$$speed = \frac{distance}{time}$$
$$force = mass x acceleration$$
$$momentum = mass x velocity$$
$$density = \frac{mass}{volume}$$
$$\text{Specific charge of a particle} = \frac{\text{charge (C)}}{\text{mass (kg)}} = \frac{Q}{m}$$
$$f = \frac{c}{\lambda} \text{frequency (Hz)} = \frac{\text{speed of light (}3.00 \times 10^8 \text{ms}^{-1}\text{)}}{\text{wavelength (m)}}$$
$$E \text{(energy of photon)} = hf \text{(Planck's constant x frequency)} = \frac{hc \text{(Planck's constant x speed of light)}}{\lambda \text{(wavelength)}}$$
$$c \text{(wave speed)} = f \lambda \text{(frequency x wavelength)}$$
$$\text{velocity} = \frac{\text{change in displacement}}{\text{time taken}} = v = \frac{\Delta s}{\Delta t}$$
$$\text{acceleration} = \frac{\text{change in velocity}}{\text{time taken}} = a = \frac{\Delta v}{\Delta t}$$
$$E_k = \frac{1}{2}mv^2$$
$$E_p = mg \Delta h$$
$$E = \frac{1}{2}k (\Delta L)^2 \text{elastic potential} = \frac{1}{2} \text{ x stiffness constant x (extension of material)}^2$$
$$E_k = hf - \phi \text{Kinetic Energy = (Planck's Constant x Frequency) - Work Function}$$
$$f_o = \frac{\phi}{h} \text{Threshold frequency} = \frac{\text{work function}}{\text{Planck's Constant}}$$
$$V_s = \frac{E_k}{e}$$
$$\lambda = \frac{h}{p}$$
$$f = \frac{1}{\text{T}}$$
$$f = \frac{1}{2l} \sqrt{\frac{T}{\mu}} \text{ where } l = \text{length of vibrating string (m), } T = \text{tension on string (N) and } \mu = \text{mass per unit length of string (kg m} ^{-1}\text{)}$$
$$w = \frac{\lambda D}{s}$$
$$d \sin \theta = n \lambda$$
$$\text{Number of slits per metre} = N = \frac{1}{d}$$
$$n = \frac{c}{c_s} \text{Absolute refractive index} = \frac{\text{speed of light in a vacuum} (\text{ms}^{-1})}{\text{speed of light in material} (\text{ms}^{-1})}$$
$$_1n_2 = \frac{c_1}{c_2} = \frac{n_2}{n_1} \text{Relative refractive index} = \frac{\text{speed of light in material 1} (\text{ms}^{-1})}{\text{speed of light in material 2} (\text{ms}^{-1})} = \frac{\text{absolute refractive index of material 2}}{\text{absolute refractive index of material 1}}$$
$$n_1 \sin i = n_2 \sin r \text{ where } n_x \text{ is the refractive index of material } x$$
$$\sin \theta _c = \frac{n_1}{n_2} \text{ where } n_1 \gt n_2$$
$$W = mg$$
$$M = F x d = \text{Force (N) x Perpendicular distance from the pivot (m)}$$
$$s = ut + \frac 1 2 at^2$$
$$s = vt - \frac 1 2 at^2$$
$$v = u + at$$
$$s = \frac 1 2 (u + v)t$$
$$v^2 = u^2 + 2as$$
$$p = m x v \text{momentum = mass x velocity}$$
$$\text{Change in momentum = mass x (final velocity - initial velocity)}$$
$$F \Delta t = \Delta (mv) \text{Impulse = Change in momentum}$$
$$W = Fs \text{Work done (J) = Force(N) x Distance moved (m)}$$
$$W = Fs \cos\theta$$
$$P = \frac{\Delta W}{\Delta T} \text{Power (W)} = \frac{\text{Work Done (J)}}{\text{Time (s)}}$$
$$\text{P = Fv Power = Force causing the motion x velocity in direction of the motion}$$
$$P = Fv \cos \theta$$
$$\text{Efficiency} = \frac{\text{useful (energy/power) output}}{\text{(energy/power) input}}$$
$$\rho = \frac{m}{V} \text{density (kgm}^{-3} \text{)} = \frac{\text{mass (kg)}}{\text{volume (m}^3\text{)}}$$
$$F = k \Delta L \text{Force (N) = stiffness constant (Nm$^{-1}$) x extension (m)}$$
$$\text{Tensile stress (Nm}^{-2}\text{)} = \frac{F}{A} = \frac{\text{force}}{\text{cross-sectional area}}$$
$$\text{Tensile strain} = \frac{\Delta L}{L} = \frac{\text{extension}}{\text{original length}}$$
$$\text{Energy} = \frac{1}{2} k (\Delta L)^2$$
$$\text{Young Modulus, } E = \frac{\text{tensile stress}}{\text{tensile strain}}$$
$$Q = It \text{Charge (C)} = \text{Current (A)} \times \text{time (s)}$$
Measurements and Uncertainties
SI Prefixes
Factor
Name
Symbol
$10^{24}$
yotta
Y
$10^{21}$
zetta
Z
$10^{18}$
Exa
E
$10^{15}$
peta
P
$10^{12}$
tera
T
$10^{9}$
giga
G
$10^{6}$
mega
M
$10^{3}$
kilo
k
$10^{2}$
hecto
h
$10^{1}$
deka
da
$10^{-1}$
deci
d
$10^{-2}$
centi
c
$10^{-3}$
milli
m
$10^{-6}$
micro
$\mu$
$10^{-9}$
nano
n
$10^{-12}$
pico
p
$10^{-15}$
femto
f
$10^{-18}$
atto
a
$10^{-21}$
zepto
z
$10^{-24}$
yocto
y
Types of Error
Random Error
These affect precision but not accuracy. These could be due to noise, or the fact you're measuring something random, or wind blowing in the wrong direction, or something else uncontrollable. Systematic Error Systematic errors affect accuracy, and happen when there is a mistake in the measurement due to the environment, the equipment or the experimental method. Accuracy or Precision:
Uncertainties
Absolute Uncertainty is the uncertainty of a measurement given as a certain fixed quantity. Fractional Uncertainty is the uncertainty given as a fraction of the measurement taken, so $\frac{absolute\space uncertainty}{measurement}$. Percentage Uncertainty is the uncertainty given as a percentage of the measurement, so $\frac{absolute\space uncertainty}{measurement} \times 100$.
Combining Uncertainties
Adding or Subtracting
When adding or subtracting data you add the absolute uncertainties. Multiplying or Dividing
When multiplying or dividing data you add the percentage uncertainties. Raising to a Power
When you raise data to a power, you multiply the percentage uncertainty by the power.
An exchange particle causes repulsion by going from one particle to another, like two people throwing a ball where the ball is the exchange particle.
An exchange particle causes attraction by going from one particle to another, but being thrown away from each other and acting like a boomerang.
These are shown below:
Photons
EM waves (and the energy they carry) can only exist in discrete packets, or quanta, known as photons ($\gamma$). The energy carried by a photon is: $E = hf$ where h = Planck's Constant ($6.63 \times 10^{-34} Js$) and f = frequency of the light (Hz)
$E = \frac{hc}{\lambda}$ where h = Planck's Constant ($6.63 \times 10^{-34} Js$), c = speed of light ($3.00 \times 10^8 ms^{-1}$) and $\lambda$ = wavelength (m)
Fundamental Forces
There are four fundamental forces: strong, weak, electromagnetic and gravity.
Only the weak interaction can change a quark's character.
EM
Virtual Photon ($\gamma$)
Infinite
0
Anything with charge
Gravity
Graviton
Infinite
0
Anything with mass/energy
Quarks and Leptons
Leptons
Quarks
$1^{\text{st}}$ gen:
$e^-$ (electron)
$\nu e$ (electron neutrino)
u (up)
d (down)
$2^{\text{nd}}$ gen:
$\mu^-$ (muon)
$\nu \mu$ (muon neutrino)
c (charm)
s (strange)
$3^{\text{rd}}$ gen:
$\tau^-$ (tauon)
$\nu \tau$ (tauon neutrino)
t (top)
b (bottom)
Charge:
-1
0
$+\frac{2}{3}$
$-\frac{1}{3}$
Click the table above to show the quark and lepton names.
There is a lepton number for each generation: $L_e$, $L_\mu$ and $L_\tau$. Each lepton has a lepton number of +1 for their generation, and antileptons have a lepton number of -1 for their generation.
e.g. $e$ and $\nu e$ have a $L_e$ number of +1, and $\overline{\mu}$ has a $L_\mu$ number of -1.
Strange quarks have a strangeness of -1, antistrange quarks have a strangeness of +1, everything else has a strangeness of 0.
- Baryons: $qqq$
e.g. $p (uud), n (udd), \overline{p} (\overline{u}\overline{u}\overline{d}), \overline{n} (\overline{u}\overline{d}\overline{d})$
Baryons have a baryon number of +1, antibaryons have a baryon number of -1 and everything else has a baryon number of 0.
Conservation
These must be conserved:
Charge
Energy/mass
Baryon number
Lepton number (for each generation of leptons)
Momentum (led to neutrino discovery as momentum wasn't conserved during $\beta^-$ decay
Strangeness (except in weak interactions)
Antiparticles
Antiparticles have opposite properties (negative if property is positive, positive if property is negative) compared to the normal particle, except mass which is the same.
Annihilation
$$p + \overline{p} \rightarrow \gamma + \gamma$$ Collision of a particle with its antiparticle will annihilate them and produce photons (two to conserve momentum). The minimum energy of the photon produced by annihilation = the rest energy of the particle type annihilated (MeV)
Pair Production
$$\gamma \rightarrow p + \overline{p}$$ A high energy photon produces a particle and its antiparticle. The minimum energy needed for pair production = the total rest energy of the particles produced (MeV)
Decay
Protons are the only stable free baryon. The leptons $e, \nu e, e^+, \overline{\nu e}$ are also stable.
The order of mass (and the order they decay in) for quarks is: $t \rightarrow b \rightarrow c \rightarrow s \rightarrow d \rightarrow u$ Strange particles are created in pairs (often pair production) by the strong interaction.
e.g. $\gamma \rightarrow s + \overline{s}$
Electromagnetic Radiation and Quantum Phenomena
The Electron Volt
$1 eV = 1.6 \times 10^{-19} J$
Energy Levels
Electrons in an atom exist in well defined energy levels. Each energy level is given a number, with n = 1 representing the lowest energy level an electron can be in, or the ground state.
Electron Transitions
Electrons can move between energy levels, either by emitting a photon containing energy equal to the difference in energies between the two levels of the transition, or by absorbing a photn with the exact energy difference between the two levels. The movement of electrons to higher energy levels is called excitation.
The energy change when an electron is excited or de-excited is $\Delta E = E_1 - E_2 = hf = \frac{hc}{\lambda}$
Excitation/Ionisation
When an electron is ionised it leaves the atom. The ionisation energy of an atom is the amount of energy needed to remove an electron from the ground state atom.
When an electron is excited it moves up one or more energy levels. Electrons can be excited by:
A current passing through it
Shining light on it
Heating it up
Collisions with other atoms
Absorption and Emission
Emission
After an electron is excited it deexcites and falls to a lower energy state, emitting energy in the form of light with $hf = \Delta E$.
Line Emission Spectra
A line emission spectrum is seen as a series of bright coloured lines against a white or black background. Each line corresponds to a particular wavelength of light emitted by the source. Line spectra provide evidence that the electrons in atoms exist in discrete energy levels. Atoms can only emit photons with energies equal to the difference between two energy levels. Since only certain photon energies are allowed, you only see the corresponding wavelengths in the line spectrum.
Absorption
When you shine white light on an atom, photons are absorbed and electrons are excited. After absorption, the electrons will deexcite and release a photon each in order to return to their ground state.
Line Absorption Spectra
You get line absorption spectra when light with a continuous spectrum of energy passes through a cool gas. The gas must be cool because at low temperatures, most of the electrons of the gas will be in their ground states. Photons of the correct wavelength are absorbed by the electrons to excite them to higher energy levels, meaning there are black lines in the emission spectrum.
Continuous Spectra
The spectrum of white light is continuous. Hot things emit a continuous spectrum in the visible and infrared. All wavelengths are allowed because the electrons are free and are not confined to energy levels in the object producing the spectrum.
Wave Particle Duality
Matter behaves sometimes as a wave and sometimes as a particle.
Photoelectric Effect $E_k = hf - \phi$ Kinetic Energy = (Planck's Constant x Frequency) - Work Function$f_o = \frac{\phi}{h}$ Threshold frequency $= \frac{\text{work function}}{\text{Planck's Constant}}$ If $E_k = 0$, electrons emerge with no kinetic energy. If $E_k \gt 0$, the photoelectric effect occurs and electrons emerge with some kinetic energy. if $E_k \lt 0$, the photoelectric effect doesn't work as electrons don't have enough energy to escape.
In the photoelectric effect, light shines on a metal. If the light is above the threshold frequency, then the atom is ionised and the electrons are emitted. Intensity doesn't affect whether they're emmitted, just at what rate.
If light was a wave then the energy could be absorbed gradually and would accumulate until there was enough for the electron to escape, but the electron must absorb enough energy from the light in one go, showing light behaves like a particle. The stopping potential is given by $V_s = \frac{E_k}{e}$ where $e =$ the charge of an electron $= 1.6 \times 10^{-19}$.Electron Diffraction
$e^-$ particle also acts as a wave since it diffracts. Maximum diffraction occurs when $\lambda \approx$ aperture ÷ obstacle size where $\lambda$ is the de Broglie wavelength. De Broglie equation:$$\lambda = \frac{h}{p}$$ where $h$ is Planck's Constant and $p$ is the momentum of the electron.
Waves
Progressive Waves
A progressive wave carries energy from one place to another without transferring any material.
Reflection
Waves reflect (bounce back) when they hit a boundary.
Refraction
Waves bend towards the normal when entering a more optically dense medium and away from the normal when entering a less optically dense medium.
Diffraction
Waves spread out after passing through a slit smaller than its wavelength.
When the gap is a lot bigger than the wavelength, diffraction is unnoticeable. You get noticeable diffraction when the gap is several wavelengths wide. When the gap width is equal to the wavelength maximum diffraction occurs. If the gap is smaller than the wavelength, the waves are mostly just reflected back.
Wave Properties
The wave properties are shown below:
Note: Peaks are also commonly called crests.
The wavelength ($\lambda$), amplitude (A) and the displacement (x) are all measured in metres.
Other properties of waves are:
Period: the time taken for one whole wave cycle, given by T and measured in seconds.
Frequency: the number of whole wave cycles (oscillations) passing a given point per second, given by $f$ and measured in hertz.
Phase: a measurement of the position of a certain point along the wave cycle.
Phase difference: the amount by which one wave lags behind another wave. If the phase difference is 0, the waves are known as 'in phase'.
Frequency and period are linked by the equation: $f = \frac{1}{\text{T}}$
Transverse Waves
In transverse waves the displacement of the particles or field is at right angles to the direction of energy propagation. All electromagnetic waves are transverse.
Longitudinal Waves
In longitudinal waves the displacement of the particles or fields is along the direction of energy propagation.
Polarised Waves
If a wave is vibrating in a mixture of directions and it passes through a polarising filter, only waves vibrating in a certain direction can get through, and the wave has been polarised. e.g. A rope being twirled around passes through a gap between vertical fence posts. Only vertical movements are allowed through the gap and the rope has been polarised.
Polarisation can only happen with transverse waves.
Superposition
The principle of superposition says that when two or more waves cross, the resultant displacement equals the vector sum of the individual displacements. The superposition at any point is the sum of the displacement of the waves at that point. If the displacements are equal and opposite, they cancel each other out. After meeting each other they continue as they were before the collision.
Interference
Constructive Interference
When two waves meet, if their displacements are in the same direction, they combine to create a bigger displacement. This is called constructive interference. Destructive Interference
When two waves meet, if their displacements are in opposite directions, they cancel to create a smaller displacement. This is called destructive interference.
Stationary Waves
A stationary wave is the superposition of two progressive waves with the same frequency (or wavelength) and amplitude, moving in opposite directions. No energy is transmitted by a stationary wave.
Resonant Frequencies
A stationary wave is only formed at a resonant frequency. First harmonic (Fundamental):
The lowest possible resonant frequency is the first harmonic. It has one loop with a node at each end. One half wavelength fits onto the string, so the wavelength is double the length of the string. Second harmonic:
The second harmonic has twice the frequency of the first harmonic. There are two loops with a node in the middle and at each end. The wavelength is the length of the string. Third harmonic:
The third harmonic has three times the frequency of the first harmonic. There are three loops with a node at either end, a node a third along and a node two thirds along. One and a half wavelengths fit on the string.
You can have as many harmonics as you like, there is an extra loop and an extra node each time, and the number of wavelengths that fit goes up by a half each time. The first five are shown below:
The formula for calculating the frequency of the first harmonic is: $f = \frac{1}{2l} \sqrt{\frac{T}{\mu}}$ $l =$ length of vibrating string (m).
$T =$ tension on string (N).
$\mu =$ mass per unit length of string (kg m$ ^{-1}$).
Single-slit Experiment
To observe a clear diffraction pattern, a monochromatic, coherent light source should be used. If the wavelength of the light is roughly equal to the size of the aperture, this produces a diffraction pattern on the screen.
The pattern has alternating bright fringes (maxima) and dark fringes (minima) due to interference. Bright fringes are due to constructive interference and dark fringes are due to destructive interference.
The central maximum in a single-slit diffraction pattern is the brightest part of the pattern and has the highest intensity. Intensity is the power per unit area. Increasing the slit width decreases the amount of diffraction, meaning the central maximum is narrower and has a higher intensity. Increasing the wavelength increases the amount of diffraction, meaning the central maximum is wider and of lower intensity.
Two-source Interference
Two-source interference can be shown either by using two light sources or by shining a monochromatic, coherent light source through two slits. The light source must be monochromatic and coherent to ensure the diffracted waves are in phase to produce a clear diffraction pattern.
The path difference is 0 at the central maximum and increases by $\lambda$ for each maxima going out from the centre. The minima next to the central maxima have a path difference of $\frac{\lambda}{2}$ and these also increase by $\lambda$ going out from the centre.
Constructive interference occurs when the path difference $ = n\lambda$ where $(n ∈ \mathbb N \text{ including } 0)$Destructive interference occurs when the path difference $ = \frac{(2n + 1)\lambda}{2} = (n + \frac{1}{2})\lambda$ where $(n ∈ \mathbb N \text{ including } 0)$
Young's Double-slit Experiment
Young's Double-slit Experiment shows two-source interference by shining a monochromatic, coherent light source (like a laser) through two slits. If the light source is not coherent a single slit is used to make the waves coherent, as shown below.
This results in an interference pattern on the screen similar to the one below.
Young's double slit formula links some properties in the double slit experiment: $w = \frac{\lambda D}{s}$ w = fringe spacing (m), the distance from the middle of one maxima to the middle of an adjacent maxima, or the middle of one minima to the middle of an adjacent minima.
$\lambda$ = wavelength (m).
D = distance between the slits and the screen (m).
s = distance between slits or slit separation (m).
Diffraction Gratings
A diffraction grating contains lots of equally spaced slits very close together. This follows the same principle as the double-slit experiment but with many more slits, making the interference pattern sharper. When monochromatic light is passed through a diffraction grating with hundreds of slits per millimetre at normal incidence (right angles to the grating), the interference pattern is very sharp as there are many beams reinforcing the pattern, meaning it is easier to take more accurate measurements.
The central maxima of the interference pattern is in line with the direction of the beam and is called the zero order. The maxima either side of the zero order are called the 1^{st} order. The maxima outside these are called the 2^{nd} order, and so on.
Diffraction grating equation: $d \sin \theta = n \lambda$ d = distance between slits or slit separation (m).
$\theta$ = angle to the normal made by the maximum (° or radians).
n = order of maximum.
$\lambda$ = wavelength of light sorce (m).
From this equation, we can see:
If $\lambda$ is bigger, $\theta$ is bigger, so the larger the wavelength the more the pattern will spread out.
If d is bigger, $\theta$ is smaller, so the coarser the grating, the less the pattern will spread out.
Values of $\sin \theta \gt 1$ are impossible, so if for a certain n the result for $\sin \theta$ is greater than 1, the order can't exist.
The number of slits per metre is: $N = \frac{1}{d}$ The maximum number of orders is when $\theta = 90°$ so $\sin \theta = 1$, so, using the diffraction grating equation, $n = \frac{d}{\lambda}$ rounded DOWN to the nearest whole number.
Refractive Index
Absolute Refractive Index of a Material $n = \frac{c}{c_s} \text{Absolute refractive index} = \frac{\text{speed of light in a vacuum} (\text{ms}^{-1})}{\text{speed of light in material} (\text{ms}^{-1})}$ The speed of light in air is only slightly less than the speed of light in a vacuum, so it is assumed the refractive index of air is 1.
Light travels slower in materials than in a vacuum as it interacts with the particles. The more optically dense a material is, the higher its refractive index.
Relative Refractive Index Between Two Materials$_1n_2 = \frac{c_1}{c_2} = \frac{n_2}{n_1} \text{Relative refractive index} = \frac{\text{speed of light in material 1} (\text{ms}^{-1})}{\text{speed of light in material 2} (\text{ms}^{-1})} = \frac{\text{absolute refractive index of material 2}}{\text{absolute refractive index of material 1}}$
Snell's Law
The angle that incoming light (the incident ray) makes to the normal is called the angle of incidence, i. The angle that the refracted ray makes with the normal is called the angle of refraction, r.
Snell's Law of refraction for a boundary between two angles is: $n_1 \sin i = n_2 \sin r$ where $n_x$ is the refractive index of material $x$.
Critical Angle and Total Internal Reflection (TIR)
The critical angle is where the refracted angle = 90° and the light is refracted along the boundary. This can happen for any boundary where the light is passing from a more optically dense material into a less optically dense material. You can work out the critical angle using: $\sin \theta _c = \frac{n_2}{n_1}$ where $n_1 \gt n_2$.
Total internal reflection is where the angle of incidence is greater than the critical angle, so refraction can't happen, meaning all of the light is reflected back into the material.
Optical Fibres
An optical fibre is a very thin flexible tube of glass or plastic fibre that can carry light signals over long distances and round corners using TIR. Step-index optical fibres have a high refractive index core surrounded by cladding with a low refractive index, which means light undergoes TIR and rebounds, staying within the fibres. The cladding also protects the fibre from scratches which could allow light to escape.
Light is shone into the fibre at one end, and it is so narrow that the light always hits the boundary at an angle greater than the critical angle, meaning TIR always occurs until the light comes out the other end.
Optical fibres are used to transmit phone and cable TV signals. They are beneficial over electricity flowing through copper cables as:
The signal can carry more information as light has a high frequency.
The light doesn't heat up the fibre so almost no energy is lost as heat.
There's no electrical interference.
Fibre optic cables are much cheaper to produce.
The signal can travel a long way very quickly and with minimal signal loss.
However, the signal can be degraded either by absorption or dispertion.
Absorption is where some of the signal's energy is absorbed by the material the fibre is made from. This energy loss results in the amplitude of the signal being reduced.
Material dispertion and modal dispertion can both degrade a signal. Both cause pulse broadening, so the received signal is broader than the initial one. Broader signals can overlap each other, resulting in data loss.
Modal dispertion is caused by light rays entering the fibre at different angles. This causes them to take different paths down the fibre, resulting in them arriving at a different time. Modal dispersion can be reduced by using a single-mode fibre, in which light is only allowed to follow a very narrow path.
Material dispersion is caused by the different amounts of refraction experienced by different wavelengths of light. Different wavelengths refract differently, and as light contains many wavelengths this results in some parts of the signal taking longer to get to the destination. Material dispersion can be stopped by using monochromatic light.
Optical fibre repeaters can be used to regenerate the signal every so often to help reduce signal degredation.
Mechanics
Scalars are quantities with no direction, just an amount. e.g. mass, speed, distance, temperature. Vectors are quantities with magnitude and direction - e.g. force, velocity, displacement.
Forces In Equilibrium
If an object is in equilibrium, all the forces acting on it are balanced, so there's no resultant force. An object in equilibrium can be at rest or moving with a constant velocity.
Moments
Weight is a force measured in Newtons: W = mg A moment is the turning effect of a force around a turning point measured in Nm: M = F x d = Force (N) x Perpendicular distance from the pivot (m) The principle of moments states that for a body to be in equilibrium, the sum of its clockwise moments must be equal to the sum of its anticlockwise moments.
Couples
A couple is a pair of forces of equal size which act parallel to each other, but in opposite directions. A couple doesn't cause any resultant linear force but does produce a moment, which can be calculated using the moment equation above.
Centre of Mass
The centre of mass is the point where you can consider all of the mass of an object to be concentrated.
You can find the centre of mass by hanging the object from a point, drawing a vertical line downwards, hanging the objct from another point and drawing another vertical lign downwards. The centre of mass is where the lines cross.
An object will topple over if the line of action of its weight (drawn vertically down from the centre of mass) falls outside its base. The higher the centre of mass, and the smaller the base area, the less stable an object is.
Uniform Acceleration
SUVAT equations:
$$s = ut + \frac 1 2 at^2$$ $$s = vt - \frac 1 2 at^2$$ $$v = u + at$$ $$s = \frac 1 2 (u + v)t$$ $$v^2 = u^2 + 2as$$ where $s$ = displacement (m), $u$ = initial velocity (ms$^{-1}$), $v$ = final velocity (ms$^{-1}$), $a$ = acceleration (ms$^{-2}$), $t$ = time (s).
Graphs
Displacement-time
On a displacement-time graph, the gradient is velocity, so the acceleration is the rate of change of the velocity (the second differential).
An object's instantaneous velocity is just its velocity at a particular moment in time, or the gradient at a particle point on a displacement-time graph. The average velocity is the total change in displacement of the object divided by the total time take, or the gradient between the first and last points on the graph. Velocity-time
On a velocity-time graph, the gradient is acceleration. Uniform acceleration is always a straight line. The steeper the gradient, the greater the acceleration.
Velocity-time graphs can have negative regions to show something travelling in the opposite direction, which speed-time graphs can't have.
The displacement on a velocity-time graph is the area under the graph. Acceleration-time
A positive acceleration means the object accelerates, a negative acceleration means the object decelerates. If the acceleration is 0 there is no acceleration and the velocity is constant. A straight horizontal line shows uniform acceleration or decelaration.
The area under an acceleration-time graph is the toal change in velocity, as $\Delta v = a \times t$
Newton's Laws of Motion
Newton's First LawAn object will have constant velocity if it has no resultant forceNewton's Second LawForce is proportional to rate of change of momentum $F \propto \frac{\Delta P}{t} F = ma$Newton's Third LawIf body A exerts a force on body B, body B exerts a force on body A, where the forces are equal in magnitude, opposite in direction, of same type, acting at the same time, with the same line of action and are acting on different bodies. Newton's 1^{st} Law: "The velocity of an object will not change unless a resultant force acts on it."Newton's 2^{nd} Law: "The acceleration of an object is proportional to the resultant force acting on it." This gives you the equation: F = m x a Force = mass x acceleration Note: acceleration is always in the same direction as the resultant force. Newton's 3^{rd} Law: "Every action has an equal and opposite reaction." This means that if an object A exerts a force on an object B, object B exerts an equal but opposite force on object A.
Acceleration Due To Gravity
An object is in freefall is when the only force acting on it is gravity, so its resultant force is its weight (m x g). It's defined as the motion of an object undergoing an acceleration of $g$.
$g = 9.81 \text{ms}^{-2}$. $g$ acts vertically downwards, so it is usually negative.
The acceleration of an object in freefall is always $g$, regardless of the mass and shape of the object.
Projectile Motion
Any object given an initial velocity and then left to move freely under gravity is a projectile. When air resistance is negligible, the horizontal velocity is constant, while the vertical velocity is affected by $g$.
Air resistance causes drag which slows the horizontal and vertical speeds of the projectile, meaning it doesn't travel as far.
Friction
Friction increases as speed increases. It always acts in the opposite direction to the motion of the object. It converts kinetic energy into heat.
Friction in a fluid is also called drag. Air resistance is a type of friction.
Lift
Lift is an upwards force on an object moving through a fluid. It happens when the shape of an object causes the fluid flowing over it to change direction. The force acts perpendicular to the direction in which the fluid is flowing, lifting the object.
Terminal Velocity
Terminal velocity happens when frictional forces (air resistance) equal the driving force (weight). If the driving force is constant, and drag increases with speed, an object will eventually reach terminal velocity. The object can't travel any faster than this without more force being applied to it.
Parachutists have a much lower terminal velocity with the parachute open than with it closed, due to an increase in air resistance, which is how parachutes allow them to survive the fall.
Momentum
p = m x v momentum = mass x velocity Assuming no external forces act, linear momentum is always conserved. Total momentum before a collision = total momentum after a collision.
Elastic and Inelastic Collisions
An elastic collision is one where momentum is conserved and kinetic energy is conserved, so there is no energy loss.
If a collision is inelastic, some energy was lost, likely to heat, sound, light etc. In the real world most collisions are slightly inelastic. Resultant force = the rate of change of momentum (kg ms$^{-2}$)Change in momentum = mass x (final velocity - initial velocity)Impulse
Impulse is defined as the product of force and time. So the impulse on a body is equal to the change in momentum of that body and is measured in Ns. Impulse is the area under a force-time graph. $F \Delta t = \Delta (mv)$ Impulse = Change in momentum
Impact Forces
The force of an impact can be reduced by increasing the time of the impact. This is how airbags work, as they increase the time taken for the person in the car to stop.
Work and Power
Work
Work is done whenever energy is transferred. Work means the amount of energy transferred from one form to another when a force causes a movement of some sort. W = Fs Work done (J) = Force(N) x Distance moved (m) This equation assumes the direction of the force is the same as the direction of the movement.
When the direction of the movement of an object is different from the direction of the force acting on it, you must find the component of the force that acts in the direction of the movement. For a force at an angle $\theta$ to the direction of motion, work can be found with: W = Fs cos$\theta$ The area under a force-displacement graph tells you the work done. Power
Power is the rate of doing work. $P = \frac{\Delta W}{\Delta T}$ $\text{Power (W)} = \frac{\text{Work Done (J)}}{\text{Time (s)}}$ The watt is defined as $W = Js^{-1}$.
Another equation for power: P = Fv Power = Force causing the motion x velocity in direction of the motion If the force and motion are in different directions, where $\theta$ is the angle at which the force acts from the direction of motion, use: P = Fv cos $\theta$
Conservation of Energy
Energy cannot be created or destroyed, it can only be transferred from one form to another.
$F = k \Delta L$ Force (N) = stiffness constant (Nm$^{-1}$) x extension (m)Limits
The limit of proportionality is the point beyond which the force is no longer proportional to the extension.
The elastic limit is the point where if you increase the force past this point, the material will be permanently stretched and will not return to its original shape when the force is removed.
The yield point is where the material starts to stretch without any load.
Stretches
A deformation is elastic if the material returns to its original shape once the forces are removed, so it has no permanent extension.
A deformation is plastic if the material is permanently stretched after the force has been removed.
Stress
If a material subject to a pair of opposite forces is stretched, the forces are tensile forces. If the material is squashed, they're compressive forces. Tensile stress (Nm$^{-2}$) $= \frac{F}{A} = \frac{\text{force}}{\text{cross-sectional area}}$Tensile strain $= \frac{\Delta L}{L} = \frac{\text{extension}}{\text{original length}}$ The breaking stress is the stress required to break the material and separate the atoms.
The ultimate tensile stress is the maximum stress that the material can withstand. $\text{Energy} = \frac{1}{2} k (\Delta L)^2$
The Young Modulus
Below the limit of proportionality, stress divided by strain is a constant, called the Young Modulus: $\text{Young Modulus, } E = \frac{\text{tensile stress}}{\text{tensile strain}}$
Brittle Materials
For brittle materials, the stress-strain and force-extension graphs don't curve, they just stop once they reaches the breaking stress.
Electricity
Circuilt Symbols
Current
Current is the rate of flow of charge, in amperes (A). $Q = It$ $\text{Charge (C)} = \text{Current (A)} \times \text{time (s)}$ Current is measured using an ammeter which must be attached in series with the component you want to measure the current of.
Potential Difference
The p.d., or voltage, between two points is defined as the work done in moving a unit charge between the points. $V = \frac{W}{Q}$ $\text{P.d. (V)} = \frac{\text{Work Done (J)}}{\text{Charge (C)}}$ P.d. is measured using a voltmeter which must be attached in parallel with the component you want to measure the p.d. of.
Resistance
Resistance is a measure of how much current you get for a p.d. across the component, in Ohms ($\Omega$). $V = IR$ $\text{P.d. (V)} = \text{Current (A)} \times \text{Resistance
(}\Omega\text{)}$ Some conductors (mostly metals) are called ohmic conductors, because the obey Ohm's Law. Ohm's Law states that, provided the physical conditions remain constant, the current through an ohmic conductor is directly proportional to the p.d. across it. Provided constant physical conditions, $I \propto V$
Diodes
Diodes are made from semiconductors and are designed to let current flow through in one direction only. Most diodes require a voltage of about 0.6V in the forward bias before they will conduct - this is known as the threshold voltage.
In reverse bias, the resistance of the diode is very high and the current that flows is very tiny.
I-V Graphs
The gradient of an I-V graph is $\frac{1}{R}$.
For ohmic conductors, the I-V graph is a straight line.
For a filament lamp (non-ohmic), the I-V graph is as below.
For diodes, the I-V graph is as below.
Resistivity
The resistivity of a material is defined as the resistance of a 1m length with a 1m$^2$ cross-sectional area, measured in ($\Omega$m). $\rho = \frac{RA}{L} \text{Resistivity} = \frac{\text{Resistance} \times \text{Cross-sectional Area}}{\text{Length}}$ The lower the resistivity of a material, the better it is at conducting electricity.
Semiconductors
Semiconductors are a group of materials that aren't as good at conducting electricity as metals as they have fewer charge carriers (i.e. free electrons) available. However, if energy is supplied to a semiconductor, it can release more charge carriers, decreasing the resistance. This can be used to create environmental sensors. Examples are thermistors, diodes and light dependent resistors (LDRs). Thermistors are most commonly Negative Temperature Coefficient (NTC) Thermistors, meaning that as the temperature increases, the resistance decreases, as below.
Superconductors
A superconductor is a material with no resistivity. If you cool some materials down to below a temperature known as the transitional temperature, their resistivity disappears and they become a superconductor. This means no electrical energy is wasted as heat. Superconductors can be used to create:
Power cables with no energy loss through heat.
Very strong electromagnets.
Very quick electronic circuits with minimal energy loss.
Power
Power, given by P, is defined as the rate of transfer of energy, measured in watts (W). $P = \frac{E}{t} \text{Power (W)} = \frac{\text{Energy (J)}}{\text{time (s)}}$$P = IV$ Power(W) = Current (A) x p.d. (V)$P = \frac{V^2}{R} \text{Power (W)} = \frac{\text{(Voltage (V))}^2}{\text{Resistance (}\Omega\text{)}}$$P = I^2R$
Energy
$E = IVt$ Energy = Current x p.d. x time$E = \frac{V^2}{R}t$$E = I^2Rt$
Internal Resistance
Internal resistance is the resistance of the battery.
Load resistance, or external resistance, is the total resistance of all the components in the circuit.
The energy wasted per coulomb overcoming the internal resistance is called the lost volts (V).
Electromotive Force (E.m.f)
E.m.f, denoted by $\varepsilon$, is the amount of electrical energy the battery produces and transfers for each coulomb of charge. $\varepsilon = \frac{E}{Q} \text{E.m.f (V)} = \frac{\text{Energy (J)}}{\text{Charge (C)}}$$\varepsilon = I(R + r)$ E.m.f (V) = Current (A) x (Load resistance + Internal resistance)$V = \varepsilon - Ir$ Terminal P.d. = E.m.f - (Current x Internal resistance)
Terminal P.d.
The terminal p.d. is the p.d. across the load resistance, or the energy transferred when one coulomb of charge flows through the load resistance. If there was no internal resistance, the terminal p.d. would be the same as the e.m.f, but in reality there is always some internal resistance. Energy per coulomb supplied by the source = terminal p.d. + lost volts.
Kirchhoff's Laws
Current, Charge and Energy are all conserved around a circuit.
Kirchhoff's first law: The total current entering a junction = the total current leaving it. Kirchhoff's second law: The total e.m.f around a circuit = the sum of the p.d.s across each component $\varepsilon = \sum IR$
$I_T = I_1 + I_2 + I_3$$V_1 = V_2 = V_3$$\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$Both Series and Parallel $\varepsilon_T = \varepsilon_1 + \varepsilon_2 + \varepsilon_3 ... \varepsilon_n$ where n is the number of cells in the circuit.
Potential Divider
A potential divider is a circuit with a voltage source and two resistors in series. The potential difference across the voltage source is split across the resistors in the ratio of their resistances.
You can use potential dividers to supply a potential difference, $V_{out}$, between 0 and $V_{in}$. This allows you to use a p.d. lower than the power supply. Using a variable resistor allows you to vary $V_{out}$. $V_{out} = \frac{R_2}{R_1 + R_2}V_{in}$ LDRs and thermistors can be used in a potential divider, meaning $V_{out}$ can vary with light level or temperature. These can be used to turn on a light when it gets dark, or to turn on a heating system when it gets cold, etc. Potentiometers
A potentiometer has a variable resistor replacing $R_1$ and $R_2$ of the potential divider. This is useful when you want to use a slider mechanism to change a voltage continuously, like in the volume control of a speaker.
Further Mechanics
Simple Harmonic Motion
Simple harmonic motion is where acceleration is proportional to displacement and acts in the opposite direction. $$a \propto - x$$ Definitions:
Amplitude - the maximum displacement at any given point.
Frequency - number of oscillations per second.
Period - $T = \frac{1}{f}$ or the time taken for one complete cycle.
Angular velocity - $\omega = 2\pi f$ or the rate of change of angle of a rotating body.
A free oscillation - where there is no or negligible friction therefore the amplitude is constant. $$T = 2\pi\sqrt{\frac{l}{g}} = 2\pi\sqrt{\frac{m}{k}}$$$$F = kx$$Resonance and Damping Damping
$$x = A\cos (\omega t) \rightarrow a = -\omega^2 A \cos (\omega t) = -\omega^2 x$$ $$a = -\frac{k}{m} x = -\omega^2 x = -\frac{g}{l} x$$ $$\omega = \sqrt{\frac{k}{m}} \text{ for mass-spring system}$$ $$\omega = \sqrt{\frac{g}{l}} \text{ for pendulum}$$ $$V_{max} = \omega A$$ $$a_{max} = \omega^2 A$$ $$E_{Potential} = \frac{1}{2}k x^2$$ $$E_{Total} = \frac{1}{2} k A^2$$ $$E_k = E_T - E_P = \frac{1}{2} m v^2 \space \therefore \space V = \omega \sqrt{A^2 - x^2}$$
Circular Motion
Velocity is always along the tangent to the circle. The direction is changing so the velocity is changing, meaning acceleration is changing, so there is constant acceleration.
Tangental velocity is the instantaneous linear velocity.
The resultant force is always centripetal at A level.
Tension in string pulls ball inwards (centripetal).
Ball pulls string outwards (centrifugal).
$$V = \frac{d}{t} \space \therefore \space \omega = \frac{\Delta \theta}{t}$$ $$V = r \omega$$ $$a = \frac{V^2}{r} = r \omega^2$$ $$F = \frac{mV^2}{r} = mr \omega^2$$ $$\omega = 2\pi f = \frac{2\pi}{T}$$ Banked Track
Vertical component = $N \cos \theta = mg$
Horizontal component = $N \sin \theta = mr \omega^2 = \frac{mv^2}{r}$
The bank helps to accelerate the car towards the centre. Without banking, centripetal force is provided only by sideways friction between the road and wheels so the car may slip outwards if the speed is too high. On banked tracks the speed can be higher while maintaining safety, as the horizontal component of N acts as a centripetal force. Conical Pendulum
Vertical component = $W = T \cos \theta = mg$
Horizontal component = $T \sin \theta = mr \omega ^2 = \frac{mv^2}{r}$ Vertical Circle
In a vertical circle, the string is most likely to break when the mass is at the bottom, as this is when the tension is greatest. The tension is greatest because the weight is directly counteracting the centripetal force, meaning more tension is needed to apply the necessary centripetal force.
The limiting acceleration occurs when the tension is 0, so where the centripetal acceleration is equal to gravity. If the centripetal acceleration is less than gravity, the mass will fall at the top of the circle and the string will slack.
Circular Motion
$\underline{a}$ is the centripetal force
It accelerates the object towards the centre
The velocity ($\underline{v}$) essentially stops it from falling. $$F = \frac{mv^2}{r} = ma$$
Angular Velocity
Angular velocity is the rate of change of displacement, and is represented by $\omega$, with the unit rad s^{-1} $$\omega = \frac{d\theta}{dt} = \frac{2\pi r}{T} = 2\pi f = \frac{v}{r}$$$$a = \omega^2 r$$
Simple Harmonic Motion
Angular velocity is used in SHM.
The solution of deriving SHM is $x = A \cos (\omega t) = A \cos (2\pi f t)$
The time period is for one oscillation $$T = \frac{2\pi}{\omega}$$ For a spring: $$T = 2\pi\sqrt{\frac{m}{k}}$$ For a pendulum: $$T = 2\pi\sqrt{\frac{l}{g}}$$ The displacement of an oscillator is $x = A\cos(\omega t)$
Velocity = $\frac{dx}{dt} = -A \omega \sin (\omega t)$
Acceleration = $\frac{d^2x}{dt^2} = -A \omega^2 \cos (\omega t) = - \omega^2 x$
Energy in SHM: A mass oscillating on a SHM system has two energy types - kinetic and potential $$E_k = \frac{1}{2} m v^2$$For a spring system: $$E_p = \frac{1}{2} kx$$For a pendulum: $$E_p = mg\Delta h$$$$\sum E = \frac{1}{2}(mv^2 + kx^2)$$
Resonance
Resonance occurs when a system is driven at a frequency close to its own natural frequency
A mass on a spring will resonate at $f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$
Additional energy doesn't change the frequency, instead it changes the maximum energies of the system.
Resonance occurs when the driving frequency of an oscillation matches the natural frequency, giving rise to large amplitudes.
Resonant frequency is the peak of a current-frequency graph.
Damping
In reality, SHM systems lose energy. Pendulums have air resistance and springs heat up.
Damping comes in different forms:
Light damping takes a long time to reduce oscillation amplitude
Heavy damping reduces amplitude a lot
Overdamping reduces amplitude fast but takes ages to return to equilibrium
Critical damping stops oscillation in the shortest possible time
A pendulum system is as shown below:
Thermal Physics
The Gas Laws
Properties of gases that can be measured:
Pressure
Volume
Temperature
Amount (moles)
Mass
Boyle's Law
Pressure and volume are inversely proportional.
Fields
Gravitational Fields
Radial fields:
Seems directly down on the surface (uniform field as the lines are essentially parallel)
Inverse square law: field strength (g) is inversely proportional to the distance squared ($g \propto \frac{1}{r^2}$)
Definition of gravitational field strength: $g = \frac{F}{m}$, gravitational field strength (N kg$^{-1}$)$ = \frac{\text{force}}{\text{mass}}$, g = Force per unit mass
Newton's law of gravitation: $F = - \frac{G m_1 m_2}{r^2}$. The $-$ sign signifies it is attracting rather than repelling
$F$ = gravitational force between masses $m_1$ and $m_2$
$G$ = gravitational constant ($6.67 \times 10^{-11} N m^2 kg^{-2}$)
$m_1, m_2$ = masses (kg)
$r$ = separation of mass centres (metres)
For $g = -\frac{GM}{r^2}$
g = gravitational field strength at a distance $r$ from centre of mass causing the gravitational field $M$
Combining equations
Centripetal force = Gravitational force $\frac{mV^2}{r} = \frac{G m_1 m_2}{r^2} \therefore v = \sqrt{\frac{G m_1}{r}}$ where V = orbital speed (ms$^-1$), M = mass of the planet being orbited and r = orbital radius
Gravitational Potential
$$\Delta E_p = mg\Delta h$$ Definition: the work done per unit mass to move a small object from infinity to that point in the field. Infinity is defined as having zero potential and all values will be less than this, so negative.
$$V = \frac{W}{m} = -\frac{Gm}{r}$$ where V = gravitational potential (J kg$^{-1}$), W = work done (J), m = mass (kg), G = gravitational constant (Nm$^2$kg$^{-2}$) and r = distance from the centre of the object in m
$G.P.E = mV$
$g = - \frac{\Delta V}{\Delta r}$ gives $g$ as the negative value of the potential gradient ($\frac{\Delta V}{\Delta r}$)
Electric Fields
Electric Potential
$$V = \frac{E}{Q}$$ Definition: the work done per unit positive charge moving a positive charge from infinity to that point in the field.
Magnetic Fields
Capacitance
Capacitor does the same job as a battery but directly stores electrical energy rather than storing it as chemical energy.
Also known as condensors particularly in the automotive industry.
Greatly improved amount of energy it can store in last 20 years.
Current (A) = rate of flow of charge (C)
1 A = 1 C/s
I = Q/t
To make a simple capacitor you need two conducting plates close to each other, with an insulator between (e.g. air) so they're parallel and they have an overlapping area.
The symbol is this or this rotated 90 degs:
Shortcircuting the capacitor discharges it, e.g. Connecting A to B will shortcircuit. Lead A is often known as a flying lead or shorting lead.
With conventional current, electrons move round the circuit from negative to positive, so moving from positive plate to negative plate: This allows current to flow even though it is not a complete circuit.
C = $\varepsilon_0 \frac{A}{d}$
Electical field is stronger when plates are close together meaning capacitor is better, but you can't go too close or the electricity would arc. Capacitor can also be improved with higher voltage and larger surface area of plates.
Material between the plates ideally has a high permativity and low conductivity.
Current won't flow indefinitely as eventually all the electrons would move from one plate to the other.
e.g. for circuit
when switch connected to A
After each time period the amount of electrons stuck to the negative plate increases/decreases, one of them
Creates these graphs for charge current and voltage in the capacitor over time:
When switch connected to B
Graphs
$Q = Q_0 e^{-\frac{t}{RC}}$ so when $t = 0, Q = Q_0 \therefore Q_0$ is the initial charge
$I = I_0 e^{-\frac{t}{RC}}$
$R = R_0 e^{-\frac{t}{RC}}$
Time Constant, T $T = RC$ $If t = T, then V = 0.37V_0$