Harmonic OscillatorsRegimes of Validity and Harmonic OscillationOscillating SpringOscillating PendulumLab 4Harmonic OscillatorsWhat do we do in this lab?This lab has three parts:1. Learn to characterize the Regime of Validity of a model.2. Examine the oscillation of a spring system and determine a Regime of Validity.3. Examine the oscillation of a pendulum system and determine a regime of validityEquipment: PhET Simulation, Microsoft ExcelKey Concepts: Simple Harmonic Oscillator, Pendulum, Regime of ValiditySafety concerns: None.4.1 Regimes of Validity and Harmonic Oscillation4.1 A. In the last lab you explored the validity of an equation relating the speed and angle ofa fired projectile to its range. In the lab’s second part, you probably found that the model wasindistinguishable from the data given your uncertainty for all the data you collected. In the lab’sthird part, you hopefully found that, as long as the drag coefficient was small enough, the modelwas indistinguishable from the data but for larger drag coefficients the model failed.This is representative of a general feature of science. When we compare a model to experimental re-sults, we (1) collect a finite amount of data and (2) see for which points, if any, the data contradictsthe model. The range of data for which the model is indistinguishable from the data is sometimescalled the Regime of Validity of the model. Inherent in such a description is a notion of uncertainty.For example, in Part 2 of Lab 3, you may have tested 5 angles between 10 and 60 degees and foundthat for all these points the distinguishability test gave indistinguishable results:12 LAB 4. HARMONIC OSCILLATORS|xpredicted− xmeasured|qδx2predicted+ δx2measured< 1 (4.1)We could then say that that the regime of validity of the model is at least 10 degrees to 60 degrees1with uncertainty given by the combined uncertaintyqδx2predicted+ δx2measuredIf we measured points outside of the range 10-60 degrees, or if we measured within this rangeto greater precision, the model might still fail. But at the level of stated precision the model isaccurate withing this range.Of course the model may have a larger regime of validity but that’s not established by the givendata. In science we can never prove that a model is right only that it has yet to fail. Even if theregime of validity covers all possible outcomes to all possible experiments, future experiments maycome along where our model fails. Richard Feynman said it best that “we can never be sure weare right, we can only ever be sure we are wrong”.In Part 3 of Lab 3, you probably found that the model was falsified for a large enough drag coeffi-cient. Perhaps you found that for a drag coefficient larger than 0.25 the model was distinguishablefrom the data but below that down to 0.04 it was not. You would report that the regime ofvalidity, in terms of the drag coefficient, is 0.04 to 0.25 with uncertainty given by the denominatorof the distinguishability test (since this is the uncertainty in the value xpredicted−xmeasured, whichyou can check using propogation fo uncertainty!)To be clear: we didn’t ask that you do this in Lab 3. We’ll use this concept in this lab now though.4.1 B. Simple Harmonic MotionThe model we’ll be examining this week is another canonical example from dynamics which iscalled Simple Harmonic Motion. (By “simple” we mean nothing pejorative, just that a minimalnumber of forces are involved).In simple harmonic motion, an object has a stable equilibrium position. For example a marble atthe bottom of a bowl. When displaced from equilibrium (the bottom of the bowl in this example),the object is subject to a force that’s proportional to its displacement:~F = −k~x (4.2)where k is what we call a proportionality constant. It relates the magnitude of the force to thedisplacement from equilibrium and has standard units of Newtons per meter. The negative signindicates that the force is always acting against the displacement from equilibrium. The force1You could also include a range of starting speed, etc. but in this lab we’ll stick to comparing one independentvariable to one dependent variable.4.2. OSCILLATING SPRING 3“wants” to restore the object to it’s equilibrium position and is typically referred to as a restoringforce. If there are no other forces acting on the object, then the net force is the following:~Fnet= m~a = −k~x (4.3)One can show that this causes the object to oscillate about the equilibrium position witha period that depends on the forces involved but which is independent of the amplitude ofdisplacement. So for example, if the marble is displaced by 1cm or 2cm it will oscillate aroundthe bottom of the bowl with the same period. While the colloquial definition of this motionis simple harmonic motion, a more useful description is to say that it is simple harmonic oscillation.This is usually only approximately accurate. Let’s investigate the validity of this model for twodifferent systems. Your goal in each case is to find a regime of validity. In each case we’ll be usinga model where:T = 2πpl/g (4.4)which gives a period which is independent of displacement and/or amplitude.For propagation of uncertainty, use the same general equation you have seen in previous labs. Thefollowing partial derivatives will be useful:∂T∂l=π√gl(4.5)∂T∂g= −πslg3(4.6)We need to know the uncertainty in gravity. Luckily, we measured gravity in lab 3 using our phonesso our uncertainty in g will be the standard deviation of the mean of our measured values from lab3.4.2 Oscillating SpringOpen the following simulation:https://phet.colorado.edu/sims/html/masses-and-springs-basics/latest/masses-and-springs-basics_en.htmlSelect “lab”4.2 A. You should see a spring system hanging from a ceiling. You can attach various masses,pick any you want, but make sure to record it. Attach it to the spring. It will begin to oscillate.4 LAB 4. HARMONIC OSCILLATORSFirst, let’s determine a predicted value based on the simple harmonic motion model.Click the boxes to see the location of equilibrium with and without the attached mass:You may find it useful later to click “Movable Line” as well.You may want to hit the stop button at the top for now to pause things.The difference between the blue and green lines gives us l. It will be different for different masses.You can measure it using the ruler.You should also record the value of g = 9.8ms2for the gravity of Earth. Use