Isaiah (10)

1 Simple Harmonic Motion Lab Online Purpose The purpose of this lab is to study some of the basic properties of Simple Harmonic Motion (SHM) by examining the behavior of a mass oscillating on a spring. Theory One type of motion is called periodic motion. In this type of motion, the behavior, called the cycle, is repeated again, again, and again over a particular time interval, AKA a period. For periodic motion, the mass will always follow the same path and return to its original location at the end of each cycle. In an ideal system this behavior would go on forever, but in reality, it goes on till the mass losses all its mechanical energy. All periodic motion has some basic properties in common. Those properties are: 1. The Cycle - The motion that is being repeated. 2. The Amplitude ( 𝐴 ) – The magnitude of the mass’s furthest displacement from its equilibrium position during the cycle. 3. The Period ( 𝑇 ) – The time it takes to complete one cycle. 4. The Frequency ( 𝑓 ) -- The number of cycles completed per unit time. (The frequency is the mathematical inverse of the period.) 𝑓 = 1 𝑇 5. The Angular Frequency ( 𝜔 ) – T he frequency multiplied by 2π. 𝜔 = 2𝜋𝑓 = 2𝜋 𝑇 One particular subcategory of periodic motion is Simple Harmonic Motion (SHM). SHM has two more properties: 1. The restoring force acting on the mass must be proportional to the displacement of the mass from its equilibrium position, and pointing in the opposite direction of the displacement. 𝐹 𝑠 = −𝑘 ∙ ∆𝑥 (The equilibrium position then is, by definition, the location where there is no restoring force acting on the mass. 𝑘 is the Spring Constant of the device applying the resorting force.) 2. The period of oscillation is independent of the value of the amplitude of oscillation. As an example, for an oscillator to be a SHM oscillator, it doesn’t matter if its amplitude is set to be 10 cm or 10 km, once set in motion the time it takes for that oscillator to complete one cycle MUST BE THE SAME.

2 We know from Newton’s Second Law that all forces can be written as 𝐹 = 𝑚𝑎 so we can set the standard force equation equal to the restoring force and see that: 𝑎 = − 𝑘 𝑚 ∙ ∆𝑥 In the particular case of a mass attached to an ideal spring, the frequency of oscillation will be related to the mass and the force constant by: 𝑓 = 1 2𝜋 √ 𝑘 𝑚 And therefore as well: 𝑇 = 2𝜋√ 𝑚 𝑘 𝜔 = √ 𝑘 𝑚 It can therefore easily be shown that the magnitude of the maximum acceleration the SHM oscillator will experience during a cycle is given by: 𝑎 𝑚𝑎𝑥 = 𝜔 2 ∙ 𝐴 Also, using the fact that linear speed is related to angular speed by 𝑣 = 𝜔 ∙ 𝑟 , we can see that the magnitude of the maximum speed the SHM oscillator will experience during a cycle is given by: 𝑣 𝑚𝑎𝑥 = 𝜔 ∙ 𝐴

3 Setup: Measuring the Spring Constant 1. Go to the following website: https://phet.colorado.edu/en/simulation/masses-and-springs 2. You should now see the following: 3. Click on “Download”, and then open the software when it has completed downloading 4. You should now see the following: