Simple Harmonic Oscillation Experiment Type: Open-Ended Overview In this experiment, students will look at three kinds of oscillators and determine whether or not they can be approximated as simple harmonic oscillators. Students should determine under what conditions they can be characterized as simple harmonic oscillators. Students should also determine if the oscillation is damped over time. Key Concepts Oscillators, oscillations, simple harmonic oscillation, damped linear oscillation, damping constant, spring constant, spring frequency Objectives On completion of this experiment, students should be able to: 1) explain the concept of simple harmonic motion 2) determine if an oscillation is simple harmonic or not 3) determine the frequency of oscillations 4) explain the concept of damped oscillations Review of Concepts Oscillations An oscillation is a back and forth motion. You can see oscillations all around you every day. It is the swaying of the fronds in a palm tree, the rocking of a rocking chair, the ticking of a grandfather clock, the motion of the suspension in your car, the swinging of a child in a playground. Sometimes we want to have oscillations (in rocking chairs, swings, etc.), other times we do not (in buildings, cars, etc.). In order to build things as scientists and engineers, we need to understand oscillations. Why do things oscillate? What are the properties of oscillations? The conditions you need in order for an object to oscillate are: • the object must be under the influence of a net restoring force • there must be at least one equilibrium position An equilibrium position is a location in space where the object has no net force operating on it. A restoring force is a force which points in the direction of (or along the path to) the equilibrium position. In other words, the restoring force tries to return the object to the equilibrium position. Simple Harmonic Oscillation The simplest kind of oscillation is the simple harmonic oscillation. It is the simplest because the restoring force can be characterized as proportional to the displacement, i.e. r r Frestoring = !kx (9-1) In this equation, k is a constant of proportionality, usually called the spring constant, and xr is a displacement. Note that the displacement need not be in the “x” direction, but can be a displacement along a path (I will give you an example shortly). Now having said this, let me tell you that there are very few real, net restoring forces which can be characterized with this formula exactly. In fact, I can think of none. I know what you’re thinking. You’re thinking “But, but … That’s the spring force, Hooke’s Law, I learned about in my text!” Well, yes, Eq. (9-1) is Hooke’s Law. That makes sense, because you already know that springs oscillate very readily. The question is how well does Hooke’s Law describe a spring? What happens if you stretch a real spring too much? (Haven’t you ever ruined a Slinky?) Does it have the same “springiness” after you do this? The answer is that there is a region of displacements for which a spring will follow Hooke’s Law fairly reasonably. Beyond that and the spring will experience a process called hysteresis. (That is a topic for another course, however.) Many restoring forces can be approximated as Eq. (9-1) under certain conditions. In your experiment today, you are going to look at a few examples of oscillators and you will tell me if the oscillator can be approximated as a simple harmonic oscillator and if so, under what conditions is it approximately simple harmonic. Let me give you an example other than the simple spring-mass system that you can find in your book. Figure 0-1 A mass sliding in a frictionless, circular bowl This is clearly an oscillator. If you move the mass to any side of the bowl, it will slosh back and forth. But is it a simple harmonic oscillator? To answer this question, let’s draw the free body diagram. Figure 0-2 The free body diagram of the sliding mass in the frictionless bowl This is an example of an oscillator whose restoring force acts along the path of motion, in this case, along the surface of the bowl. The restoring force in this case is the component of the gravitational force which acts along the surface of the bowl. Frestoring = mg sin! (9-2) Well, that doesn’t look like Hooke’s Law at all! So this is not a simple harmonic oscillator in general. However, for small angles, sin θ ≈ θ. Also, the displacement r along the path is related to θ by the formula ! = sr / R . That means that for small angles this restoring force becomes Frestoring = mg' & s # (9-3) = mg$ ! % R " Eq. (9-3) has the form F = (constant)*displacement. The “spring constant” in this case is mg/R. So for small angles, this mass sliding in a frictionless bowl is a simple harmonic oscillator! That’s all well and good for theoretical determinations, but how do we find out if the oscillator is simple harmonic experimentally? To find out experimentally, we need to know the equation of motion for an object undergoing simple harmonic motion. The equation of motion for simple harmonic oscillation is a cosine function. x(t) = Acos("t +! 0 ) (9-4) In this equation, A is the (constant) amplitude of the oscillation, ω is the frequency of the oscillation, and δo is the initial phase of the oscillation. Be careful! X(t) is the distance along the path of the motion! That means the arc distance, s, in the case of the mass sliding in the bowl. The frequency of the oscillation for simple harmonic motion is k ! = (9-5) 0 m If you can verify both (9-4) and (9-5) are true for your system, then you have successfully shown that the oscillator can be approximated as a simple harmonic oscillator. Damped Oscillations Of course, there’s no such thing as a perfectly lossless mechanical system. If we were to look at a real spring oscillating over some period of time, the graph of its motion would never be a perfect cosine function. After a while, the oscillations would die down. In other words, the oscillator would lose energy over time. A force which causes the oscillator to lose energy is called a damping force. Some common damping forces are friction and air drag. Since we have already discussed friction in a previous lab, I will be very brief. Sometimes the damping force can be modeled as proportional to the velocity of the mass, i.e. Fdamping = !bv (9-6) where b is a constant of proportionality called the damping constant. When this is true, the equation of motion becomes a sinusoidal function which is attenuated by a decreasing exponential, Figure 0-3 A plot of the displacement vs. time of a damped harmonic oscillator #(b / 2m)t x(t) = Ae cos("'t + ! o ) (9-7) The frequency is changed from the natural frequency (the frequency for simple harmonic oscillation) by the relation 2 & b # ' 1 $ ! (9-8) ( = (0 ' $ ! % 2m(0 " For small damping forces, this shift in the frequency is very small, so equation (9-8) is not very helpful for most cases. It is sufficient that you show that the motion of your oscillator is Eq. (9-7) in form. Procedure You will come up with a procedure in your group! Read the task (below) carefully and come up with a procedure that you think will accomplish it. You have two weeks to complete this task. Your task: • You need to determine whether or not each of the oscillators you have chosen can be approximately described as a simple harmonic oscillator. This means you must determine: o Under what conditions it can be described as such. (E.g. small timescale, small oscillations, etc.). Please don’t test extreme amplitudes in the spring-mass system; it may damage the spring o What parameters describe the motion. For example, what determines the frequency of oscillation in each system? o Whether or not each of the oscillators is damped over time. • The spring has an added parameter: the effective mass. This will be discussed in class during the 2nd week. You must pick three oscillators to investigate. You only need to fully complete the task for the first two. The last one may prove unreasonable to model. • One of your oscillators must be a simple spring mass system. Choose either a mass hanging on a spring vertically or a mass attached to a spring horizontally. There are many different sets of equipment you can use for the horizontal case (an air track, a car track, etc.) • One of your oscillators must be a simple pendulum (a mass on a string). The last oscillator can be anything you can dream up with the equipment we have. I challenge you to come up with something that is very non-linear!!! (I.e. something that cannot be approximated well as a simple harmonic oscillator). For this last case, all you need to do is find the position as a function of time, rather than analyzing it carefully. Think of something wacky—the wackier the better! Two springs attached to a single mass diagonally and bouncing on a table? Pendulums attached to pendulums? Have fun with this last one! .