45
Creating Creating Models Models Chapter 10 Chapter 10
Creating Creating ModelsModels
Chapter 10Chapter 10
Lesson ObjectivesLesson Objectives
Know what a capacitor is and how Know what a capacitor is and how it worksit works
Introduce the idea of capacitanceIntroduce the idea of capacitance Know that the unit of capacitance Know that the unit of capacitance
is the Farad (F) = C Vis the Farad (F) = C V-1-1
EquationsEquations
Ndt
dN RC
Q
dt
dQ RC
t
eQ
Q
0
teN
N 0
k
mT 2
sm
k
dt
ds
dt
d
dt
sd
2
2
ftAs 2sin
C
QCVQV
22
2
1
2
1
2
1
kxF 2. 2
1kxE EP
What if…What if…
Chapter 10.1Chapter 10.1
Smoothed out radioactive decay
Consider only the smooth form of the average behaviour.In an interval dt as small as you please:probability of decay p = dtnumber of decays in time dt is pNchange in N = dN = –number of decays
dN = –pNdN = –N dt
time t
t
N
Nt
Actual, random decay
Simplified, smooth decay
time t
rate of change= slope
probability p of decay in short time t is proportional to t:
p = taverage number of decays in time t is pNt short so that N much less than Nchange in N = N = –number of decays
N = –pNN = –N t
N = –Nt
dNdt=
dN = –Ndt
Clocking radioactive decay
Half-life
time t
N0
N0 /2
N0 /4
N0 /8
t1/2 t1/2 t1/2 t1/2
Activity
time tt1/2
halves every half-life
slope = activity =dNdt
Measure activity. Activity proportional to number N left
Find factor F by which activity has been reduced
Calculate L so that 2L = F
L = log2F
age = t1/2 L
Radioactive clock
In any time t the number N is reduced by a constant factor
In one half-life t1/2 the number N is reduced by a factor 2
In L half-lives the number N is reduced by a factor 2L
(e.g. in 3 half-lives N is reduced by the factor 23 = 8)
number N ofnuclei halvesevery time tincreases byhalf-life t1/2
Stocks and FlowsStocks and Flows
Chapter 10.2Chapter 10.2
Charge, current and timeCharge, current and time
Q Q = Charge transferred in = Charge transferred in Coulombs (C)Coulombs (C)
I I = Current in Amps (A)= Current in Amps (A) tt = Time in seconds (s) = Time in seconds (s)
ItQ
Stores of water and electric charge
dam filledwith water
pressure differenceincreases as amountof water in damincreases
Electric charge
conducting plates withopposite chargesconcentrated on them potential difference V
increases as amount ofcharge stored increases
–Q +Q
Capacitors store electric charge. The larger the capacitance the largerthe charge stored at a given potential difference
to calculate Q or V:
CV = Q
Q = CV
charge stored per volt
define capacitance:
C = QV
units:
charge Qpotential difference Vcapacitance C
coulomb Cvolt Vfarad F = C V–1
Exponential water clock
height h
volume of water V pressure differencep across tube
fine tube to restrict flow
what if....volume of water per second flowingthrough outlet tube is proportionalto pressure difference across tube,and the tank has uniform crosssection?
flow rate f = dVdt
Pressure differenceproportional to height h.Constant cross section soheight h proportional to volumeof water V:
p V
Rate of flow of waterproportional to pressuredifference:
f = dV pdt
Flow of water decreases water volumerate of change of water volume proportional to water volume:
dVdt
–V
t
Time to half empty islarge if tube resistsflow and tank haslarge cross section
Water level decays exponentially if rate of flow proportional to pressuredifference and cross section of tank is constant
Exponential decay of charge
What if....current flowing through resistance is proportional to potentialdifference and potential difference is proportional to chargeon capacitor?
capacitance C
current Iresistance R
potential difference, V
Potential difference Vproportional to charge Q
V = Q/C
Rate of flow of chargeproportional to potentialdifference
I = dQ/dt = V/R
flow of charge decreases chargerate of change of charge proportional to charge
dQ/dt = –Q/RC
time for half chargeto decay is large ifresistance is largeand capacitance islarge
Charge decays exponentially if current is proportional to potentialdifference, and capacitance C is constant
Q
t
current I = dQ/dt
charge Q
Radioactive decay times
Time constant 1/at time t = 1/N/N0 = 1/e = 0.37 approx.t = 1/is the time constant of the decay
Half-life t1/2
0
N0
N0/2
N0/e
dN/dt = – N N/N0 = e–t
t = 0 t = t1/2 t = time constant 1/
In 2 = loge 2
Half-life is about 70% of time constant 1/. Both indicate thedecay time
N/N0 = = exp(– t1/2)
In = –t1/2
t1/2 = ln 2 = 0.693
12
at time t1/2 number N becomes N0/212
Leonhard Euler 1707-1783
01ie
Equations for energy stored
E = Q0V0
E = CV02
E = Q02/C
12
12
12
Energy stored on capacitor = QV12
Energy delivered at p.d. V when a small charge Q flows E = V Q
Energy E delivered by same charge Q falls as V falls
Energy delivered = charge average p.d.
Energy delivered = Q0 V0
capacitor discharges add up strips to get triangle
V1
V2
V0
V0/2
Q Q Q0charge Q charge Q
Capacitance, charge and p.d.
Q0 = CV0
V0 = Q0/CC = Q/V
Q0V012
energyEdelivered= V2 Q
energyEdelivered= V1 Q
energy = area =
12
Clockwork Clockwork ModelsModelsChapter 10.3Chapter 10.3
Phasors and WavesPhasors and Waves
http://www.surendranath.org/Applethttp://www.surendranath.org/Applets/Waves/Twave01A/Twave01AApplets/Waves/Twave01A/Twave01AApplet.html.html
Galileo Galilei 1564-1642
Language to describe oscillations
+A
0
–Aperiodic time Tphase changes by 2
Aangle t
Sinusoidal oscillation
time t
amplitude A
f turns persecond
= 2f radian per second
2 radianper turn
Phasor picture
s = A sin t
Periodic time T, frequency f, angular frequency :
f = 1/T unit of frequency Hz = 2f
Equation of sinusoidal oscillation:
s = A sin 2ft s = A sin t
Phase difference /2
s = A s in 2fts = 0 when t = 0
s = A cos 2fts = A when t = 0
t = 0
sand falling from a swinging pendulum leavesa trace of its motion on a moving track
k
mT 2
fT
22
maF
sa 2
ksF
g
lT 2
Spring Pendulum
Angular Frequency Acceleration
Motion of harmonic oscillator
large force to left
large displacement to right
zero velocity
mass m
displacementagainst time
velocityagainst time
forceagainst time
right
left
small displacement to right
small velocityto left
mass msmall force to left
right
left
large velocityto left
mass mzero net force
right
left
small displacement to left
small velocityto left
mass m
small force to right
right
left
large displacement to left
mass mlarge force to right
zero velocityright
left
John Harrison 1683-1766
AppletApplet
Oscillations/Spring MassOscillations/Spring Mass http://www.surendranath.org/Applethttp://www.surendranath.org/Applet
sJ2.htmlsJ2.html
Dynamics of harmonic oscillator
How the graph starts
force changesvelocity
0
time
t
zero initialvelocity
velocity would stayzero if no force
How the graph continues
force of springs accelerates mass towardscentre, but less and less as the mass nears thecentre
time
0
trace straighthere because nochange ofvelocity
no force at centre:no change of velocity
trace curvesinwards herebecause ofinwardschange ofvelocity
change of velocitydecreases asforce decreasesnew velocity
= initial velocity+ change ofvelocity
Constructing the graph
if no force, same velocityand same change indisplacementplusextra change indisplacement fromchange of velocity dueto force
= –(k/m) s (t)2
change in displacement = v t
t
t
extra displacement= v t
change of velocity v= acceleration tv = –(k/m) s t
because of springs:force F = –ks
acceleration = F/macceleration = –(k/m) s
Health warning! This simple (Euler) method has a flaw. It always changesthe displacement by too much at each step. This means that the oscillatorseems to gain energy!
extra displacement
Changing rates of change
change in ds = d(ds) = dv dt
= a dt2
The first derivative ds/dt says how steeply the graph slopesThe second derivative d2s/dt2 says how rapidly the slope changes
dtds = v d t
v = dsdt
s
t
dtds = v dt
dt
ds = (v + dv) dt
a = dvdt
t
s
dtv dt
dtv dt
dv dt
change in ds = d(ds) = dv dt = a dt2
s
t
new slope = new rate of change of displacement
= new velocity (v + dv)
new ds = (v + dv) dt
dv = a dt
slope = rate of change of displacement
= velocity v
= acceleration a
d ds d2sdt dt dt2 = a( )=
rate of changeof velocity
rate of changeof slope
=
AppletApplet
Oscillations/PendulumOscillations/Pendulum http://www.surendranath.org/Applethttp://www.surendranath.org/Applet
sJ2.htmlsJ2.html
Force, acceleration, velocity and displacement
If this is how the displacement varieswith time...
... the velocity is the rate of changeof displacement...
... the acceleration is the rate ofchange of velocity...
...and the acceleration tracks the forceexactly...
... the force is exactly opposite tothe displacement...
Phase differences Time traces varies with time like:
/2 = 90
/2 = 90
= 180
zero
displacement s
force F = –ks
displacement s
cos 2ft
same thing
–sin 2ft
–cos 2ft
–cos 2ft
cos 2ft
acceleration = F/m
velocity v
Simple models compared
Exponential growth Exponential decaydQ / dt = + kQ dQ / dt = – kQ
positivefeedback
time
negativefeedback
time
population+
k
population–
k
Harmonic oscillator d2s / dt2 = – (k / m)s or v = ds / dta = dv / dta = – (k / m)s
accelerationa = F / m
velocity displacement
springconstant
kforce
F = – ksmass m –
time
rate of changeof velocity
rate of change ofdisplacement
rate of change ofpopulation
rate of change ofpopulation
Homework – For MondayHomework – For Monday
Research Foucault’s PendulumResearch Foucault’s Pendulum What is it?What is it? What is it for?What is it for? Could we make one?Could we make one?
Research the “Millennium Bridge”Research the “Millennium Bridge” Why did it wobble?Why did it wobble? How did they fix it?How did they fix it?
ResonatingResonating
Chapter 10.4Chapter 10.4
Lesson ObjectivesLesson Objectives Introduce the concept of resonance when Introduce the concept of resonance when
an oscillator is driven by a sourcean oscillator is driven by a source Explain and measure how resonance is a Explain and measure how resonance is a
result of the driving frequency being result of the driving frequency being similar to natural frequencysimilar to natural frequency
Be able to use F=ks to calculate spring Be able to use F=ks to calculate spring constant k and calculate periodicityconstant k and calculate periodicity
Understand how the resonance peak is Understand how the resonance peak is usually very narrow without dampingusually very narrow without damping
Retrofitted Mass Dampers
AppletApplet
Forced OscillationsForced Oscillations http://www.surendranath.org/Applethttp://www.surendranath.org/Applet
s.htmls.html http://www.walter-fendt.de/ph14e/rehttp://www.walter-fendt.de/ph14e/re
sonance.htmsonance.htm
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 200 400 600 800 1000 1200
Frequency (Hz)
Sh
ort
est
Res
on
ance
Len
gth
(m
)
No weight With Weight
ksF k
mT 2
g
lT 2
Energy stored in a stretched spring
no forcework F1 xforce F1
x
larger force
extension x
area below graph= sum of force change in displacement
extra areaF1 x
total area Fx
21
unstretched
F1
Energy supplied
energy stored in stretchedspring = kx2
21
small change xenergy supplied = F x
stretched to extension x by force F:energy supplied = Fx2
1
spring obeysHooke’s law: F = kx
F = 0x = 0
F = kx
Energy stored in stretched spring is kx221
00
x
Energy flow in an oscillator
displacement
time
time
s = A sin 2ft
PE = kA2 sin22ft
0
0
potential energy= ks21
2
potential energy
energy in stretched spring
energy carried by moving mass
time
time
0
0
kinetic energy= mv21
2
velocity
mass andspringoscillate
vmax
A
Avmax vmax
The energy stored in an oscillator goes back and forth between stretched spring and movingmass, between potential and kinetic energy
from spring tomoving mass
from movingmass to spring
from movingmass to spring
from spring tomoving massenergy in
stretched spring
energy inmoving mass
kinetic energy
vmax = 2fA
v = vmax cos 2ft
KE = mvmax cos22ft212
12
Resonant response
10
5
01
Example: ions in oscillating electric field
ions in a crystalresonate andabsorb energy
Oscillator driven by oscillating driver
electricfield
+ – + –
low damping:large maximum responsesharp resonance peak
frequency/natural frequency0 0.5 1 1.5 2.0
10
5
01
frequency/natural frequency0 0.5 1 1.5 2.0
more damping:smaller maximum responsebroader resonance peak
Resonant response is a m axim um w hen frequency of driver is equal to natural frequency of oscillator
narrow rangeat peakresponse
12
wider rangeat peakresponse
12
The Tacoma Narrows Bridge
Question HintsQuestion Hints
Hints for 110SHints for 110S
1.1. You will have to do quite a bit of You will have to do quite a bit of rearranging of equations.rearranging of equations.
6.6. Think of the definition of Think of the definition of potential difference when you are potential difference when you are trying to work out how much energy trying to work out how much energy is involved in passing charge is involved in passing charge through a battery.through a battery.
Hints for 150SHints for 150S
2.2. Write an equation for distance Write an equation for distance travelled. Distance is the area under travelled. Distance is the area under your sketched velocity–time graph.your sketched velocity–time graph.
Hints for 220SHints for 220S
3.3. To calculate the amount of To calculate the amount of stretch, consider the loss of stretch, consider the loss of gravitational potential energy of the gravitational potential energy of the jumper, all changed to elastic strain jumper, all changed to elastic strain energy. You will end up with a energy. You will end up with a quadratic equation to solve.quadratic equation to solve.
Hints for 210DHints for 210D
1.1. Just remember that the kinetic Just remember that the kinetic energy at any point = total energy – energy at any point = total energy – stored energy.stored energy.
