Chapter 10 Flashcards
1
Q
Computational model
A
- Tackling questions scientists have e.g “when will a forest fire start?” you can roughly how they will decay
- For instance below a certain density fires never spread right through the forest; above that density they always do.
2
Q
Computational model example
A
-Will a material conduct? graphite grains in an insulting ceramic; at this density there is no conducting path across the material
3
Q
Rabbits + radioactivity
A
-A rabbit breeds a new rabbit, with a certain probability
-A rabbit dies with a certain probability
more rabbits = faster population growth
-This is an exponential change; when the rate of change of something is proportional to the amount of that something there is.
4
Q
Radioactive decay
A
-A genuinely random event
important to know how to deal with radioactive waste from power generators, industries and hospitals.
-It matters how long the material lasts + how active it is
5
Q
Faster decay =
A
more active, but less time it lasts
6
Q
Decay of nucleus
A
quantum event
7
Q
Number of decaying nuclei proportionality relationship (graph)
A
- This is proportional to the number left to decay
- This is why a graph of time against number of nuclei always falls
8
Q
Tracking substances (medically)
A
-By attaching a small amount of radioactive material into the body, it will decay and therefore we can see where the active material goes
9
Q
Activity of radioactive material
A
- Activity is linked to the probability of decay
- Number of decaying = pN
where p= probability
and N = nuclei
10
Q
Activity of radioactive material is measured in…
A
Becquerels (Bq)
Number of decays or counts per second
11
Q
Number of decays is on average proportional to…
A
-The interval Δt
12
Q
Equation for the probability of decay
A
p = λΔt
p= probability of decay Δt = in a time λ= decay constant
13
Q
Beta constant (λ)
A
-Probabililty of decay in a fixed time
14
Q
Equation for the activity of a radioactive substance
A
a = λN
-If you know the activity + the number of nuclei then you can calculate the decay constant
15
Q
Number of nuclei & activity relationship
A
-The number of nuclei present at any one moment decreases at a rate equal to the activity
16
Q
Differential of N with respect to t
A
dN/ dt = -λN (with λn = a)
17
Q
Relationship of probability of decay and t
A
-Probability of p decay in short time Δt is proportional to Δt
18
Q
Half life t1/2
A
-The time of radioactive material to be reduced by a factor of two.
19
Q
Greater activity =
A
Shorter half life
20
Q
Half life equation
A
t1/2 = ln2/ λ
21
Q
What does half-life tell you?
A
-It tells you how long the radioactive substance will last
22
Q
What does the decay constant tell you?
A
-How rapidly it decays
23
Q
Working out half lives
A
- In any one time, the number (N) is reduced by a constant factor
- In one half-life t1/2 the N is reduced by a factor of two
- So in L half lives, the number n is reduced by a factor 2^L
(e. g. in 3 half lives N is reduced by the factor 2^3 =8)
24
Q
Working out activity
A
-Measure the activity
-Activity is proportional to the number N left
-Find factor F by which activity has been reduced
-Calculate L so that 2^L = F
=> L = log(the base 2)F
age = t1/2L
25
Models to simplify problems
- A model is a set of assumptions that simplifies a problem
- Topics may be unconnected but all based on models that use the differential equations to describe the rate of change of something
- e.g. radioactive decay and capacitor charge can both be modelled in similar ways
26
Unstable atoms are radioactive
- If a atom has too many/not enough neutrons or too much energy in the nucleus it may be unstable
- The unstable atoms break down by releasing energy and/or particles until they reach a stable form
- Radioactive decay is a random process
27
Radioactive decay can be modelled by exponential decay
- By using an exponential decay you can predict decay
- A large enough sample of unstable atoms shows a behaviour pattern; you can predict how many atoms will decay in a given time
- Plotting a graph of number of atoms (nuclei) decaying each second against time shows an exponential decay curve
28
Activity of a sample is the...
The number of atoms that decay each second
- It is proportional to the size of the sample
- This is why activity-time graphs are exponential
- So the activity falls and the graph gets shallower and shallower
29
Longer half-life of an isotope =
Longer it stays radioactive
30
Finding the half-life from a graph
- Read off the value of count rate (decay rate) where t=0
- Go to half of the original value
- Draw a horizontal line from there to the curve
- Read off the half-life where the line crosses the x-axis
- Could repeat again by finding half of the second value and then adding your answers and dividing by two
31
When measuring the half-life of a source, remember...
-To subtract the background radiation from the activity readings to give the source activity
32
The number of radioactive atoms remaining, N, depends on the number originally present, N0, The number remaining calculation is...
N= N0e^(- λt)
33
Example: A sample of the radioactive isotope 13N contains 5 x 10^6 atoms. The decay constand is 1.16 x 10^-3 s^-1.
(a) What is the half life of this isotope?
(b) How many atoms of 13N will remain after 800 seconds
(a) T1/2 = ln2/ λ so => ln2/ (1.16 x 10^-3) = 598 s
| (b) N= N0e^(- λt) so => (5x 10^6) x e^(-(1.16x10^-3)(800) = 1.98x10^6 atoms.
34
Storing electric charge: capacitors
-In a camera electric charge is stored on a capacitor by connecting it to a large p.d. This is then discharged through the flash tube.
35
Example of a natural capacitor
Lightening: air currents carry charged iced crystals in storm clouds slowly building up large p.d between the top + bottom of the cloud. Then the air conducts and a lightening flash discharges the cloud.
36
What is a capacitor?
-Capacitor is a pair of electrical conductors close together. 'Charging' a capacitor means pulling those charges apart and getting a lot of positive charge on one conductor and negative on the other. Capacitors are often made of sheets of metal foil with an insulating layer between them.
37
Capacitors are used to store...
Electrical charge
38
How does a capacitor work?
-A battery will transport charge from one plate to the other until the voltage produced by the other charge build up is equal to the battery voltage.
39
What is a capacitor defined as?
-Capacitance is defined as the amount of charge stored per volt.
40
What is capacitance a measure of?
-It is a measure of how much charge a capacitor can hold; defined as the amount of charge per unit volt.
41
Capacitance in electric charge
-Conducting plates with opposite charges; p.d increases as the amount of charge stored increases.
42
Equation for capacitance
C=Q/V
(or Q=CV)
Where Q= charge (coulombs), C= capacitance (farads), V=p.d (volts)
43
What are the units for capacitance?
Farad F or CV^-1
44
Microfarad
μF (x10^-6)
45
Nanofarad
nF (x10^-9)
46
Picofarad
pF (x10^-12)
47
For some calculations you may also the equation for charge...
Q= IT
| or I = Q/T
48
Capacitors in defibrillators
-The circuit in a defibrillator can be programmed to vary how much charge is stored depending on the size of the patient. The charge is stored on a capacitor and then released in a short, controlled burst.
49
Capacitors in back-up power supplies
-Computers are often connected to back-up power supplies to make sure that you don't lose any data if there's a power cut. These often use large capacitors that store charge while the power is on then release that charge slowly if the power goes off. The capacitors are designed to discharged over a number of hours, maintaining a steady flow of charge.
50
Experiment: Investigating the charge stored on a capacitor
-Set up a circuit to measure current and p.d. (voltmeter over the capacitor, ammeter, variable resistor and cell)
-Constantly adjust the variable resistor to keep the charging current constant for as long as possible.
-Record the p.d regularly until it equals the battery p.d
-Data for current and time, use Q=IT to work out charge
-Then plot charge against p.d (volts)
Therefore Q/V= C (gradient)
51
Equation for time constant
T =RC
52
Larger RC means...
Slower charge decay
53
What is the capacitor discharge through an ohmic resistor
-The rate of flow of charge is proportional to the p.d driving the flow
54
What happens in a circuit which is connected to a capacitor and where the switch can flick between the cell and the bulb (or outlet source)
- When the switch is flicked to the left (cell), the charge builds up on the plates of the capacitor. Electrical energy provided by the battery is stored by the capacitor
- When flicked to the right (bulb) the energy stored on the plates will discharge through the bulb, converting electrical energy into light and heat.
55
How is work done in a capacitor circuit?
-Work is done removing charge from one plate and depositing charge onto the other one. The energy for this must come from the electrical energy of the battery, and is given by the charge x p.d.
56
Current is the rate of change of charge with respect to time
I = dQ/ dt
57
Flow of charge
-Flow of charge decreases charge
-Rate of change is proportional to charge
dQ/dt = -Q/RC
58
When will time of decay for large?
-Time for half of charge to decay is large if resistance is large and capacitance is large.
59
Graph of Q against t
-Charge decays exponentially if current is proportional to p.d and capacitance, C is constant
60
Graph of Volts (p.d) against Charge (Q)
This is a straight line
-The p.d across the capacitor is proportional to the charge stored on it, so the graph is a straight line through the origin.
61
How do you find the energy stored from a graph of Volts (p.d) against Charge (Q)?
-You can find the energy stored by the capacitor from the area under a graph of p.d against charge stored on the capacitor.
62
Equation for energy stored using p.d and charge
E = 1/2QV
63
Equation for energy stored using p.d and capacitance
E =1/2CV^2
64
Equation for energy stored using charge and capacitance
E= 1/2Q^2 /C
65
Energy delivered at p.d when a small charge
E = VQ
66
Finding energy Energy from graph of V against Q
Energy = are of triangle
| Area under graph
67
Charging a capacitor by connecting it to a battery
- When a capacitor is connected to a battery, a current flows until the capacitor is fully charged
- The electrons flow onto the plate connected to the negative terminal of the battery, negative charge builds up
- Negative charge build up repels electrons off the plate connected to the positive terminal
- These electrons are then attracted to the positive terminal
- An equal but opposite charge builds up on each plate, causing a p.d between the plates
68
Charging a capacitor by connecting it to a battery; current in the circuit
- Initially the current through the circuit is high, as the charge builds up on the plates, electrostatic repulsion makes it harder and harder for more electrons to be deposited
- When the p.d across the capacitor is equal to the p.d across the battery, the current falls to zero. The capacitor is fully charged.
69
To discharge a capacitor
- Take out the batter and reconnect the circuit
- Connect across a resistor and the p.d drives current through the circuit
- This current flows in the opposite direction from the charging current
- The capacitor is fully discharged when the p.d across the plates and the current in the circuit are both zero.
70
Charging or discharging a capacitor depends on two things
- The capacitance (C). This affects the amount of charge that can be transferred at a given voltage
- The resistance (R). This affects the current in the circuit
71
Discharge rate is proportional to what?
-Discharge rate is proportional to the charge remaining
72
Exponential change of Q
-The rate of change of a quantity Q is proportional to the amount of that quantity
dQ/ dt = KQ
Where K may be positive (growth) or negative (decay)
73
Growth and decay of Q
-The quantity Q grows or decays so that its amount charges by a constant factor in equal intervals of time. Adding to the time multiplies the quantity.
74
Graph of Q against t
-The amount of charge initially falls quickly, but the rate slows as the amount of charge decreases- the rate of discharged is proportional to the charge remaining
75
The graph of dQ/dt against Q
-This is the rate of discharge against the charge remaining which gives a straight line
76
Equation for the rate of discharge
dQ/dt = -Q/ RC
Where Q= charge remaining
R= resistance
C= capacitance
77
Charge on a capacitor decreases exponentially
- When a capacitor is discharging, the amount of charge left falls exponentially with time
- It always take the same length of time for the charge to halve, no matter how much charge you start with.
78
Equation for charge left on the plates of a capacitor discharging from full
Q= Q0 x e^(-t/ RC)
79
Graph of V against t: charging
```
Exponential graph ( upside down)
(Same graph for current-time)
```
V = V0 - V0 x e^(-t/ RC)
80
Graph of V against t: discharging
Exponential graph (Same graph for current-time)
V = V0 x e^(-t/ RC)
81
Time constant
T =RC units s
82
Time constant in Q= Q0 x e^(-t/ RC)
Becomes Q = Q0 x e^-1
so when t = T => Q/ Q0 = 1/e
83
What is the time constant?
-The time constant is the time taken for the charge on a discharging capacitor to reduce by a factor of 1/e
84
The larger the resistance in a series with the capacitor...
the longer is takes to discharge or charge.
85
Discharging capacitors
- Decay equation: Q= Q0 x e^(-t/ RC)
- The quantity that decays is Q, the amount of charge left on the plates of the capacitor
- Initially the charge of the plates is Q0
- It takes RC seconds for the amount of charge remaining to reduce by a factor of 1/e
- The time taken for the amount of charge left to decay by helf (half-life) is t1/2 = ln2 x RC
86
Radioactive isotopes
- Decay equation: N= N0 x e^(-λt)
- The quantity that decays is N, the number of unstable nuclei remaining
- Initially, the number of nuclei is N0
- It takes 1/ λ seconds fro the number of nuclei remaining to reduce by a factor of 1/e
- The time taken for the number of nuclei to decay by half (half-life) is t1/2 = ln2 / λ
87
Graph of number of undecayed nuclei of radioactive sample against time (N against t)
-Exponential curve
88
Graph of the natural log of the number of undecayed nuclei against time (ln(N) against t)
-Straight line
89
Y = mx + c form of N= N0 x e^(-λt)
N= N0 x e^(-λt)
```
ln(N) = -λt + ln(N0)
y = mx + c
```
90
Graph of the natural log of the number of undecayed nuclei against time (ln(N) against t): Gradient
-The gradient is the decay constant -λ
| From this you can calculate the half-life of the sample
91
Y = mx + c form of Q= Q0 x e^(-t/ RC)
```
ln(Q) = -(1/RC)t + ln(Q0)
y = mx + c
```
92
Working out half-life from a N against t graph
-Find when N0 then half this value and go across the graph to find where t is at this point
93
Equation for half life
t1/2 = ln2 / λ
94
Time constant of decay
t = 1/ λ
95
Simple harmonic motion
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.
96
Restoring force
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.
97
SHM (simple harmonic motion)
- The motion of oscillating systems is defined by simple harmonic motion
- A object moving in harmonic motion has a point of equilibrium (midpoint)
- The distance of the object from the midpoint is the displacement
- There is a restoring force pulling or pushing the object back to the midpoint
- The size of that force depends on the displacement and the force that makes the object accelerate towards the midpoint.
98
Restoring force exchanges PE and KE
- The type of PE depends on what it is that's providing the restoring force; gravitational PE for a pendulum or elastic PE for a mass on a spring
- As the object moves towards the midpoint, the restoring force does work on the object and so transfers PE to KE. When moving away from the midpoint all the KE is transferred back to PE again.
99
PE at equilibrium
-The objects PE is zero and its KE is maximum
100
PE at maximum displacement
-On both sides of the equilibrium the objects KE is zero and its PE is maximum
101
Mechanical energy
The sum of the potential and kinetic energy and stays constant (as long as there is no damping involved)
102
SHM graph of displacement
-Displacement, x, varies as a cosine or sine wave with a maximum value, A. It's a cosine wave when the stopwatch is started with the mass at maximum displacement.
103
SHM graph of velocity
-Velocity, v, is the gradient of the displacement-time graph (dx/dt). It has a maximum value of A(2πf) or Aω where f is the frequency and is a quarter of a cycle in front of displacement.
104
ω =
2πf
| also f = 1/t so = 2π/ t
105
SHM graph of acceleration
Acceleration, a, is the gradient of the velocity-time graph (d^2x/ dt^2). It has a maximum value of A(2πf)^2 or Aω^2, and is in antiphase with the displacement.
106
Cycle of oscillation
-Maximum positive displacement to maximum negavtive displacement
107
Frequency in SHM
-In SHM the frequency and period are independent of the amplitude, so a pendulum clock will keep ticking in regular time intervals even if its swing becomes very small.
108
SHM acceleration and displacement relationship
a ∝ -s
or 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
109
Starting from maximum displacement
x = Acos(2πft)
110
Starting from equilibrium
x = Asin(2πft)
111
Mass on a spring in SHM
-When a mass is pushed left or right of equilibrium, there is a force exerted on it
F= kx
where k= spring constant (stiffness) and x= displacement
112
Equation for the period of a mass oscillating on a spring
T = 2π√ (m/k)
113
Relationship between time and mass
T ∝ √m
So T^2 ∝ m
114
Relationship between time and the spring constant
T ∝ √1/k
So T^2 ∝ 1/k
115
Elastic potential energy from compressing or stretching a spring
-You can find how much energy is stored by plotting a force-extension graph; the area under the graph is the elastic potential energy
116
Equation to work out the elastic potential energy
E= 1/2kx^2
| triangle with base force kx and extension x
117
Simple pendulum is an example...
of SHO (Simple harmonic oscillator)
118
Equation for period of a pendulum
T = 2π√ (l/g)
where l= length of the pendulum in m
and g is gravity
119
Relationship between and SHO and displacement
-The force on an SHO is proportional to its displacement
120
Resistive forces gradually...
Take energy from the oscillator so that tis amplitude decreases until energy is fed back to compensate
121
What type of curve is the time trace for a oscillation
Sinusoidal
122
The swings of the pendulums are said to be
isochronous
123
What the graph looks like of displacement-time
Cosine graph
124
What the graph looks like of velocity-time
upside down sine graph
125
What the graph looks like of force-time
upside down cosine graph
126
The frequency of the oscillation can be given by
ω^2 = m/k
127
Equation for displacement of SHM
A
128
Equation for velocity of SHM
Aω
129
Equation for acceleration of SHM
Aω^2
130
Stretching a mass on a spring
- If you stretch and release a mass on a spring, it oscillates at its natural frequency
- If no energy's transferred to or form the surroundings, it will keep oscillating with the same amplitude
131
Total energy of a freely oscillating mass on a spring
E = 1/2mv^2 + 1/2kx^2
| KE + PE
132
External forces on a system
- A system can be forced to vibrate by a periodic external force
- The frequency of this force is called the driving force
133
Resonance
- When the driving force approaches the natural frequency , the system gains more energy from the driving force and so vibrates with a rapidly increasing amplitude
- This is resonance
134
Examples of resonance
- Organ pipe; the column of air resonates driven by the motion of air at the base
- Swing; it resonates if it's driven by someone pushing it at it's natural frequency
- Glass smashing; A glass resonates when driven by a sound wave at the right frequency
135
Damping
- An oscillating system will lose energy to its surroundings, this is usually down to frictional forces like air resistance
- These are called camping forces
136
Why are systems often deliberately damped?
Systems are often deliberately damped to stop them oscillating or to minimise the effect of resonance
137
Shock absorbers
-Shock absorbers in a car suspension provide a damping force by squashing oil through a hole when compressed
138
Critical damping
-This reduces the amplitude in the shortest possible time
| graph decreases very quickly
139
Why are car suspension systems damped ?
- So that they don't oscillate but return to equilibrium as quickly as possible
- absorbs energy so the oscillation will become smaller.
140
Energy stored in a spring- equations and graph
Graph: area under graph
| Total area = 1/2Fx
141
Energy stored in spring equation
E= 1/2kx^2
| E= 1/2Fx
142
What is the equation for the force needed to extend a spring?
F=kx
| force is proportional to x
143
Energy in an oscillating system
-Energy stored in a mechanical oscillator is continually shifting back and forth from KE to PE
144
In a harmonic oscillator the stretch of the spring is the...
Displacement (s)
| so E = 1/2ks^2
145
What are the relationships between KE and PE in oscillating systems
- KE reaches its max as PE reaches min.
- Exact opposite phases
- Sum of the two is exactly constant at all times
146
Energy stored in an oscillator goes between...
Stretched spring and moving mass
| between potential and kinetic energy
147
How can damping be measured?
-Damping can be measured by the number of oscillations until the vibration has died away after a sharp impulse (the Q-factor)
148
Less damping=
-Narrower + sharper resonance response
149
More damping=
-Greater range of frequencies to which the resonator responds
150
Greater damping=
-Smaller maximum repsonse
151
How can you avoid the vibration of a car?
-Can use wadding packed behind it to absorb energy
152
Mechanical energy =
PE + KE
| = 1/2kx^2 + 1/2mv^2
153
What happens to the curve if you increase damping? (graph)
-The width of resonant response curve increase as damping increases
154
(Graph) Low damping
- Large max response
| - Sharp resonance peak
155
(Graph) More damping
- Smaller max response
| - Broader resonance peak
156
When is resonance response at a maximum?
-Resonance is a maximum when frequency of driver is equal to natural frequency of oscillator
157
Max extension + 0 velocity =
Energy stored in spring
158
0 velocity + max velocity =
Energy carried by moving mass
