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Class 15: Calculation of natural frequency Class 15: Calculation of natural frequency Old Slide Origin of simple harmonic motion U(x) = kx 2 Total energy Total energy F = - kx x V=0 a is max V=0, a is max. V=0 a is max V=0, a is max. a=0, v

Class 15: Calculation of natural frequency Class 15: Calculation of natural frequency
Old Slide Origin of simple harmonic motion U(x) = ½ kx 2 Total energy Total energy F = - kx x V=0 a is max V=0, a is max. V=0 a is max V=0, a is max. a=0, v is max.
Arbitrary potential Arbitrary potential Taylor expansion of arbitrary potential about x=x 0 : Taylor expansion of arbitrary potential about x=x : 1 = + + + 2 L U(x) ( ) U(x ( ) ) U' ( (x )( )( x - x ) ) U" (x ( )( )( x - x ) ) 0 0 0 0 0 0 0 0 0 0 2 For simple harmonic motion, F i l h i i U(x 0 ) ‐‐‐ Irrelevant, just an offset U’( ) 0 U’(x 0 )=0 ‐‐‐ Simple harmonic motion Si l h i i always occur at the minimum potential i i t ti l
Simple harmonic motion of arbitrary potential Given arbitrary potential U(x) Suppose U(x )=0 Given arbitrary potential U(x). Suppose U(x 0 )=0 1 ≈ 0 + + 2 U(x) U(x) U(x U(x ) ) U" U" (x (x )( )( x x - x x ) ) 0 0 2 ∴ = k U" (x ) 0 k U" (x ) ω = = 0 0 m m

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