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Week 9: Oscillations

Oscillation Summary

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Springs obey Hooke’s Law: F = −k~x (where k is called the spring constant. A perfect spring (with no damping or drag force) produces perfect harmonic oscillation, so this will be our archetype.

A pendulum (as we shall see) has a restoring force or torque proportional to displacement for small displacements but is much too complicated to treat in this course for large displacements. It is a simple example of a problem that oscillates harmonically for small displacements but not harmonically for large ones.

An oscillator can be damped by dissipative forces such as friction and viscous drag. A damped oscillator can have exhibit a variety of behaviors depending on the relative strength and form of the damping force, but for one special form it can be easily described.

An oscillator can be driven by e.g. an external harmonic driving force that may or may not be at the same frequency (in resonance with the natural frequency of the oscillator.

The equation of motion for any (undamped) harmonic oscillator is the same, although it may have di erent dynamical variables. For example, for a spring it is:

d2x

 

k

d2x

+ ω2x = 0

 

 

+

 

x =

 

 

(767)

dt2

 

dt2

 

m

 

 

where for a simple pendulum (for small oscillations) it is:

d2θ

 

g

d2θ

+ ω2

 

 

 

+

 

x =

 

θ = 0

(768)

dt2

 

dt2

 

 

 

 

(In this latter case ω is the angular frequency of the oscillator, not the angular velocity of the mass dθ/dt.)

The general solution to the equation of motion is:

x(t) = A cos(ωt + φ)

(769)

p

where ω = k/m and the amplitude A (units: length) and phase φ (units: dimensionless/radians) are the constants of integration (set from e.g. the initial conditions). Note that we alter the variable to fit the specific problem – for a pendulum it would be:

θ(t) = Θ cos(ωt + φ)

(770)

p

with ω = g/ℓ, where the angular amplitude Θ now has units of radians.

The velocity of the mass attached to an oscillator is found from:

v(t) =

dx

= −Aω sin(ωt + φ) = −V sin(ωt + φ)

(771)

dt

(with V = vmax = ).

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