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Physics 2 > Oscillating Systems > Flashcards

Oscillating Systems Flashcards

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1
Q

complex number basics

A

a=rcos theta
b=rsin theta

r = sqrt(a^2+b^2)

theta = arctan(b/a)

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2
Q

derivation of euler’s identity

A

a+ib
rcostheta+irsintheta
sub in taylor expansion of costheta and sintheta

terms cancel
left with re^itheta

e^ipi=cospi+isinpi=-1=i^2

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3
Q

if i represents a rotation by 90 degrees into the imaginary plane, complex numbers can be represented using

A

polar coordinates

(r,theta)

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4
Q

phase

A

how much something leads/lags the main motion

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5
Q

in complex numbers, phase is angle theta to the

A

imaginary plane

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6
Q

if z=A0e^iwt, the angle is changing with

A

time

therefore vector is rotating

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7
Q

rotation when wt=2pi

A

2pi

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8
Q

T=

A

2pi/w

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9
Q

features of SHM

A

motion confined within +/-A

T between 2 successive occasions where x and dx/dt repeat

has relative phase phi

sinusoidal variations

small displacements

restoring force directly prop. to displacement from eqm

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10
Q

Hooke’s law

A

F=-kx

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11
Q

large angles

A

period gets longer and is a function of initial starting angle

no longer SHM

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12
Q

Fourier’s Theorem

A

any function that repeats regularly can be built up from a set of sinusoidal functions of appropriate periods and amplitudes

superposition

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13
Q

fourier series: more terms=

A

more square the wave

can better approximate system with more terms

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14
Q

expression for mass on a spring

A

F=ma
F=-kx

equate and sub in a=d2x/dt2

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15
Q

solving DE:

m(d2x/dt2)=-kx

A

guess a sinusoidal solution and differentiate twice

x=Asin(wt+phi 0)
v=…
a=-w^2x

use w^2=k/m

d2x/dt2=-k/m x so solution

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16
Q

what turns cosine to sine

A

phase of pi/2

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17
Q

what variables completely define SHM?

A

A,w,phi0

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18
Q

derivation of phi0 = arctan(-v0/wx0)

A

take x=Acos(wt+phi0) when t=0

dx/dt=v0=…

v0/w=…

v0/x0w= -Asin phi0 / A cosphi0= -tanphi0

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19
Q

derivation of A=sqrt(x0^2+v0^2/w^2)

A

square x0 and v0

v0^2/w^2=…

x0^2+v0^2/w^2

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20
Q

to solve equation of motion of harmonic oscillator, need to find function for which

A

the double derivative leads back to original function

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21
Q

x in complex form

A

Z=Ae^i(wt+phi0)

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22
Q

SHM may be described by the projection of

A

a particle in uniform circular motion

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23
Q

velocity on argand diagram

A

perpendicular to position vecotr

angle of wt+phi0+pi/2

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24
Q

acceleration on argand diagram

A

anti-parallel to position (phase 180)

angle wt+phi0+pi

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25
Q

total energy of system

A

kinetic + potential

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26
Q

have x=Acos(wt+phi0), v=-wAsin(wt+phi0)

potential energy, V

A

V = - integral Fdx = integral kx dx

=1/2 kx^2

sub in x

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27
Q

potential energy V for a linear force F is

A

quadratic

for small displacements approximate taylor expansion as linear

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28
Q

have x=Acos(wt+phi0), v=-wAsin(wt+phi0)

kinetic energy, T

A

=1/2 m (dx/dt)^2

sub in w^2=k/m

cancel terms

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29
Q

total energy

A

E=T+v

sub in previous expressions

sin^2+cos^2=1

E=1/2kA^2

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30
Q

why is total energy constant throughout motion

A

only depends on A and k

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31
Q

time dependence PE and KE oscillate out of phase with each other by

A

180 degrees

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32
Q

potential energy maximal if

A

displacement maximal

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33
Q

kinetic energy maximal if

A

velocity is at its peak/trough

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34
Q

idealised system for pendulum

A

massless string
extenstionleess string
no air resistance
no friction at pivot

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35
Q

restoring force for a displacement of s=l phi

A

F=-mg sin phi

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36
Q

how to get DE:

d2phi/dt2 + g/l sin phi =0

A

equate restoring force and f=ma

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37
Q

primary bonds

A

ionic
covalent
metallic

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38
Q

secondary bonds

A

van der waals

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39
Q

ionic

A

exchanging of e- forming ion

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40
Q

covalent

A

sharing of e-

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41
Q

metallic

A

sea of e- shared by all atoms

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42
Q

VDW

A

irregularities in e- distribution creates dipoles that attract one another

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43
Q

what 2 forces determine the potential of an ionic molecule

A

coulomb attraction between + and - ions

quantum mechanical effect within Pauli exclusion principle

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44
Q

quantum mechanical effect within Pauli exclusion principle

A

as two ions go towards each other, electron clouds overlap, e- move into states of higher energy

seen as repulsive force, potential parameterised as B/r^9

45
Q

Vr=

A

Fcoulomb + B/r^9

46
Q

ionic potential

A

must be a position where distance R between ions where sum of attractive and repulsive forces is zero

must be minimum

diff to find turning point

(around tp approximate as quadratic)

47
Q

how to find force constant k

A

second derivative of potential

find dv/dr where R=r
get an expression for B

find d2v/dr2 and sub in B

48
Q

k is only dependent on

A

bond length R

49
Q

how could bond length be measured

A

if vibration frequency measured

would allow a measurement of how far atoms are from each other

50
Q

reduced mass

A

μ = m1m2/m1+m2

51
Q

general method for verifying something is a solution to a DE

A

take solution and differentiate twice and try to simplify to required form

52
Q

as long as system is linear, the result of two or more harmonic vibrations is

A

the sum of individual vibrations

53
Q

superposition angle

A

alpha 2 - alpha 1

result of rotation by (wt+alpha1)

54
Q

if rotation of frequencies is incommensurable eg root2

A

result will not be periodic

55
Q

combination of frequencies close to each other

A

effect called beats

56
Q

resulting vibration has frequency that is average of the two initial frequencies known as

A

beat frequency/ fast oscillation

57
Q

resulting amplitude varies periodically with time and is known as

A

modulating envelope/ slow oscillation

58
Q

modulating envelope depends on

A

difference between both frequencies

59
Q

how to show modulating envelope depends on difference between frequencies

A

assume two vibrations have equal amplitude

set w1t=a+b, w2t=a-b

set x=x1+x2

use trig identities on formula sheet with
a=w1t-b
b=a-w2t

sub into 2cosacosb

get expression: x=(modulation)(beat)

60
Q

superposition in two dimensions

if w1 and w2 not commensurable

A

movement of point limited to rectanglel of 2A1 by 2A2

entire rectangle ‘filled’

61
Q

superposition in two dimensions

w1 and w2 are commensurable

A

have periodic 2D orbit known as Lissajous figure

62
Q

lissajous figures - straight line

A

identical frequency, no relative phase

y=A2/A1 x

63
Q

lissajous figure - cirlce

A

x=A1coswt
y=A2cos(wt+pi/2)

square x and y and add gets cirlce with radius A

64
Q

lissajous figure for identical frequency but arbitrary amplitude and phase

A

ellipse

65
Q

homogeneous eqn

A

RHS=0

66
Q

general solution of a non-homogeneous eqn

A

sum of general solution of related homogeneous eqn and particular solution of non-homogeneous

67
Q

SHM with damping DE

A

assume forces linear and equate

md2x/dt2=-kx-bv

rearrange and divide by m

b/m=2 gamma, k/m=w^2 (sub in)

use integrating factor z(t)=e^lambda t

diff integrating factor and sub in for a,v,x

diide by e^lambda t to get characteristic eqn

find lambda by taking roots of quadratic

68
Q

if gamma^2 < w0^2

A

root -ve and therefore imaginary

69
Q

if gamma^2=w0^2

A

root 0 so toor vanishes, resulting in one special solution

70
Q

if gamma2>w0^2

A

root +ve so real

71
Q

dissipative term

A

represents resistive force which is velocity dependent

(expect term prop. to dx.dt)

72
Q

damping characterised by

A

gamma which has same dimensions as frequency

73
Q

light damping

A

root -ve and imaginary

74
Q

total energy for light damping

A

E=1/2kA^2 e^-2gammat

decay constant of 2 gamma

75
Q

quality factor

A

Q=w/2 gamma

describes how damping factor acts on system

as gamma approaches 0, very little damping

76
Q

w^2=

A

w0^2-gamma^2

77
Q

beat frequency

A

difference between 2 individual frequencies

think of as one wave laps the other

78
Q

beat period

A

point where both waves line up

79
Q

heavy damping

A

system experiences no oscillations

once excited, slowly moves back to eqm position

80
Q

critical damping possible solution

A

z(t)=te^-gamma t

81
Q

what type of damping takes longer to reach eqbm

A

light

82
Q

although heavy damping is more aggressive in initial steepness…

A

it takes longer than critical damping to reach eqbm

(depending on initial conditions, may pass eqbm point once before settling)

82
Q

what type of damping reaches eqbm first

A

critical

useful applications in mechanics

83
Q

resonances

A

amplitude of an oscillation can become very large even if periodic driving force is small, if the driving frequency w is close to the natural frequency w0

84
Q

example of resonance

A

tapping on wine glass till it smashes

pushing a swing: need to push in phase to keep going higher

85
Q

resonance catastrophe

A

amplitudes beyond what physical system can allow

destroys system

86
Q

equation of motion for a damped oscillator with a periodic external force

A

m d2x/dt2 = -kx - bdx/dt +Focoswt

87
Q

equation of motion for a damped oscillator with a periodic external force in exponential form

A

d2z/dt2 + 2gamma dz/dt + wo^2z = F0e^iwt

88
Q

after some time, solution to the homogeneous case will have been

A

damped away

just leaves special solution corresponding to external force

89
Q

transition period

A

addition of a damped oscillator at the natural frequency to the forced oscillation at the driving frequency

90
Q

steady state

A

transient effects die away and see only the effects of the driving force (solutions of inhomogeneous equation)

91
Q

resonance curve

A

frequency dependence of A and delta

92
Q

resonance frequency

A

max amplitude can be obtained

93
Q

how to find resonance frequency

A

differentiating to find turning point

94
Q

w &laquo_space;wres

A

system oscillates in phase with A=F/mw0^2 approx = Fo/k

95
Q

w=wres

A

system oscillates with A much larger than that of driving force but pi/2 out of phase

96
Q

w»wres

A

system is pi out of phase and oscillates in opposite direction to driving force with oscillations having small A

97
Q

for small damping coefficient, phase change

A

prominent, strong peak and fast phase change around the natural frequency

98
Q

for large damping coefficient, phase change

A

resonance frequency moves closer to zero and resonance peak vanishes also resulting in slow phase change

99
Q

higher quality factor means higher A at wres and

A

sharper resonance curve

100
Q

mass on string for w approaching 0

A

direction of motion pi/2 out of phase with direction of force applied

101
Q

mass on string for w approaching infinity

A

pi out of phase

102
Q

largest amplitude when

A

w=w0

103
Q

power absorbed by driven oscillator

A

P=dW/dt = Fdx/dt = Fv

104
Q

for undamped oscillator, no dissipative effects so

A

mean power absorbed P bar =0 (steady state solution for x(t))

105
Q

power only depends on

A

Q

eg: high Q glass absorbs more power than low

system with Q=50 will absrob 50x as much power as Q=1

106
Q

good at absorbing power =

A

good at oscillating = low damping coefficient

107
Q

Tacoma bridge collapse

A

during storm, wind blowing against the bridge caused it it start twisting

wind provided external periodic force that matched natural frequency of the bridge

caused amplitude of torsion to become so great the bridge collapsed

108
Q
A

Physics 2 (8 decks)

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