Lots of things vibrate or oscillate. A vibrating tunign fork, a playground swing

Question

Lots of things vibrate or oscillate. A vibrating tunign fork, a playground swing, and the diaphragms of speakers are all examples of physical vibrations. Thre are also electrical and acoustical vibrations, such as radio signals and the sound created when you blow across the top of an open bottle.

One simple system that oscillates is a mass hanging from a spring. The force applied by an ideal spring is proportional to how much it is stretched or compressed. Given this force behavior, the up and down motion of the mass is called simple harmonic and the position can be modeled with

y=Acos(?t+?)

In this equation, y is the vertical displacement from the equilibrium position (the height at which the mass hangs when left alone), A is the amplitude of the motion, ?=2? f is the angular frequency of the oscillation (f alone is the frequency), t is the time, and ? is a phase constant. The latter is related to when and how you start off the oscillation. This experiment will clarify each of these terms further.

For now, we will focus on a mathematical review of the cosine function, which is clearly a key function to describe simple harmonic motion.

For this problem, either use a graphing calculator if you have one and know how to use it to graph. Otherwise, a neat, simple online tool is fooplot (fooplot.com).

Lots of things vibrate or oscillate. A vibrating tunign fork, a playground swing, and the diaphragms of speakers are all examples of physical vibrations. There are also electrical and acoustical vibrations, such as radio signals and the sound created when you blow across the top of an open bottle. One simple system that oscillates is a mass hanging from a spring. The force applied by an ideal spring is proportional to how much it is stretched or compressed. Given this force behavior, the up and down motion of the mass is called simple harmonic and the position can be modeled with y=A cos(?t+?) In this equation, y is the vertical displacement from the equilibrium position (the height at which the mass hangs when left alone), A is the amplitude of the motion, ?=2? f is the angular frequency of the oscillation (f alone is the frequency), t is the time, and ? is a phase constant. The latter is related to when and how you start off the oscillation. This experiment will clarify each of these terms further For now, we will focus on a mathematical review of the cosine function, which is clearly a key function to describe simple harmonic motion. For this problem, either use a graphing calculator if you have one and know how to use it to graph. Otherwise, a neat, simple online tool is fooplot

Explanation / Answer

A is teh maximum amplitude.

Thus, A = 1

Now, T = 1s

f = 1/T = 1 hz

w = 2*pi*f = 6.28 rad/s

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