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Honors Physics 1Class 18 Fall 2013

Harmonic motion

Unit circle and the phasor

Solution to the spring differential equation

Initial and boundary conditions

Conservation of energy

The damped harmonic oscillator

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Mass on Spring

2

2

2

2

Substitute,

F kx

d xF ma m

dt

d x kx

dt m

where x is displacement from equilibrium.

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Second-order linear differential equation

2

2

2

2

0

Assume a solution of the form: ( )

0

4

2

If b=0, then =

If c=- , then =real, ( )

If c=+ , then ( )

t

c t c t

i ct i ct

d x dxb cxdtdt

x t e

b c

b b c

c

c x t Ae Be

c x t Ae Be

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Physical problems of the same form– Mass on spring– Torsional oscilator– Simple pendulum– Physical pendulum– Atoms– Molecules– LC oscillator

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Differential Equations

General solutions and Sums of solutions– Sturm-Liouville problems

Sturm-Liouville problems have a standard form and have the unique characteristic that all possible solutions over the defined range can be generated from a linear sum of the orthogonal functions that are the general solutions to the equation.

Orthogonality of solutions and Completeness of the set of solutions allows us to express any possible solution in terms of sums of the orthogonal functions.

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Simple harmonic motion basics

xm Amplitude (meters)

t + Phase ([radians])

Initial phase ([radians])

Angular Frequency ([rad]/s, s-1)

T Period (s)

f Frequency (Hz, [oscillations]/s)

cos

cos 2

2cos

m

m

m

x x t

x ft

tx

T

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Circular motion – Phasor representation

In uniform circular motion, the x and y

components of the position can be

described by sin and cos functions.

For a particle moving at angular frequency at

distance R from a central point (x,y)=(0,0).

( ) cos cos

( ) sin sin

x t R R t

y t R R t

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Circular motion

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Initial, or Boundary Conditions

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Class Activity

Initial conditions in harmonic motion

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Mass on a spring

Note: xm , and are determined by details of the specific system.

cos where, m

kx x t

m

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Energy of mass+spring system

2 2 2 21 1sin

2 2 mK mv m x t

2 2 21 1cos

2 2 mU kx kx t

2 2 2 ,Note that: som mm x kx

2 2 2 21 1sin cos

2 2m mE U K kx t t kx

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Energy oscillates between U and K

x=Max, v=0, K=0, U=Max

x=0, v=Max, K=Max, U=0

x=Max, v=0, K=0, U=Max

x=0, v=Max, K=Max, U=0

x=0, v=Max, K=Max, U=0

Tim

e

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Damping – a correction to many physical models

/2

2

2

0

2( ) cos( ' );

'2

Note special case 2

bt mm

mx bx kx

mx t x e t

b

k b

m m

b k

m m