Simple Harmonic Motion

Daniel Kim 11/3/23 PHY 133.L69 TA: Yikai Wu Simple Harmonic Motion Lab

Introduction: In this lab we have three objectives. Experimentally finding the angular velocity of a simple harmonic oscillator and comparing this value to the theoretical value, Using the Fast Fourier Transform (FFT) function to find the peak frequency, and Experimentally testing the relationship between the mass of the object and the angular frequency of the oscillator. Simple harmonic motion is a type of oscillatory motion that occurs when a mass on a spring is subject to a restoring force. The restoring force is directly proportional to the displacement of the mass from the equilibrium position and will always pull the mass back in the direction of the equilibrium position which can be derived using Newton’s Second Law in an equation. Procedure: 1) Measure the mass of just the iolab device. 2) Connect the long spring to the screw on the io lab, hang the device, and let it oscillate. 3) Add another object and measure the mass of the device + the object 4) Let the device with the new mass oscillate 5) Add one more object and measure the mass of the device + two objects 6) Let the device with the new mass oscillate 7) Use the Fast Fourier Transform function to find the peak frequency 8) Plot the angular frequency of the device versus the mass of the device Results: Figure 1. Device at rest

Figure 4. Mass of device with an added object Figure 5. Device at rest with 2 added objects/mass

Figure 6. Mass of device with 2 added objects Figure 7. Oscillation of only the iolab

Table 1. Acceleration, Force, and Total Mass of device + objects Fg=mg Force Due to gravity (N) Force due to acceleration (m/s^2) Mass (kg) Iolab -1.988 -9.811 0.2026 Iolab + 1 mass -2.025 -9.813 0.2064 Iolab + 2 mass -3.137 -9.811 0.3197 Table 2. Iolab + masses oscillating data Time for 5 peaks (s) Time for 1 period(s) (s) Angular Frequency (w) From Period (rad/s) Peak Frequency (Hz) Angular Frequency from Peak Frequency (rad/s) Iolab 3.095 0.773 8.570 2.345 14.734 Iolab + 1 mass 3.460 0.865 7.263 1.368 8.595 Iolab + 2 mass 4.181 1.045 6.013 1.567 9.846 Table 3. Values used for mass vs. angular frequency plot Mass (kg) Angular Frequency (rad/s) Iolab 0.2026 14.734 Iolab + 1 mass 0.2064 8.595 Iolab + 2 mass 0.3197 9.846 Graph 1. Graph of mass (kg) vs. Angular Frequency

Calculations: Iolab device mass: Fg = mg (-1.988N) = m(-9.811m/s^2) m= 0.2026kg Iolab device mass + 1 mass: (-2.025N) = m(-9.813m/s^2) m= 0.2064kg Iolab device mass + 2 masses: (-3.137N) = m(-9.811m/s^2) m=0.3197kg Time for 1 period(s): Iolab: 3.095 / 4 = 0.773 Iolab with 1 mass: 3.460 / 4 = 0.865 Iolab with 2 mass: 4.181 / 4 = 1.045 Angular Frequency (w) From Period (rad/s): Iolab: 1/0.773 = 1.364 2π(1.364) = 8.570 Iolab with 1 mass: 1/0.865 = 1.156 2π(1.156) = 7.263 Iolab with 2 mass: 1/1.045 = 0.957 2π(0.957) = 6.013