phy293_l03.page2

Phy293_l03.page2 - ◦ This is a special case of critical damping(come back next time 3 Underdamped Solution • This is the only solution that

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Unformatted text preview: ◦ This is a special case of critical damping (come back next time) 3. Underdamped Solution • This is the only solution that exhibits oscillations • Roots α 1 , 2 are complex since for ω 2 > γ 2 / 4 the argument of the square-root is negative α + 1 , 2 =- γ 2 ± r γ 2 4- ω 2 =- γ 2 ± i r ω 2- γ 2 4 =- γ 2 ± iω • We’ll see that ω is the damped oscillation frequency ω 2 = ω 2- γ 2 4 • The general solution can be written in several ways x = C 1 e- γt/ 2 e iω t + C 2 e- γt/ 2 e- iω t x = e- γt/ 2 [ D 1 sin( ω t ) + D 2 cos( ω t )] x = Ae- γt/ 2 cos( ω t + φ ) • In each case there are two constants to be determined from initial conditions ( C 1 , 2 ; D 1 , 2 ,A ; φ ) • The first practice problems suggest you find the relationship between these different forms of the solution for practice ◦ ie. the relationship between the different constants • The last form of the solution makes manifest how damping modifies our original (un-damped) SHO solution (see figures)...
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This note was uploaded on 07/10/2011 for the course PHY 293 taught by Professor Pierresavaria during the Fall '07 term at University of Toronto- Toronto.

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