Lab #X: Deriving the Spring Constant of an Ideal Spring
Purpose:
The purpose of this lab is to experimentally derive the spring constant, k, of an ideal, or hookean spring, derive the potential energy relationship of the spring, and, in doing so, better understand spring motion and properties.
Equipment:
- 1 Table Clamp Stand
- 1 Metal Rod with a Screw Thread
- 1 Attachable Beam
- 1 Ideal Spring
- 5 0.2 kg Masses
- 1 Meter Stick
Procedure:
First, place the table clamp stand hole side up so that the edge of the table is between the two sides of the clamp. Tighten the clamp until firmly secured. Screw in the metal rod securely, and then attach the beam firmly to the rod at a point 3/4 the way up the rod from the table. Attach the spring to the beam so that it will not fall off. Place the meter stick so that it is perpendicular to the table and up against the beam next to the spring. Measure where the bottom of the spring is with no masses attached and record it. Then attach one 0.2 kg mass to the bottom of the spring and measure the new distance from the table to the base of the spring. Add an additional 0.2 kg mass to the bottom of the spring and repeat, continuing until 1 kg is on the base of the spring.
Data:
Mass (kg) | Distance (m) | Weight (N) | Displacement (m) |
---|---|---|---|
0.0 | .43 | 0.0 | 0.0 |
0.2 | 0.416 | 1.962 | 0.014 |
0.4 | 0.337 | 3.924 | .093 |
0.6 | 0.256 | 5.886 | 0.174 |
0.8 | 0.179 | 7.848 | 0.251 |
1.0 | .098 | 9.81 | 0.332 |
Data Analysis:
Now given that F = 27.356 N/m * x + .9657, one can find the relationship between the potential energy and displacement using the principle dU/dx = -F.
(-27.356 N/m * x - .9657 = dU/dx) * dx
S (-27.356 * x - .9657) dx = S dU
-13.678 * x^2 - .9657 * x = U
(-27.356 N/m * x - .9657 = dU/dx) * dx
S (-27.356 * x - .9657) dx = S dU
-13.678 * x^2 - .9657 * x = U
Conclusion:
In conclusion, the spring constant of an ideal spring was measured by finding the displacement given by a specific force, and the spring constant for my specific spring (the red spring) was found to be 27.356 N/m. Additionally, the relationship between the potential energy and the displacement of the spring was derived, and found to be -13.678 N/m * x^2 - .9657 * x = U. The actual spring constant of my spring was 25 N/m (+- 10%), so a source of error may be at work. One error that may have affected the results would be the ignoring of the weight of the spring itself in the measurements of weight over distance. Another error that may have affected the results is the assumption that these springs are perfectly ideal springs.