Chapter 10 - creating models Flashcards
What is a model?
A set of assumptions which simplifies and idealises a problem
Why are models useful?
They allow you to write equations and make calculations and predictions for a particular situation, without having too many factors to consider.
One model can often be extended or changed so that it can be applied to a different process.
What makes an atom unstable/radioactive?
Too many neutrons, too many protons or too much energy in the nucleus
What is radioactive decay?
When atoms release energy and/or particles until they reach a stable form
What is meant by radioactive decay being a random process?
You can’t tell which atom will decay next or when a given atom will decay, you can only work with averages
The decay is unaffected by any external condition such as temperature
What can radioactivity be modelled by?
Exponential decay
When can a random process be predicted using a model?
With a large enough sample (ie, number of atoms) the overall behaviour shows a pattern, so can be predicted with a model
What can an exponential decay model for radioactive decay let you predict?
The number of atoms which will decay in a given period of time (eg one second)
NOT when an individual atom will decay
What is activity?
The number of unstable atoms which decay per second
What is the activity of a sample proportional to?
The size of the sample
What makes something exponential decay?
The rate of change of a quantity is proportional to the size of that quantity
What is the decay constant?
The probability of a given nucleus decaying in any given second
What are the units of the decay constant?
s^-1
What is the symbol for the decay constant?
λ
What is activity?
The number of unstable atoms which will decay in a second
What is the symbol for activity?
A
What is the equation for activity?
A = λN Activity = decay constant x number of radioactive nuclei remaning in the sample
In radioactive decay, what is N?
The number of radioactive nuclei remaining in the sample
What are the units of activity?
Becquerels, Bq
What does one Becquerel mean?
One decay per second
What are the equivalent SI units for a Becquerel?
s^-1
What is the differential equation for radioactive decay?
dN/dT = - λN
What does dN/dT represent?
The rate of change in number of unstable nuclei remaining
What are differential equations used for?
Describing the rate of change of a quantity
What is the symbol for half life?
T1/2
What is the half life of an isotope?
The average time taken for the number of undecayed atoms in a sample to halve
How is half life measured?
Measuring the time taken for the activity of a sample to halve
What do you have to remember to do when measuring the activity of a radioactive source?
Subtract background radiation from readings
What is the equation for the half life of an isotope?
T1/2 = ln2/λ
What is the equation for the number of unstable nuclei remaining at a given time?
N = N0^-λt
What is the exact solution of the differential equation dN/dT = -λN
N = N0^-λt
What is the definition of capacitance?
The charge stored per volt
What are capacitors?
Things which can store electrical charge
What is the symbol for capacitance?
C
What is the equation for capacitance?
C = Q/V
What are the units of capacitance?
Farads, F
What are the equivalent SI units of a farad?
CV^-1
What is the equation which links current, charge and time?
I = ΔQ/Δt
What are two uses for capacitors?
flash photography and defibrillators
How can the charge on a capacitor be investigated experimentally?
connect a capacitor to a battery, voltmeter, ammeter and variable resistor
Constantly adjust the variable resistor to try and keep the current constant (which will not be possible when it is nearly fully charged)
Record the pd across the voltmeter at regular intervals until it equals the battery pd
Charge stored = current * time
capacitance = Q/V (gradient of Q vs V graph)
How could capacitance be found graphically?
The gradient of a Q vs V graph
What do capacitors store as well as charge?
electrical energy (initially provided by the battery)
What happens when a capacitor discharges?
It is disconnected from the battery and the circuit is closed
Work is done (by converting electrical energy stored by the capacitor) moving charge from one plate to the other until the charge is equal on both plates
there is zero pd across the capacitor and zero current in the circuit
What are the two equations for energy stored on a capacitor?
E = 1/2 QV E = 1/2 CV^2
How is a capacitor charged?
It is connected to a battery
The electrons flow onto the plate connected to the battery’s negative terminal, so a negative charge builds up
The build up of negative charge repels electrons from the plate connected to the positive terminal
An equal but opposite charge builds up on each plate, giving a potential difference across the capacitor
Why does no charge flow between the plates of a capacitor?
They are separated by an insulator
What two factors does time taken for a capacitor to charge or discharge depend on?
- The CAPACITANCE of the capacitor (the amount of charge which can be transferred at a given voltage)
- The CIRCUIT RESISTANCE of the circuit (this affects the current, so how quickly the charge can be transferred)
Why is capacitor charging an exponential growth and capacitor discharge an exponential decay?
The rate of change of charge is proportional to the charge, giving an exponential relationship
What is the differential equation for capacitor discharge?
dQ/dt = -Q/RC (Q = charge remaining)
What is the equation for charge remaining on a capacitor at a given time?
Q = Q0e^-t/RC
What is the equation for pd across a capacitor at a given time (when charging)?
V = V0 - V0e^-t/RC
What is the equation for pd across a capacitor at a given time (when discharging)?
V = V0e^-t/RC
What is the symbol for the time constant?
Τ (tau)
What is the equation for the time constant?
Τ = RC
What is the time constant?
The time taken for the charge on a discharging capacitor to fall to 37% of the original value
(or for to rise to 63% of the Q0 in a charging capacitor)
If the resistance in series with a capacitor is larger, what will happen?
The capacitor will take longer to charge or discharge
How many seconds does it take for the number of unstable nuclei remaining to fall to 37% of their original value?
1/λ seconds
How would you plot a logarithmic graph to find the decay constant?
Plot the natural log of the number of undecayed nuclei (N) against time
The y-intercept is the natural logarithm of initial number of undecayed nuclei (N0)
The gradient is -decay constant (λ)
(the same method works for activity of a sample)
What is the equation used for a logarithmic graph used to find the decay constant?
lnN = -λt + lnN0
If the natural log of Q (for a capacitor) was plotted against time, what would the gradient of the straight line graph be?
-1/RC
RC = time constant
What does SHM stand for?
simple harmonic motion
What is the definition of simple harmonic motion?
An oscillation in which the acceleration of an object is directly proportional to its displacement from the midpoint, and is directed towards the midpoint
What is a restoring force?
The force which pushes or pulls an object moving with simple harmonic motion back to the midpoint
The force is directly proportional to object displacement
What is the displacement in simple harmonic motion?
The distance of an oscillating object from the midpoint of oscillation
What does the work done on an oscillating object do to the energy of the object?
Causes potential energy to be transferred to kinetic energy
What is the mechanical energy?
The total energy of a SHM system
This stays constant (if there is no damping) and is the sum of kinetic and potential energy
At the midpoint of an oscillation, what is the potential energy and what is the kinetic energy?
The potential energy is zero and the kinetic energy is at its maximum
At the amplitude (maximum displacement) of an oscillation, what is the potential energy and what is the kinetic energy?
The kinetic energy is zero and the potential energy is at its maximum
For simple harmonic motion, which two properties do not depend on the amplitude?
frequency and period
What is the differential equation linking acceleration and displacement in SHM?
d^2x/d^2t = -ω^2 x
a = -ω^2 x
What is the symbol for displacement?
x
What is the equation for displacement with regards to time for an object released at maximum displacement?
x = Acosωt
What is the equation for displacement with regards to time for an object released at equilibrium?
x = Asinωt
What is the name for the midpoint of an oscillation in SHM?
equilibrium
What is ω?
The angular speed
What is the equation for angular speed?
ωt = 2πf
How far out of phase is a velocity time graph with a displacement time graph for SHM?
The velocity time graph is 90 degrees ahead of the displacement time graph
How far out of phase is an acceleration time graph with a displacement time graph for SHM?
They are in antiphase
180 degrees out of phase, maximums correspond to minimums
What are the equations for displacement, velocity and acceleration when an object is released from amplitude (SHM) ?
x = Acosωt v = -Aωsinωt a = -Aω^2cosωt
What are the equations for displacement, velocity and acceleration when an object is released from equilibrium (SHM) ?
s = Asinωt v = -Aωcosωt a = -Aω^2sinωt
What is the maximum displacement in SHM?
A
amplitude
What is the maximum velocity in SHM?
v = ωA
What is the maximum acceleration in SHM?
a = ω^2A
What are two examples of a simple harmonic oscillator (SHO)?
A mass on a spring or a simple pendulum
What is the equation for a force exerted on a simple harmonic oscillator when it is moved away from equilibrium?
F = kx force = spring constant * displacement (distance from equilibrium)
What is the equation for the period of a mass on a spring?
T = 2π√m/k
What are the units of the spring constant?
Nm^-1
How can the energy(elastic potential) stored by extending or compressing spring be found graphically?
The area under a force vs extension graph
What is the equation for energy stored in an extended or compressed spring?
E = 1/2kx^2
What is the equation for the period of an oscillating pendulum?
T = 2π√l/g
What is the differential equation used for estimating the displacement of an SHO after a given time period, t?
d^2x/dt^2 = -kx/m
What are free vibrations?
No transfer of energy to or from surroundings
(so the total energy of the system remains constant)
used for a spring oscillating in air
At what frequency will a mass on a spring oscillate if stretched and released?
Its natural frequency
What is the formula for the total energy of a freely oscillating mass on a spring?
E = 1/2mv^2 + 1/2Kx^2 E = KE + PE
When does resonance happen?
When the driving frequency matches the natural frequency
When do forced vibrations happen?
When there is an external, periodic driving force
What is the frequency of the driving force called?
The driving frequency
How does resonance happen?
When the driving frequency approaches the natural frequency, the system gains more and more energy from the driving force, so oscillates with a rapidly increasing amplitude, causing it to resonate
Give three examples of resonance
- The column of air in an organ pipe
- Pushing a swing
- Smashing a glass with sound waves
- A radio’s electric circuit tuned to resonate at the same frequency as the radio station
What is damping?
When energy from an oscillating system is lost to the surroundings
Why would a system be deliberately damped?
To minimise the effects of resonance
What are damping forces?
Forces, usually frictional forces, which cause an oscillating system to lose energy to its surroundings
How is the amplitude of oscillation affected by damping?
The amplitude is reduced over time, until it reaches zero
How is the change in amplitude affected when damping is heavier?
The amplitude will reach zero more quickly
What is critical damping?
Damping which will reduce the amplitude to zero in the shortest possible time (no oscillation)
Name something which is deliberately overdamped
A car suspension system
What is it called when a system is more than critically damped?
Overdamping
What happens to a system which is overdamped?
The system takes longer to return to equilibrium than a critically damped system
What can reduce a system’s amplitude in the same way as damping?
Plastic deformation
Energy is absorbed as the shape changes, reducing the amplitude
How does a lightly damped system behave when resonating?
The resonance peak is very sharp
The amplitude only increases dramatically when the driving frequency is very close to the natural frequency
How does a heavily damped system behave when resonating?
The response is much flatter
The amplitude doesn’t increase much even near to the natural frequency, it isn’t as sensitive to driving frequency
What is a resonance peak?
The peak in a graph of amplitude against driving frequency which happens when the system resonates and the amplitude increases suddenly
