Homework Problem Set 5 Solutions ( )2

3. continued
7
b.) Using the analytic solution to Newton's equations for the harmonic oscillator, plot the
bond displacement coordinate x as a function of time.
For simple harmonic motion and the provided initial conditions, the analytic expression for the position is
x(t) = v(0)
ω
sin( ω t)
.
Using the harmonic vibrational frequency ν o calculated in part (a), the angular frequency ω can be
determined,
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ω = 2π ν o
( )
= 2π 2.52 ×10 13 s −1
ω = 1.58 ×10 14 s −1 .
The initial velocity is
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v( 0) = 8400 m/s . Thus, the position as a function of time is given by the equation
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( ) = v(0)
x t
sin( ω t)
ω
⎛⎛ 8400 m/s ⎞⎞
= ⎜⎜
⎝⎝ 1.58 ×10 14 s −1 ⎟⎟ sin 1.58 ×10 14 s −1 t
⎠⎠
= 5.31×10 −11 m
( )
( ) sin( 1.58 ×10 14 s −1 t)
( ) sin( 1.58 ×10 14 s −1 t) .
x( t) = 0.531Å
A plot of this function versus time is shown in the figure below.
0.6
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0.4
0.2
x(t) (Å)
0.0
0.0 0.1 0.2 0.3 0.4 0.5
-0.2
-0.4
-0.6
time (ps)