Spring Constant Report Template

Carleton University Laboratory Report Course #: 1007 Experiment #: 2 Spring constant Ali Khelil (101258560) Date Performed: Oct 16, 2023 Date Submitted: Oct 30, 2023 Lab Period: L1 Partner: Goodness Station #: 16 TA: Teresa

Purpose The purpose of this experiment was to determine the spring constant of a hanging spring using two different methods. The first method that we attempted was the static method to figure out the spring constant. In the experiment we had used a motion detector, the spring, and a force sensor. The second method of determining the spring constant is by using the dynamic method. The static method was determined by suspending masses of different weights and then measuring how much the spring extended from rest, using a motion sensor to determine the extension. The dynamic method was determined by allowing the spring to oscillate with different masses on it. The spring constant is represented as k which informs us that the force on a spring is directly proportional to the extension of the spring so long as the spring does not become deformed when it reaches its maximum extension. For the dynamic part of the experiment, simple harmonic motion was used. For the static portion of the experiment, Hooke’s Law was used. Observations/Graphs Use templates from previous labs to copy/paste any Figure and Table numbers for your captions. Static Method: Figure 1: A graph of the spring constant using the static method. A linear fit was done for this data set to show how the extension of the spring compared to the force in newtons are linear. Table 2: the slope and the Y-intercept collected from the experiment using logger pro. The slope and Y-intercept uncertainties were calculated in logger pro. Unites are shown below the values. Slope Y-intercept m σ m b σ b Value 20.47 0.2061 0.04798 0.03060

F = M ×g = 0.050 kg× 9.8 m s 2 = 0.490 N If the spring extension is equal to the slope of the graph, then the uncertainty for k s and σ s is equal to the uncertainty of the slope. therefore, the spring constant can be written as: k ( ¿¿ s±σ s ) N s 2 ¿ ¿ ( 20.47 ± 0.2061 ) N s 2 determining the acceleration due to gravity on the Planet: F = kx F = Mg therefore kx = Mg g = kx M g = ( 20.47 )( 12.5 ) 50 g = 5.118 m / s 2 Calculating the t-test to compare the expected y-intercept: ¿ x 1 − x 2 ∨ ¿ √ σ 2 + σ 2 t = ¿ ¿ 0.04798 − 0 ∨ ¿ √ 0.0306 2 + 0 t = ¿ t = 1.568 Since the value for t is less than 2, this shows that the results are consistent. Part Two: Dynamic Method Calculating the spring constant:

m = 4 π 2 k k = 4 π 2 m k = 4 π 2 0.05002 s 2 / rad 2 / kg k = 789.25 kg s 2 Converting the Kd units to the SI units: N m = kg m s 2 = kg s 2 Calculating the uncertainty: σ 1 ω 2 = 2 σ ω av ω av 3 σ 1 ω 2 = 2 ( 5.363 ) 5.363 ❑ 3 σ 1 ω 2 = 10.726 154.249 σ 1 ω 2 = 0.0695 Calculating the t-test to compare the two values ( 𝑘? ± 𝜎𝑘? ) and ( 𝑘? ± 𝜎𝑘? ): ¿ x 1 − x 2 ∨ ¿ √ σ 2 + σ 2 t = ¿ ¿ 789.25 − 0.04798 ∨ ¿ √ 0.0695 2 + 0.0306 2 t = ¿