10 Creating models
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Revision Notes
Capacitance ................................................................................................................................
5
Exponential decay processes .....................................................................................................
5
Simple harmonic motion .............................................................................................................
6
Damping and resonance ............................................................................................................
6
Summary Diagrams (OHTs)
Energy stored on a capacitor ......................................................................................................
8
Exponential decay of charge ......................................................................................................
9
Radioactive decay ................................................................................................................... 10
Half-life and time constant ....................................................................................................... 11
Describing oscillations ............................................................................................................. 12
Motion of a simple harmonic oscillator .................................................................................... 13
Graphs of simple harmonic motion .......................................................................................... 14
Computing oscillator motion step by step ................................................................................ 15
Elastic potential energy ........................................................................................................... 17
Energy flows in an oscillator .................................................................................................... 18
Resonance ............................................................................................................................... 19
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capacitance as the ratio C = Q / V the energy stored on a capacitor E
 1
2
QV
Revision Notes: Capacitance
Summary Diagrams: Energy stored on a capacitor
decay of charge on a capacitor modelled as an exponential relationship between charge and time; with the rate of removal of charge proportional to the quantity of charge remaining
Revision Notes: Exponential decay processes
Summary Diagrams: Exponential decay of charge
radioactive decay modelled as an exponential relationship between the number of undecayed atoms, with a fixed probability of random decay per atom per unit time
Revision Notes: Exponential decay processes
Summary Diagrams: Radioactive decay ;
Half-life and time constant
simple harmonic motion of a mass m subject to a restoring force F
  kx proportional to the displacement
Revision Notes: Simple harmonic motion
Summary Diagrams: Describing oscillations ;
Motion of simple harmonic oscillator ;
Graphs of
SHM ;
Computing oscillator motion step by step
changes of kinetic energy 1
2 mv
2
and potential energy 1
2 k x
2
during simple harmonic motion
Summary Diagrams: Elastic potential energy
; Energy flows in an oscillator
free and forced vibrations (oscillations) of an object damping of oscillations resonance (i.e. when natural frequency of vibration matches the driving frequency)
Revision Notes: Damping and resonance
Summary Diagrams: Resonance
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for capacitors: half-life, time constant for radioactivity: half-life, decay constant, random, probability
Revision Notes: Exponential decay processes
simple harmonic motion, amplitude, frequency, period, resonance
Revision Notes: Simple harmonic motion ; Damping and resonance
Summary Diagrams: Describing oscillations ;
Resonance
relationships of the form d x /d t = – kx , i.e. where a rate of change is proportional to the amount present
Revision Notes: Exponential decay processes
radioactive decay against time (plotted both directly and logarithmically)
Summary Diagrams: Radioactive decay ;
Half-life and time constant
decay of charge, current or potential difference with time for a capacitor ( plotted both directly and logarithmically )
Summary Diagrams: Exponential decay of charge
charge against voltage for a capacitor as both change, and know that the area under the curve gives the corresponding energy change
Summary Diagrams: Energy stored on a capacitor
displacement –time, velocity–time and acceleration–time for simple harmonic motion (showing phase differences and damping where appropriate)
Summary Diagrams: Graphs of SHM
variation of potential and kinetic energy in simple harmonic motion
Summary Diagrams: Energy flows in an oscillator
variation in amplitude of a resonating system as the driving frequency changes
Summary Diagrams: Resonance
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10 Creating models
small difference methods to build a numerical model of a decay equation small difference methods to build a model of simple harmonic motion
Revision Notes: Exponential decay processes
Summary diagrams: Computing oscillator motion step by step
data to calculate the time constant

= RC of a capacitor circuit data to calculate the half-life of a radioactive source
Revision Notes: Exponential decay processes
Summary Diagrams: Exponential decay of charge
; Half-life and time constant
the relationships for capacitors:
C = Q/V
I =

Q /
 t
E = (1/2) QV = (1/2) CV
2
Revision Notes: Capacitance
Summary Diagrams: Energy stored on a capacitor
the basic relationship for simple harmonic motion d
2 x
2
 a
 
( k m
) x d t the relationships x = A sin 2
 ft and x = A cos 2
 ft for harmonic oscillations the period of simple harmonic motion:
T

2
 m k and the relationship F = -kx
Revision Notes: Simple harmonic motion
Summary Diagrams: Describing oscillations ;
Motion of simple harmonic oscillator ;
Graphs of
SHM ;
Computing oscillator motion step by step
the conservation of energy in undamped simple harmonic motion:
E total
 1
2 mv
2  1
2 k x
2
Summary Diagrams: Energy flows in an oscillator
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Capacitance is charge stored / potential difference, C = Q/V .
The SI unit of capacitance is the farad (symbol F).
One farad is the capacitance of a capacitor that stores a charge of one coulomb when the potential difference across its terminals is one volt. This unit is inconveniently large. Thus capacitance values are often expressed in microfarads (

F) where 1

F = 10
–6
F.
Relationships
For a capacitor of capacitance C charged to a potential difference V:
Charge stored Q = C V.
Energy stored in a charged capacitor E = 1/2 Q V = 1/2 C V
2
.
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In an exponential decay process the rate of decrease of a quantity is proportional to the quantity remaining.
Capacitor discharge
For capacitor discharge through a fixed resistor, the current I at any time is given by I = V / R , where V = Q / C . Hence I = Q /RC.
Thus the rate of flow of charge from the capacitor is
I
 d d
Q
 
Q t RC where the minus sign represents the decrease of charge on the capacitor with increasing time.
The solution of this equation is

Q

Q
0 e
 t RC
.
The time constant of the discharge is RC .
Radioactive decay
The disintegration of an unstable nucleus is a random process. The number of nuclei

N that disintegrate in a given short time
 t is proportional to the number N present:

N = – 
N
 t , where

is the decay constant. Thus:
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10 Creating models


N t
  
N .
If there are a very large number of nuclei, the model of the differential equation d d
N
  
N t can be used. The solution of this equation is
N

N
0 e
  t
.
The time constant is 1 /

. The half-life is T
1/2
= ln 2 /

.
Step by step computation
Both kinds of exponential decay can be approximated by a step-by-step numerical computation.
1. Using the present value of the quantity (e.g. of charge or number of nuclei), compute the rate of change.
2. Having chosen a small time interval d t , multiply the rate of change by d t , to get the change in the quantity in time d t .
3. Subtract the change from the present quantity, to get the quantity after the interval d t .
4. Go to step 1 and repeat for the next interval d t .
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Simple harmonic motion is the oscillating motion of an object in which the acceleration of the object at any instant is proportional to the displacement of the object from equilibrium at that instant, and is always directed towards the centre of oscillation.
The oscillating object is acted on by a restoring force which acts in the opposite direction to the displacement from equilibrium, slowing the object down as it moves away from equilibrium and speeding it up as it moves towards equilibrium.
The acceleration a = F/m , where F = - ks is the restoring force at displacement s . Thus the acceleration is given by: a = –( k / m ) s ,
The solution of this equation takes the form s

A sin ( 2
 ft
 
) where the frequency f is given by
 
2  k m , and

is a phase angle.
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In any oscillating system, energy is passed back and forth between parts of the system:
1. If no damping is present, the total energy of an oscillating system is constant. In the mechanical case, this total energy is the sum of its kinetic and potential energy at any instant.
2. If damping is present, the total energy of the system decreases as energy is passed to the surroundings.
If the damping is light, the oscillations gradually die away as the amplitude decreases.
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1
0
1
Damped oscillations time
Lightly damped oscillations heavy damping
0 critical damping time
Increased damping
Forced oscillations are oscillations produced when a periodic force is applied to an oscillating system. The response of a resonant system depends on the frequency f of the driving force in relation to the system's own natural frequency, f
0
. The frequency at which the amplitude is greatest is called the resonant frequency and is equal to f
0
for light damping. The system is then said to be in resonance. The graph below shows a typical response curve.
0 oscillator frequency resonant frequency
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capacitor discharges add up strips to get triangle
V
0
V
1
V
2

Q energy

E delivered
= V
2

Q charge Q

Q energy

E delivered
= V
1

Q
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
V
0
/2 energy = area =
1
2
Q
0
V
0
Q
0 charge Q
Energy delivered = charge

average p.d.
1
Energy delivered = Q
0
V
0
Capacitance, charge and p.d.
Equations for energy stored
C = Q / V
Q
V
0
0
= CV
= Q
0
/
0
C
E = Q
0
V
0
E = CV
0
2
E = Q
0
2 / C
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Exponential decay of charge
What if....
current flowing through resistance is proportional to potential difference and potential difference is proportional to charge on capacitor?
capacitance C charge Q current I resistance R potential difference, V current I = d Q /d t
Potential difference V proportional to charge Q
Rate of flow of charge proportional to potential difference
V = Q/C I = d Q /d t = V / R flow of charge decreases charge rate of change of charge proportional to charge d Q /d t =
–
Q / RC
Q time for half charge to decay is large if resistance is large and capacitance is large t
Charge decays exponentially if current is proportional to potential difference, and capacitance C is constant
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Smoothed out radioactive decay
Actual, random decay

N
 t

N
 t time t probability p of decay in short time
 t is proportional to
 t : p =
  t average number of decays in time
 t is pN
 t short so that

N much less than N change in N =

N =
–number of decays

N =
– pN

N = – 
N
 t

N
 t

N
Simplified, smooth decay rate of change
= slope d N
= d t time t
Consider only the smooth form of the average behaviour.
In an interval d t as small as you please: probability of decay p =

d t number of decays in time d t is pN change in N = d N =
–number of decays d N =
– pN d N = – 
N d t
N
t

N
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Radioactive decay times d N /d t = – 
N
N
0
N / N
0
= e
–  t
N
0
/2
N
0
/e
0 t = 0 t = t
1/2 t = time constant 1/

Time constant 1/
 at time t = 1/

N / N
0
= 1/e = 0.37 approx.
t = 1/
 is the time constant of the decay
Half-life t
1/2 at time t
1/2
number N becomes N
0
/2
N / N
0
= = exp( –  t
1/2
)
In = –  t
1/2 t
1/2
= ln 2

=
0.693

In 2 = log e
2
Half-life is about 70% of time constant 1/

. Both indicate the decay time
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Language to describe oscillations
Sinusoidal oscillation
+ A
0
– A amplitude A sand falling from a swinging pendulum leaves a trace of its motion on a moving track
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t = 0
Phasor picture s = A sin
 t
A angle
 t time t periodic time T phase changes by 2
 f turns per 2

radian second per turn

= 2
 f radian per second
Periodic time T, frequency f, angular frequency

: f = 1/ T unit of frequency Hz

= 2
 f
Equation of sinusoidal oscillation: s = A sin 2
 ft s = A sin
 t
Phase difference

/2 s = A sin 2
 ft s = 0 when t = 0 s = A cos 2
 ft s = A when t = 0
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Motion of harmonic oscillator displacement against time velocity against time large displacement to right right zero velocity force against time mass m large force to left left small displacement to right small velocity to left mass m small force to left right left large velocity to left mass m zero net force right left small displacement to left right small velocity to left mass m small force to right left large displacement to left right zero velocity mass m large force to right left
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Force, acceleration, velocity and displacement
Phase differences Time traces displacement s varies with time like:
If this is how the displacement varies with time...
cos 2
 ft

/2 = 90
 velocity v
... the velocity is the rate of change of displacement...
–sin 2  ft

/2 = 90
 same thing zero acceleration = F / m force F = – ks
... the acceleration is the rate of change of velocity...
–cos 2  ft
...and the acceleration tracks the force exactly...
–cos 2  ft

= 180
 displacement s
... the force is exactly opposite to the displacement...
cos 2
 ft
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These two diagrams show the computational steps in solving the equation for a harmonic oscillator.
Dynamics of harmonic oscillator
How the graph starts zero initial velocity velocity would stay zero if no force force changes velocity new velocity
= initial velocity
+ change of velocity
How the graph continues force of springs accelerates mass towards centre, but less and less as the mass nears the centre change of velocity decreases as force decreases trace curves inwards here because of inwards change of velocity
0
 t time
0 trace straight here because no change of velocity no force at centre: no change of velocity time
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Constructing the graph
 t change in displacement = v
 t
 t if no force, same velocity and same change in displacement plus extra change in displacement from change of velocity due to force extra displacement = –( k / m ) s (
 t )
2 because of springs: force F = – ks acceleration = F / m acceleration = –( k / m ) s change of velocity
 v
= acceleration
  t
 v = –( k / m ) s
 t extra displacement
=
 v
 t
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The relationship between the force to extend a spring and the extension determines the energy stored.
Energy stored in a stretched spring
F
1 area below graph
= sum of force
 change in displacement extra area
F
1
 x total area
1
2
Fx unstretched
0
0 no force
 x extension force F work F
1
1
 x x larger force
Energy supplied
F = 0 x = 0
F = kx small change
 x energy supplied = F
 x x stretched to extension x by force F : energy supplied = Fx spring obeys
Hooke ’s law: F = kx energy stored in stretched spring =
2 kx
2
2
2
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The energy sloshes back and forth between being stored in a spring and carried by the motion of the mass.
Energy flow in an oscillator displacement potential energy
= ks
2
0 energy in stretched spring s = A sin 2
 ft time potential energy 0
1
PE = kA
2
sin
2
2
 ft time mass and spring v max oscillate
A v max
A v max kinetic energy
0 energy carried by moving mass
KE = mv
2
2 max
cos
2
2
 ft velocity kinetic energy
= mv
2 energy in stretched spring
0 v = v max
cos 2
 ft v max
= 2
 fA from spring to moving mass time time from spring to moving mass energy in moving mass from moving mass to spring from moving mass to spring
The energy stored in an oscillator goes back and forth between stretched spring and moving mass, between potential and kinetic energy
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Resonance occurs when driving frequency is equal to natural frequency. The amplitude at resonance, and just away from resonance, is affected by the damping.
Resonant response
Oscillator driven by oscillating driver Example: ions in oscillating electric field electric field
+
–
+
– ions in a crystal resonate and absorb energy low damping: large maximum response sharp resonance peak
10 more damping: smaller maximum response broader resonance peak
10
5 narrow range at peak
2 response
1
0
0 0.5
1 1.5
frequency/natural frequency
2.0
5 wider range at peak
2 response
1
0
0 0.5
1 1.5
frequency/natural frequency
2.0
Resonant response is a maximum when frequency of driver is equal to natural frequency of oscillator
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