# Experiment Type: Open-Ended

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1 Simple Harmonic Oscillation Overview Experiment Type: Open-Ended 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

2 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 = kx (9-1) In this equation, k is a constant of proportionality, usually called the spring constant, and x r 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 -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.

3 Figure -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. = 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 r along the path is related to θ by the formula = s / R. That means that for small angles this restoring force becomes = mg' & = mg\$ % s R " (9-3) 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 + ) (9-4) ( In this equation, A is the (constant) amplitude of the oscillation, ω is the frequency of the oscillation, and δ o 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

4 k = (9-5) 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. F damping = 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 -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

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