Lab #5: Deriving a Relationship Between the Potential Energy of a Spring and its Compression.
Purpose:
The purpose of this lab is to experimentally derive a relationship between the distance a spring is compressed and the resulting potential energy that the spring contains from being compressed past its neutral state, finding the spring constant k from the collected data, and derive the relationship between the distance a spring is compressed and the resulting force produced by the spring.
Equipment:
- Spring-Loaded Projectile Launcher
- 1 Marble
- 1 Video
Recorder - 1 10-Meter-Long or similarly lengthy sheet of paper, marked at every decimeter and
meter
Procedure:
Experimental
Determine the mass of the marble that is to be launched. Then, in an area with a high ceiling, place the spring-loaded projectile launcher pointed straight up, preferably just in front of a wall. Just behind the launcher preferably on the wall, hang the 10-meter-long marked sheet of paper so that the zero mark is at the floor. Then, placing the marble in the launcher, perform three launches of the marble for each spring compression setting, and record the height that the marble travels using the video recorder.
Mathematical
Taking the measurements from the recorded video and the marble, and assuming that acceleration due to gravity at the experiment site is 9.81 m/s^2, find the potential energies of the marble during each trial using the formula Ug = mgh, with Ug being potential energy, m being the mass of the marble, g being acceleration due to gravity, and h being the height measured during the experiment. Next, plot the potential energy against the compression distance of the spring and, given that the relationship is a power function, fit a least squares regression line to the plot. This is the potential energy as a function of spring compression distance. Next, knowing that Us = kx^2/2, with x being the compression distance, find the spring constant k for the spring of the launcher. Finally, knowing that Fx = -dU/dx, find the function of the force of the spring as a function of spring compression distance.
Collected Data:
Mass of the marble: .0063 kg ; Acceleration due to gravity: 9.81 m/s^2
Trial | Spring Setting | Spring Compression | Height | Potential Energy |
---|---|---|---|---|
1 | 5 | .111 m | 2.45 m | .151 J |
2 | 5 | .111 m | 2.85 m | .176 J |
3 | 5 | .111 m | 2.7 m | .167 J |
1 | 4 | .095 m | 2.2 m | .135 J |
2 | 4 | .095 m | 2.3 m | .142 J |
3 | 4 | .095 m | 2.3 m | .142 J |
1 | 3 | .079 m | 1.9 m | .117 J |
2 | 3 | .079 m | 1.65 m | .102 J |
3 | 3 | .079 m | 1.65 m | .102 J |
1 | 2 | .063 m | 1.12 m | .069 J |
2 | 2 | .063 m | 1.25 m | .077 J |
3 | 2 | .063 m | 1.03 m | .064 J |
1 | 1 | .047 m | 2.03 m | .064 J |
2 | 1 | .047 m | .95 m | .059 J |
3 | 1 | .047 m | 1 m | .062 J |
Data Analysis:
The least squares regression line reveals the relationship between spring compression distance and potential energy stored in the spring to be Us = 7.8173x^2 + .4953x + .0168, where x is the distance the spring is compressed and Us being the potential energy in the spring. Assuming that Us = kx^2/2, then k = 2Us/x^2, so that, using an x value of .1 m and a potential energy value of .144503 J (attained from the derived equation), k = 28.9006 N/m. Finally, using the derived equation, Us, the force of the spring in relation to the distance it is compressed is Fs = -dUs/dx, which is Fs = -15.6346x - .4953.
Conclusion:
In conclusion, using an experimental method and knowledge of gravitational potential energy as a function of height, a relationship between the potential energy of a spring and its compression distance was derived, and was found for this particular spring launcher to be Us = 7.8173x^2 + .4953x + .0168, with x being the compression distance and Us being the potential energy of the spring. The spring constant k was found to be 28.9006 N/m from the collected data and the derived equation knowing that Us = kx^2/2. The relationship of the force of a spring as a function of spring compression distance was found to be Fs = -15.6346x - .4953, with Fs being the force produced by the spring and x being the spring compression distance, knowing that Fs = -dUs/dx. Errors in the lab design include the failure to take into account the force of air resistance against the marble when calculating the potential energy it, and not actually knowing the acceleration due to gravity in the area (as gravity can vary greatly in an area due to differing densities of the earth, elevation, concentration of land mass in an area, et cetera).