Forces in Equilibrium

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Data

The mass of the object hanging was

mobject = .0600 ± .0005 kg

Figure 1. This graph displays the motion of the spring with the mass attached as it moves through 2 cycles of

oscillation. The graph is obviously some form of a sine function, which is normal for objects with oscillating motion, but its characteristics are all altered due to the motion of the unique spring and mass system in this lab. The vertical shift of the graph (D) is .343 because this is the equilibrium position of the spring. The amplitude (A) is .1935 because this is the maximum distance that the mass moved in either direction from equilibrium. The angular frequency (B) is 7.050 and equal to (2π)/T, when T equals the time elapsed through one full cycle. And finally, the horizontal shift (C) ensures that the function lines up correctly with the data points regardless of what point in the sine wave data collection began.

Figure 1

[pic 1]

Analysis

From the data extracted, several things can be noticed. First of all, motion of a spring can be almost perfectly represented by sine functions. Although the data points are all gathered separately, the form they take in Figure 1 clearly shows the curvature and symmetry that sine functions are known for. Thanks to Motionlab data analysis software, the values of position collected in hand with the time elapsed in between each point were used to calculate instantaneous velocities of the mass throughout its motion. These velocities were then used with equation #4 to calculate the kinetic energy of the mass at any given time. The position values were also used with equation #3 to calculate the potential energy of the mass at any given time.

In order to find a value of the internal energy of the spring from the data, the total energy of the system and the total mechanical energy of the mass at each time needed to be found. The internal energy would equal the difference of these two values at any given time. The total energy of the system was found by simply calculating the initial potential energy (Equation #4) of the mass before testing began. This is because it’s at rest so kinetic energy is zero and the spring is at equilibrium (not stretched at all), so its internal energy is also zero. In figure 3, this value is constant due to the fact that only 2 seconds of motion were analyzed, and no significant energy dissipation was seen. Using Equation #5 for all times throughout the analyzed time frame, the total mechanical energy of the mass was found. With these two values known at all times, the internal energy of the spring was calculated. It can also be noted that in Figure 3, at any given point, the sum of the value of internal energy of the spring and mechanical energy of the mass is equal to the total energy of the system. This proves the fact that practically no energy was dissipated throughout the testing.

Finally, once the internal energies of the spring at all points are calculated, the relationship between this and a function of position was determined. Since the purpose of this lab is to find a linear equation to represent this relationship and it’s known that in equation #2 (½)k is a constant, the assumption was made that (Δx)2 must be the independent variable x. In figure 2, this assumption is illustrated, and the correlation of the line best fit is .987 which shows that the calculated data points from the data collected are relatively accurate compared to this linear model. This leaves .8897 ± .012 or (½ k) as the slope of the function. When the slope was multiplied by 2, the k value was equated and a value of approximately 1.7794 was found. There are also alternate ways to solve for this k constant besides the method used in this lab.

In lab 3.6, this k constant was analyzed and calculated by hanging several different masses from the same spring used in this lab. By hanging different masses on the spring, the force of gravity applied to the spring (Fg) and the change in position of the spring (Δx) could be calculated, and equation #1 was used in order to solve for the k constant. By using a variety of masses (.090 ± .0005 kg, .070 ± .0005 kg, and .050 ± .0005 kg), the calculated k values were taken and averaged to get a spring constant of 2.624.

Obviously the difference between the two k values is .8446, which doesn’t necessarily make sense considering the same spring should have the same k constant regardless of its setting; but, there are several errors that could have possibly affected the calculations that lead up to the k value found in this lab. Primarily, due to the limited capabilities of the recording device and Motionlab, the limited number of data points allowed and the blurriness of the video during the placement of data points could’ve resulted in inconsistencies pertaining to position values of the mass. In turn, these inconsistencies would then offset the instantaneous velocities and position values used to calculate the mechanical energy of the mass at each time throughout its motion. With possible variations in the mechanical energy, the internal energy of the spring could have been miscalculated, which would affect the slope of the line in figure 2 that produced the k value. Along with this, another possible source of error in this lab would be the fact that only two full cycles of oscillation were analyzed. Since these cycles happened in a time frame of only 1.75 seconds, no significant dissipation of energy could be seen. But this may not have been the case if more cycles were observed.

Conclusion

In this lab, it was noticed that a linear graph of the internal energy of a spring can be created by using the independent variable (Δx)2. The accuracy of this equation (the correlation of .987 provided by Logger Pro) assures this method of graphical analysis is reliable. This conclusion, in turn, validates the initial prediction of the lab. It was also discovered that the k constant of any spring could be found using this lab procedure, which could be helpful in any application where a spring of unknown properties is being used.

In the case of using springs for a suspension system in a car, this lab implies that springs would be great at dissipating the energy of the car bouncing up and down. By compressing and expanding while forces are applied to them, much of the energy will be contained within the springs, and then dissipated naturally to avoid an excessive amount of movement of the car itself. This is why advanced spring