graphs are shown below:
The GPE vs. position graph shown in Figure 7 was also expected. As the position (distance from the motion detector to the mass) increases, the GPE increases. It was also expected that the graph of GPE and position is linear because, looking at the equation for GPE (mgh), the change in "h" results in a proportional change in GPE. The GPE vs velocity graph was not expected to be an ellipse, but after some observing it made sense to be an ellipse. In the oscillating mass spring system, the velocity constantly changes from zero to +max, then back to zero and then to -max, then to zero back again. This results in a constant rotation in the slope of the graph, causing an elliptical arc. In addition, the points of minimum and maximum GPE being at zero velocity also made sense because, when the velocity is zero, the spring is either at its maximum stretch (lowest GPE of the mass) or at its maximum compression (highest GPE of the mass). Lastly, the EPE graphs are shown below:
The EPE vs. position graph seen in Figure 8 was expected and made sense. As the mass-spring system reached the unstretched position of the spring (0.857 m), the EPE of the spring would decrease and would be transferred to KE of both the spring and mass, as witnessed in the experiment and seen in the graph. In addition, the graph takes the shape of half a parabolic arc, which makes sense because x is equal to the unstretched position minus the position obtained from the data collected by the motion detector as well as squared. Interpreting this, x can never be larger than 0.857 and x^2 can never be negative, resulting in EPE always being positive and taking the shape of a parabola with the "slope" of the parabola equal to 0.5k, where k is the spring constant. Lastly, the EPE vs velocity graph was not expected but upon further observation made some sense. The EPE would be greatest at a velocity of 0 since that is when the spring is stretched or compressed at its maximum. In addition, for the same reason the GPE vs velocity graph is circular, it is too. However, one anomaly is that the lowest EPE is also shown by the graph at a velocity of 0 m/s. It is not known why, but one guess would be that it is affected by GPE, since at those points of minimum EPE it is the highest energy of the three.
Putting all of the above graphs together, the following two graphs are obtained:
Looking at the total energy, it is nearly constant with around a 3.4% change throughout the graph, but, looking at the uncertainties present in the experiment and the fact that the system and the world is not ideal, it shows that the system was a very accurate model for proving that the conservation of energy principle is in fact true. Therefore, it can be said that the energy in this system was conserved.
Conclusion:
There were some sources of error and uncertainty in this experiment, but they did not have a very large effect on the results of the experiment, with those results already being expected. For example, the 3.4% error in the total energy not being constant results from the fact that an ideal spring was not used yet it was still assumed that it was. Assuming so does not take into account that the spring does not symmetrically stretch and compress, resulting in different areas of the spring being in different conditions (variation in dm and elastic energy, making 0.5x^2 an inaccurate potential function for EPE). In addition, the stretching and compressing of a spring that is not ideal results in the loss of energy through non-conservative ways, such as the formation of heat arising from friction in the spring and the loss of energy due to vibration in the spring.
Also, the uncertainty in the measurement of the unstretched position is the cause of the elastic potential energy not being exactly zero at the unstretched position. The uncertainty in the unstretched position is caused by both uncertainty in the equipment and the fact that it is difficult for a vertical spring that has mass to be kept in the unstretched position.
In addition, the uncertainty in the value of the spring constant had an effect on the error in this experiment. In order to lower the error from measuring the k value, better equipment could have been used to find k, such as the availability of an accurate force sensor, a better version of LoggerPro that can better monitor and connect with the equipment that collects data. Lastly, more time for the experiment could have been given to allow for the running of more trials and collection of more data, lowering the error in the experiment. Overall, though, the uncertainty/error in the experiment was very low and had little effect on the success of proving that the system is an accurate model for the conservation of energy.
Then, the following experimental plots were found using Logger Pro: KE vs. position, KE vs. velocity, EPE vs. position, EPE vs. velocity, GPE vs. position, and GPE vs. velocity, The KE graphs are shown below:
Figure 6: KE vs position and KE vs velocity graphs of the mass-spring system
The KE graphs shown in Figure 6 are as expected. For the position graph, as the position of the mass-spring system reaches the unstretched position of the spring, the KE is the greatest (max velocity), and as the position of the system reaches the maximum stretch or compression of the spring, the KE is zero (zero velocity), which was expected and witnessed in the oscillation of the graph. Looking at the velocity graph, the kinetic energy is proportional to the velocity squared, as expected and seen from the result of a parabolic graph. When the velocity is zero, the KE is zero, and when the velocity is highest (in both directions), the KE is the highest, as expected. Next, the GPE graphs are shown below:
Figure 7: GPE vs position and GPE vs velocity graphs of the mass-spring system
The GPE vs. position graph shown in Figure 7 was also expected. As the position (distance from the motion detector to the mass) increases, the GPE increases. It was also expected that the graph of GPE and position is linear because, looking at the equation for GPE (mgh), the change in "h" results in a proportional change in GPE. The GPE vs velocity graph was not expected to be an ellipse, but after some observing it made sense to be an ellipse. In the oscillating mass spring system, the velocity constantly changes from zero to +max, then back to zero and then to -max, then to zero back again. This results in a constant rotation in the slope of the graph, causing an elliptical arc. In addition, the points of minimum and maximum GPE being at zero velocity also made sense because, when the velocity is zero, the spring is either at its maximum stretch (lowest GPE of the mass) or at its maximum compression (highest GPE of the mass). Lastly, the EPE graphs are shown below:
Figure 8: EPE vs position and EPE vs velocity graphs
The EPE vs. position graph seen in Figure 8 was expected and made sense. As the mass-spring system reached the unstretched position of the spring (0.857 m), the EPE of the spring would decrease and would be transferred to KE of both the spring and mass, as witnessed in the experiment and seen in the graph. In addition, the graph takes the shape of half a parabolic arc, which makes sense because x is equal to the unstretched position minus the position obtained from the data collected by the motion detector as well as squared. Interpreting this, x can never be larger than 0.857 and x^2 can never be negative, resulting in EPE always being positive and taking the shape of a parabola with the "slope" of the parabola equal to 0.5k, where k is the spring constant. Lastly, the EPE vs velocity graph was not expected but upon further observation made some sense. The EPE would be greatest at a velocity of 0 since that is when the spring is stretched or compressed at its maximum. In addition, for the same reason the GPE vs velocity graph is circular, it is too. However, one anomaly is that the lowest EPE is also shown by the graph at a velocity of 0 m/s. It is not known why, but one guess would be that it is affected by GPE, since at those points of minimum EPE it is the highest energy of the three.
Putting all of the above graphs together, the following two graphs are obtained:
Figure 9: Energy vs position and energy vs time of the mass-spring system
Looking at the total energy, it is nearly constant with around a 3.4% change throughout the graph, but, looking at the uncertainties present in the experiment and the fact that the system and the world is not ideal, it shows that the system was a very accurate model for proving that the conservation of energy principle is in fact true. Therefore, it can be said that the energy in this system was conserved.
Conclusion:
There were some sources of error and uncertainty in this experiment, but they did not have a very large effect on the results of the experiment, with those results already being expected. For example, the 3.4% error in the total energy not being constant results from the fact that an ideal spring was not used yet it was still assumed that it was. Assuming so does not take into account that the spring does not symmetrically stretch and compress, resulting in different areas of the spring being in different conditions (variation in dm and elastic energy, making 0.5x^2 an inaccurate potential function for EPE). In addition, the stretching and compressing of a spring that is not ideal results in the loss of energy through non-conservative ways, such as the formation of heat arising from friction in the spring and the loss of energy due to vibration in the spring.
Also, the uncertainty in the measurement of the unstretched position is the cause of the elastic potential energy not being exactly zero at the unstretched position. The uncertainty in the unstretched position is caused by both uncertainty in the equipment and the fact that it is difficult for a vertical spring that has mass to be kept in the unstretched position.
In addition, the uncertainty in the value of the spring constant had an effect on the error in this experiment. In order to lower the error from measuring the k value, better equipment could have been used to find k, such as the availability of an accurate force sensor, a better version of LoggerPro that can better monitor and connect with the equipment that collects data. Lastly, more time for the experiment could have been given to allow for the running of more trials and collection of more data, lowering the error in the experiment. Overall, though, the uncertainty/error in the experiment was very low and had little effect on the success of proving that the system is an accurate model for the conservation of energy.
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