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Undergraduate Classical Mechanics Spring 2017
Physics 319
Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 9
Undergraduate Classical Mechanics Spring 2017
Hooke’s Law
• For one dimensional simple harmonic motion force law is
• Corresponding potential function
• Significance
– Near an equilibrium Fx = ‒dU/dx = 0, so a quadratic
approximation is the approximation to the potential
with leading significance if k ≠ 0 by Taylor’s theorem
– If k > 0, the motion exhibits, and is the quintessential
example of, strong stability (motion under a
perturbation stays near the motion without the
perturbation)
2
02
xkx
U x kx dx
xF kx
Undergraduate Classical Mechanics Spring 2017
Simple Harmonic Motion
• Energy Diagram
• The amplitude of the motion is denoted by A
Undergraduate Classical Mechanics Spring 2017
Solutions for Simple Harmonic Motion
• Equation of motion
• ω = 2π f is the angular frequency of the oscillation
• Period of oscillation is τ = 2π / ω
• General solution
• Equation is linear, and so superposition applies. Another way to
write the general solution is
• C+ and C‒ in general complex, and need for a real solution
2 0
mx kx
x x
2 2
2 2 2
i t i t
i t i t
i t i t
x t C e C e
x t i C e i C e
C e C e x t
cos sinc sx t B t B t
*C C
Undergraduate Classical Mechanics Spring 2017
Relation Between Expansion Coefficients
• Equating the two forms of the solution
• Or going in the other direction
• In other words
• Complex C+ (C‒) allows one to handle the oscillation phase!
cos sin2 2
/ 2
/ 2
i t i t i t i t
c s c s
c s
c s
e e e eB t B t B B
i
C B iB
C B iB
/
c
s
B C C
B C C i
2Re 2Re
2Im / 2Im /
c
s
B C C
B C i C i
Undergraduate Classical Mechanics Spring 2017
Solution in Amplitude-Phase Form
• Suppose we wish to write the general solution in amplitude-
phase form
• Expression with complex representation even easier!
2 22
1 1
4
tan Im / Re tan Im / Re
2Re Re
Re
i t i t
c s
i i t
A C C C C C C
C C C C
x t C e B iB e
Ae e
2 2 2 2 2 2
cos sin cos
cos cos sin sin
cos sin
tan /
c s
c s
s c
B t B t A t
A t A t
B B A A
B B
Undergraduate Classical Mechanics Spring 2017
Solution in Pictures
• Solution in amplitude phase form
• Computing the phase shift
A
δ
A Bs
Bc
Undergraduate Classical Mechanics Spring 2017
Bottle in Bucket Example
Undergraduate Classical Mechanics Spring 2017
Energy Considerations
• Conserved total energy
2 2
2 2 2 2 2
2
2 2
sin cos2 2
2
E T U
m kx x
m kA t A t
kA
Undergraduate Classical Mechanics Spring 2017
2D Isotropic Oscillator
• Oscillations in two directions at same frequency
Undergraduate Classical Mechanics Spring 2017
An-isotropic oscillations
• Lissajous figures when motion repeats itself (periodic)
Undergraduate Classical Mechanics Spring 2017
Damped Motion
• Add a friction force (linear drag)
• ω0 is the frequency without damping
• Solution ansatz to linear ordinary differential equation
(LCR circuit in electrical engineering)
2 2
02 0
i t
i t
x t Ae
i Ae
2
02 0
2
frict
tot
f bx
f kx bx
k bmx x x x x x
m m
b
m
Undergraduate Classical Mechanics Spring 2017
• For solutions to homogeneous equation
• If β = 0 (undamped) reduces to case before
• If β << ω0 (called the underdamped case), the square root
is real, the angular frequency is adjusted to
and the oscillation damps with exponential damping rate β
Homogeneous Solution
2 2 2 20 0
2 2
0
2 2
0
2 0
4 4 / 4
i t i tt t
i
i
x t C e e C e e
0 0i t i tx t C e C e
2 20
2 2
0costx t Ae t
Undergraduate Classical Mechanics Spring 2017
• If β >> ω0 (called the overdamped case), the square root is
imaginary, the damping has two rates and no oscillation
• If β = ω0 (called the critically damped case), the square
root vanishes. Need another method to determine second
solution
• Motion dies out most quickly when the damping is critical
Over Damping and Critical Damping
2 2 2 2
0 0t tt tx t C e e C e e
0
0 0
2
0 0
2 2 2
0 0 0 0 0
2 0
2 2 2 0 0
t
t t
x x x
x t f t e
f f f f
x t Ce Dte
Undergraduate Classical Mechanics Spring 2017
Solutions Qualitatively
