Lab 6 Report.docx

LAB 6: HARMONIC MOTION Angela Eller, John Rios Texas A&M University College Station, TX 77843, US. Abstract This report explores the concept of harmonic motion. By using the tracking camera, the vertical position of an oscillating mass on a spring is captured for the duration of the experiment. These measurements are used to compute the spring constant of three springs carrying different masses with the implementation of Hooke’s Law. As a continuation of this calculation, the expected position of the mass is also determined, providing an insight into the effect of the environment on results. This experiment displays the effect of different spring constants and different masses on the harmonic motion of a spring. Keywords: oscillation, spring constant, mass, Hooke’s Law 1. Introduction The primary goal of this experiment was to determine the spring constant of three different springs using the given data. Some of the physics concepts learned in class help in finding the angular frequency of both spring constant and period of oscillation, as well as harmonic oscillator as a function of time, Angular velocity was gathered using the period of each spring’s motion. Using the angular velocity, the spring constant was then calculated. Equation 1 𝑘 = 𝑚ω 2 For this equation, is the spring constant, , is the angular velocity, and is the added mass of the 𝑘 ω 𝑚 spring-mass system. Equation 2 ω = 2π 𝑇 For this equation, is the angular velocity that is to be calculated, is the period of the spring, the time ω 𝑇 between each oscillation as it bounces up and down in simple harmonic motion. Equation 3 ω = 𝑣 𝑟 Where means angular velocity, V is the velocity of the object, and r is the radius of the object ω Equation 4 𝑣 = 𝑉 ? 2 + 𝑉 ? 2 Where V is the velocity of margin, is the radius in the x-direction, and is the radius in the y direction 𝑉 ? 𝑉 ? Equation 5 𝑟 = 𝑟 ? 2 + 𝑟 ? 2 Where r is the radius of margin, is the radius in the x-direction, and is the radius in the y direction 𝑟 ? 𝑟 ? Equation 6 ω = 𝑑θ 𝑑𝑡 = θ 2 −θ 1 𝑡 2 −𝑡 1 = 𝑣 𝑟 Where is the angular velocity, v is the velocity, and r is the radius of the function. ω Equation 7 σ = Σ(? 1 − µ)/𝑁

Where means standard deviation, N is the number of measurements, is the measurements, and is the mean of σ 𝑋 𝑖 µ the measurements. 2. Experimental Procedure In the experiment, three different springs, Red, Green, and white, a mess set and a mass hanger were used for each test experiment. Each spring was individually screwed onto a stationary horizontal bar and allowed to hang vertically in front of a tracking camera. Once the green spring is secured, 400 g or 0.400 kg of mass with a trackable position sticker is hooked onto the spring using the mass hanger. Once the new spring-mass system comes to an equilibrium, the system is lightly pulled downward by the hanging mass. The mass system was then released so that the system oscillated up and down, whilst being tracked by the camera. This process was repeated for the Red spring and the same process was repeated for the white spring but with 300 g of weight instead of 400. Using the recorded position data and timestamps, sinusoidal position graphs were created for each of the three springs. Because the lowest position of the masses would only be reached once per oscillation, the troughs of each wave were used to determine the period of each mass-spring system. To determine the time for each oscillation, Equation 2 had to be solved. This was done by solving the amount of time it took the spring to reach the same y value in the graph. These values were then plugged into Equation 2 to calculate the angular velocity. Upon completion, Equation 1 was used to solve for the spring constant (k). After calculating the average of all spring constants for the oscillation, Equation 7 was used to solve for uncertainty. This method was used for all three spring types. 3. Results and Analysis The recorded position data of each mass-spring system was compiled into graphs to help visualize the data and provide evidence for the period of each system. Figure 1 represents the graph created from data using the green spring, Figure 2 represents the graph created from data using the Red spring, and Figure 3 represents the same using the white spring. Using period data from each graph, Equation 2 was used to compute each spring’s angular velocity data. This data was then plugged into Equation 1 to find data regarding each spring’s constant. The averages of these constants were then finally calculated, using Equation 7 to account for the uncertainty of the process. With this, the experimentally calculated spring constant of the Green Spring was calculated to be 45.647 0.435, while the ± spring constant for the Red Spring was determined to be 20.863 0.349, and the spring constant for the White ± Spring was found at 32.668 0.0367. ± Figure 1. Graph of y-position vs time using the green spring Figure 2. Graph of y-position vs time using the yellow spring