P11_standwaves3

P11_standwaves3 - Standing Waves Name: Teammates:...

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Standing Waves Name: Teammates: Fig. 1: Tacoma Narrows Bridge Introduction For a spring mass system or a pendulum, when release from a position away from the equilibrium position, the position, velocity, and acceleration of the mass are all described by single sinusoidal functions. () cos( ) =+ xt A t ω ϕ s in ( ) =− + vt A t ωϕ 22 cos ( ) + at A t ωω Because the motion repeats at one particular frequency, called harmonic, the system is referred to as a simple harmonic oscillator (SHO). It is noteworthy that the motion of a great many systems, such as the shock absorbers in cars or a wrecking ball on the end of a crane, can be modeled fairly well by SHO. This is not to say that all systems that oscillate can by modeled by a single SHO. Take, for instance, the motion of a pinecone on the end of a branch of a tree. The small branch to which the cone is attached operates very much like a pendulum, which is a SHO. However, this branch is attached to larger branches, which are themselves attached to the tree trunk. All of these branches and trunks by themselves operate like SHO’s. When coupled together to form a tree, the motion of the cone will be quite complicated, as each SHO will have its own distinct frequency at which it will want to oscillate. Our arms act in a somewhat similar manner to this. The forearm and the humerous operate as two pendula that are attached at the elbow, albeit with the limitation that the forearm is not free to swing completely backward relative to the humerous due to bicep tendons. Driven Oscillators and Resonance Of course, the effects of friction and drag will dampen any oscillatory behavior. Some systems have very small retarding forces acting upon them, and will oscillate for a long time without much decay in their motion others will quickly stop oscillating after they are pulled from equilibrium.
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P11_standwaves3 - Standing Waves Name: Teammates:...

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