Physics Flashcards
(140 cards)
1
Q
What is the notation for vectors and scalars?
A
Vector v̲
Scalar v
2
Q
What should we consider all atoms and molecules as?
A
Point particles
3
Q
What is Newtons first law?
A
A body remains at rest or at constant velocity when no net external force acts upon it
a̲ = 0 ⇔ ∑F̲ = 0
where F̲ is the vector sum of all external forces acting on a body
4
Q
What is Newtons second law?
A
The rate of change of momentum of a body is proportional to the force that acts on the body
dp̲/dt = ∑F̲
If mass is constant then:
∑F̲ = ma̲
5
Q
What is Newtons third law?
A
For two interacting bodies, if object a exerts a force on object b there is an equal and opposite force on a from b
F̲₁₂ = F̲₂₁
6
Q
What do Newtons laws lead directly to?
A
The conversation of linear momentum
7
Q
How do you calculate force from momentum?
A
Force is change in momentum over time
8
Q
What is the conservation of linear momentum?
A
If no external force acts on the particles in a system then the sum of all the particle momenta will remain constant
P̲₁ + P̲₂ + … = const
Get by combining N2L and N3L
9
Q
What is the conservation of energy?
A
For any isolated system, where energy cannot enter or leave the total energy within the system must remain constant
Energy cannot be created or destroyed only transferred
10
Q
How do you calculate kinetic energy, K?
A
K = 1/2mv²
11
Q
How do you calculate force from potential?
A
F(x) = -dU(x)/dx
12
Q
How can we only define potential energy and what about potential energy is important?
A
Can only define a potential energy associated with a force that only depends on position
We are only interested in the change of potential energy
13
Q
Why are we only interested in change in potential energy?
A
It is reflected in the change of the particles kinetic energy
14
Q
When can we say a force has done work on a particle?
A
If a force moves a particle
15
Q
What is the total mechanical energy, E?
A
E = K + U = const
K1 + U1 = K2 + U2
Can derive this conservation law from N2L and definition of work
16
Q
On potential curves what are places where F = 0 known as?
A
Equilibrium points
17
Q
What is Leonard Jones potential?
A
Used to describe interaction between atoms or molecules
18
Q
What is true for a solid or a gas relating to the separation between molecules?
A
For a solid: Mean separation is small
- Translation K must be small
For a gas: Separation is large
- U must be small
19
Q
What is the physical meaning of the two parts of Leonard Jones?
A
r⁻¹² term = repulsion (dominates at very small distance)
r⁻⁶ term = attraction (represents vdW forces which dominate at intermediate distances)
20
Q
When considering collisions between particles, what scale are we working with and why?
A
Tend to work on a macro scale where we only care about momentum of each particle before and after they have interacted
21
Q
What are the types of collisions on a macroscale?
A
Elastic (ΔK = 0)
Inelastic (ΔK < 0)
Superelastic (ΔK > 0)
All of which total linear momentum is conserved
22
Q
What is true about velocity in an elastic collision?
A
Δu = -Δv
23
Q
What is the equation for total linear momentum conservation in a head on collision?
A
m1u1 + m2u2 = m1v1 + m2v2
24
Q
What is an elastic collision?
A
If a collision between particles conserves total mechanical energy
Can understand this collision on a microscope by considering any of the inter-particle potential curves we’ve already seen
25
How do you explain inelastic collisions?
Kinetic energy can be lost to or taken from:
- molecular vibrations
- molecular rotations
Inelastic collisions can lead to dissociation
26
When will a collision look superelastic?
If some internal energy is released
27
What is the position/velocity of the CoM?
Weighted average of all particles positions/velocities
28
Why is the idea of CoM velocity helpful?
It can help understand molecular systems
The molecule as a whole will move at the CoM velocity but it may also be vibrating or rotating
29
What are the two important frames of reference?
The lab frame (coordinate system that is fixed with respect to us)
The centre of mass frame
30
What is there to know about the centre of mass frame?
It has its origin at the centre of mass position of a system
Moves with respect to the lab frame at the centre of mass velocity
The centre of mass doesn't move - we see relative positions of particles
31
How do we define particle positions and velocities in the centre of mass frame?
Denote them with an asterix
32
What is true for all inertial reference frames?
Laws of physics are the same
Hence K = 1/2mv^2 holds for all inertial reference frame
However measured values of physical quantities (K or V) are frame dependent
33
What do elastic collisions look like in the CM frame?
They collide at the origin of the frame and are symmetrical
34
What is true in a totally inelastic collision in the CM frame?
There is no movement after the collision as CoM is no longer moving
35
What is true about the total momentum in the CM frame?
Total momentum is zero in the CoM frame
36
What is true about two body problems?
They will rotate/vibrate etc about the centre of mass
It is no longer in an inertial frame
Change in direction = Change in velocity = Acceleration
37
What is an inertial vs non inertial frame of reference?
Accelerating frames of reference are non inertial frames of reference
- Need to take that into account when describing equation of motion
38
How many equations of motion are there to solve for a system of N bodies?
A system of N bodies involves solving N equations of motion
39
What can we do for a two body problem?
Reduce the problem to an equivalent one body problem
- where reduced mass orbits/vibrates about a fixed total mass at a distance r
40
What equation of motion describes the separation between two objects?
F̲ = μr̲
41
What are polar coordinates?
A coordinate system based on distance and angle from the origin
42
What does the vector product produce?
A third vector that is perpendicular to the other two vectors
For the direction of a̲xb̲ we use the right hand rule
- Curl fingers from the first vector to the second vector and see where thumb points
43
What is true if angular velocity and angular acceleration are in the same direction?
The rotation is speeding up
If point in opposite directions then the rotation is slowing down
44
What is true for circular particle motion at constant speed?
Angular velocity is also constant, even though velocity is not, meaning angular acceleration must be zero
45
How to work out the direction of angular velocity?
Using the right hand rule
46
How do you convert between velocity and angular velocity?
vᵢ = rᵢω
47
What is the rotational analogue of mass?
Moment of inertia
48
What does the moment of inertia depend on?
Distribution of mass
Axis of rotation
49
What is the moment of inertia for a diatomic?
I = μr²
50
What is true about centripetal force?
It acts on a particle to keep it rotating and is directed towards the axis of rotation
51
What is true about centrifugal force?
It is a 'fictitious' force that we need to introduce when we work in a rotating frame (non inertial)
Acts away from axis of rotation
52
What are the SI units for newtons?
1 N = 1 kgms-2
53
How do we define the effective potential?
Ueff(r) = U(r) + Krot(r)
54
Why do we need effective potential in rotational mechanics?
To account for the additional source of energy from rotational kinetic energy
55
What is a stable point of effective potential?
At a large value of r (capturing effect of centrifugal distortion)
56
What is the centrifugal distortion of a diatomic?
If molecule starts rotating at a constant ω, at moment it begins to rotate there is no centripetal force
As no centripetal force, atoms start to move in a straight line -> r increases
r increases until new value of r gives a force -dU/dr that is the centripetal force required to sustain circular motion
Result is a new equilibrium with larger interatomic separation
57
What are the rotational analogues of force and momentum?
Torque and angular momentum
58
How do you calculate torque?
τ̲ = r̲ x F̲
torque is vector quantity
- Measure of the ability of a force to cause rotation
59
Why is the cross product needed to calculate torque and angular momentum?
Only components of F and p perpendicular to r are relevant
60
How do you calculate angular momentum?
L̲ = r̲ x p̲
61
What is true about the component of the force applied to a particle that is parallel to the r̲ direction?
It will not lead to any torque
62
What is true about the central force for rotation?
We need a non zero F for rotation
If central force is zero then torque is also zero
63
What is a central force?
A force that acts along the line between the object and the origin
64
How do you calculate angular momentum from moment of inertia?
L = Iω
65
How can we relate angular momentum to rotational kinetic energy?
Krot = L²/2I
66
What is conservation of angular momentum?
If there is no net torque on the system then its angular momentum must remain constant
67
What is Hooke's law?
F = -kx
where F is restoring force, x is distance from point of equilibrium and k is spring constant
68
What is the simplest type of vibration and when does it occur?
Simple harmonic motion, SHM
Occurs whenever a body is displaced from equilibrium and is subject to a linear restoring force
69
Can you have positive and negative values of x?
Yes backwards or forwards from equilibrium
70
What will the motion of a particle attached to a spring be?
Oscillation about the equilibrium position
71
What does the phase constant in SHM set?
The initial condition of the system
72
What is ω₀?
Natural angular frequency of the oscillation (in rads-1)
73
What is the period of oscillations?
T = 2π/ω₀
74
What is the total vibrational energy of SHM?
Evib = 1/2kA² where A is amplitude and k is spring constant
75
What is the equation for potential in SHM?
U = 1/2kx²
where x is displacement about the equilibrium position
When deriving can set c to zero as interested in change in potential
76
What do we need to do to displacement if we want to work with respect to an origin that is not at the equilibrium position?
Need to transform x to (x-x₀) where x = x₀ at equilibrium
77
What is the realistic potential experienced by one atom/molecule when in the presence of another approximated by?
The Lennard-Jones potential
Has stable equilibrium where x = x₀
78
What can we say if there is a small perturbation from equilibrium in the Lennard-Jones potential?
If the particle stay nears equilibrium then the potential curve is approximately parabolic
We can therefore say that if the perturbation is sufficiently small then the subsequent oscillation is well approximated by SHM
79
What happens if there is a large perturbation from equilibrium in Lennard-Jones potential?
There are two possible scenarios
1. Particle doesn't have enough energy to escape potential well (E < 0) and so will oscillate in an asymmetric fashion
2. Particle will escape to x = ∞ (E > 0)
80
How can we be more rigorous in characterising oscillations about equilibrium in a realistic potential well?
We can use a Taylor series expansion
81
What can you use the Taylor series for in oscillations?
To see if potential curve seen is parabolic for small perturbations from equilibrium
82
What is the Taylor series?
An approximation: Takes any function and approximates it as an infinite sum of terms. (polynomial of order n)
83
What are some ways you could assess if the perturbation away from equilibrium is small enough for the parabolic approximation to hold?
1. Expand to n=10 or higher and see if oyur result is very different from going to n=2
2. Compare magnitude of n=2 and n=3 terms in Taylor series
3. Take potential and calculate U(x₀ - δx) and U(x₀ + δx), then compare their values
84
What is the type of harmonic motion that includes damping as a result of energy loss?
Damped harmonic motion (DHM)
85
How can you introduce a simple form of damping?
What is the extra thing in DHM?
By placing our system in a viscous liquid
There is now a force acting on the oscillator backwards
86
What does the drag force always act to do?
Oppose motion
87
What is the angular frequency in DHM relative to the natural angular frequency of SHM?
It is less in DHM than in SHM
Means damping increases the period of the oscillation
88
What is the quality factor?
Describes how 'good' the oscillator is
Can be used to categorise the DHM oscillator
Q = ω₀/ɣ
89
What are the three types of DHM and how are they defined by the quality factor?
Underdamped: Q > 1/2
Critically damped: Q = 1/2
Overdamped: Q < 1/2
90
What is the damping term?
q = ɣ/2
91
What are the values of ω (frequency of oscillations for DHM) for each different level of damping?
Underdamping: ω is real
Critical damping: ω is zero (no oscillation - will come to rest first)
Overdamping: ω is imaginary (no physical oscillation
92
How do the periods of oscillation vary with damping?
Periods of oscillations increase with increased damping
ω = 2π/T
93
What is the difference between critical damping and overdamping? (graphs)
Overdamped is quicker to reduce its displacement to start with
However critical damping will come to rest first
94
What happens to harmonic motion if we introduce a driving force?
We get forced harmonic motion (FHM) and things are more complicated
95
What are the two parts to the solution to FHM?
Transient oscillations and steady state oscillations
96
What is the difference between transient and steady state in FHM?
In early stage of oscillation, energy in per cycle is greater than energy out per cycle - amplitude increases over time
Say that the system is transient
Eventually amplitude will have grown to a point where energy balances out - amplitude now constant with time
Say the system is in steady state
97
What is true about energy in the oscillator at steady state?
Amplitude is constant so total energy is also constant
Meaning the power dissipated through damping is equal to power absorbed by the driver
98
What happens to Ass when ωdr << ω₀?
Ass ≈ F₀/k
99
What can we do if ωdr is close to ω₀?
We can maximise the amplitude
- Then we can use resonance to increase amount of energy in system
100
When does resonance happen?
When the frequency of the driving force matches the systems natural frequency
Leading to constructive interference of the oscillations
101
What is a wave?
A disturbance that propagates through a medium
They transport energy but not matter
102
How are transverse and longitudinal waves distinguished?
By the direction the medium is preturbed
103
What is the difference between a transverse wave and a longitudinal wave?
Transverse: Vibration at right angles to wave direction e.g. light
Longitudinal: Vibration in same direction as wave e.g. sound
104
What do all waves satisfy?
The wave equation
105
What happens in combined transverse and longitudinal waves?
Particles move in circular motion even when wave direction is left to right e.g.
Example of these waves are water waves
106
What is the distance over which a travelling wave repeats called?
The wavelength, λ
107
What can ω and k be thought of in the travelling wave solution?
Frequencies
ω is how many waveforms pass in a unit of time (ω = 2π/T)
If we freeze time k reflects how many waveforms over a unit of space (k = 2π/λ)
108
How do you calculate vp (phase speed)?
vp = ω/k
109
What are Rayleigh waves?
Waves that are a combination of transverse and longitudinal that propagate along the surface of a solid
- Can be used to determine physical properties of crystals and minerals
110
What is the wave superposition principle?
If two solutions of the wave equation are added together, the result is also a solution
111
How is cavity ring-down spectroscopy performed?
Performed by creating a laser pulse and measuring how strongly it is absorbed by a gas sample
112
What is the superposition of two waves at a point in time and space also called?
Interference
113
What is the difference between constructive and destructive interference?
Destructive interference occurs when the phase difference between the waves is an odd multiple of π (out of phase)
Constructive interference occurs when the difference in an even integer multiple of π (in phase)
114
What happens if the difference in phases of two waves lies somewhere between constructive and destructive interference?
The magnitude of displacement caused by the summed waves will lie somewhere between max and min values
115
What are points that don't move called in waves?
Fixed nodes
116
What is the temporal coherence of a wave?
How accurately we can predict the phase of the wave and thus the disturbance in the medium
At the same point in space at some later time, t + Δt
117
What is the coherence time?
Tc: the duration over which we are able to predict the future phase when we know the phase at some point in time
118
What is a standing wave?
A wave that appears to be 'standing still'
Formed by the superposition of two waves moving in opposite directions
119
What is true about all standing waves?
They have a much higher amplitude than either of the others (2A vs A)
120
What is spatial coherence?
Refers to how well we can predict the phase of the wave at a position y + Δy if we know the phase at position y
121
Analogous to temporal coherence what can we define for spatial coherence?
The spatial coherence length Lc
122
What is true at a closed boundary?
What about an open boundary?
u(xb,t) = 0
∂u(x,t)/∂x = 0 where x = xb
123
What is the difference between an open boundary and a closed boundary?
Closed boundary - Displacement is zero at the end
Open boundary - Displacement is not zero (derivative of displacement is zero)
124
What are the three types of boundary conditions?
Closed-closed (displacement at both ends is zero)
- e.g. a string tied at both ends
Open-open (derivative of displacement is zero at both ends)
- e.g. vibration of a free pipe
Closed-open (derivative at one end is zero and other end has zero displacement)
125
What is true about closed-closed boundary conditions?
You need a whole number of half wavelengths
126
What is the difference between temporal coherence and spatial coherence with respect to x and y axis?
Temporal coherence - when looking along x axis at certain point on y you can predict the wave (equally spaced)
Spatial coherence - when looking along y axis at certain point on x you can predict wave (not straight line)
127
How do you create coherence?
To create spatial coherence we can pass the light through a slit that isolate part of the 2D waveform
To create temporal coherence we can pass the light through a filter (e.g. coloured glass) to isolate a narrow frequency range
128
With twin source interference how can we tell if there will be constructive or destructive interference?
Δx = dsinθ
This is the optical path length difference
If dsinθ/λ = n then maxima condition and constructive
If dsinθ/λ = (2n+1)/2 then minima condition and constructive
129
What is Bragg's law for constructive interference of crystal structures?
2dsinθ = nλ
Looking at difference in path length
Integer number of wavelengths or difference = coherence
130
What do you need to create standing waves?
Waves travelling in opposite directions that product standing waves (have same angular frequency and wave number)
131
What do you get when two waves with slightly different angular frequencies and wave numbers interfere?
We get a higher frequency wave bounded by a lower frequency 'beat' wave
132
How do you explain interference between waves with slightly different properties?
Consider a position where the peaks of the two waves line up (constructive interference)
As we move away from this the waves move out of phase until they reach a point where they destructively interfere
As we both further they move back in phase and we get another peak
Interference produces a modulation at the beat frequency
133
Where does a phenomenon known as 'dispersion' come from?
Fundamentally because each underlying waveform has a different speed
Occurs when you have multiple waves making up a wave form that all have different angular frequencies and wave numbers
134
What is the relationship between ω and k called?
Dispersion relation (characteristic of any wave)
135
What is the other important speed we can attribute to a wave?
Its group velocity
vg = dω/dk
136
What is true if vg = vp and if vg ≠ vp?
If vp = vg then the pulse retains its shape as it moves and the medium is 'non dispersive'
If not then the underlying waves travel at different phase speeds and pulse will therefore spread out over time and then medium is 'dispersive'
137
What does the c and b mean in beat waves?
c comes from carrier wave properties
b comes from beat wave properties
138
How fast does a beat wave propagate?
At the group velocity
139
Does a pulse stay the same over time?
Dependent on if it is in a dispersive medium or not
140
What are the two different types of dispersion?
Normal dispersion, vg < vp
e.g. light in glass
Anomalous dispersion, vg > vp
e.g. ripples on the surface of a pond
