The Equation for Simple Harmonic Motion

The Equation for Simple Harmonic Motion - frequency. This...

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The Equation for Simple Harmonic Motion From the equation for simple harmonic motion we can tell a lot about the motion of a harmonic system. First of all, x is maximum when the cosine function is equal to 1, or when x = a . Thus a in this equation is the amplitude of oscillation, which we have already denoted by x m . Secondly, we can find the period of oscillation of the system. At t = 0 , x = x m . Also, at t = 2 Π , x = x m . Since both these instances have the same position, the time between the two gives us our period of oscillation. Thus: T = 2 Π and ν = = finally, σ = 2 Πν = Note that the values of period and frequency depend only on the mass of the block and the spring constant. No matter what initial displacement is given to the block, it will oscillate at the same
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Unformatted text preview: frequency. This concept is important. A block with a small displacement will move with slower velocity, but with the same frequency as a block with a large displacement. Notice also that our value for σ is the same as what we called the constant b in our original equation. So now we know that a = x m and b = σ . In addition we can take the time derivative of our equation to generate a full set of equations for simple harmonic motion: x = x m cos( σt ) v =- σx m sin( σt ) a =- σ 2 x m cos( σt ) Thus we have derived equations for the motion of a given simple harmonic system....
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This note was uploaded on 02/09/2012 for the course PHY PHY2053 taught by Professor Davidjudd during the Fall '10 term at Broward College.

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