# AS Core Physics 2

## 49 Density:

• The density of a material is a measure of its mass per unit volume
• $\rho =$ $\frac{m}{V}$ where
- $\rho$ is the density ($\mathrm{kg}\ \mathrm{m}^{-3}$)
- $m$ is the mass ($\mathrm{kg}$)
- $V$ is the volume ($\mathrm{m}^{3}$)

## 50 Upthrust:

• Upthrust is a force that a fluid exerts on a body floating in it, opposing its weight
• It is equal to the weight of fluid displaced, this is Archimedes' principle

## 51 Fluids:

• Liquids and gases are fluids
• Viscosity is a fluid's resistance to flow - e.g. honey is more viscous than water
• Viscosity varies with temperature
- They have an inverse relationship in liquids, so an increase in temperature decreases viscosity and vice-versa
- In gases, viscosity and temperature are directly proportional
• In addition to upthrust, an object moving through a fluid will also experience viscous drag
- At low speeds, there is very little drag. But as speed increases, so does viscous drag
- Viscous drag is proportional to the size of the object
• So at terminal velocity, weight $-$ drag $-$ upthrust $=$ 0

## Flow:

• There are several types of flow which describe how the particles in a fluid travel:
- Turbulent flow: the speed and direction of fluid particles varies with time and change randomly
- Laminar flow: particles of different streamlines may move at different speeds
• A streamline is a way of representing the direction of flow on a diagram
• Streamlines never cross
• These two types of flow are shown below: ## Stokes' law:

• Stokes' law is used to calculate the resistive force of a ball travelling through a fluid
• $F = 6\pi \eta rv$ where
- $F$ is the force ($\mathrm{N}$)
- $\eta$ is the viscosity of the fluid ($\mathrm{N}\ \mathrm{s}\ \mathrm{m}^{-2}$)
- $r$ is the radius of the sphere ($\mathrm{m}$)
- $v$ is the velocity ($\mathrm{m}\ \mathrm{s}^{-1}$)
• When an object falls through a fluid, it accelerates as weight is greater than upthrust. Viscous drag increases with speed and eventually, it will reach terminal velocity once the resultant force reaches zero
• Stokes' law only applies to small spherical objects moving at low velocities through laminar flow

## 52 Determining the viscosity of a liquid with a falling ball (CP 4):

• Drop a ball bearing of known diameter (measure it with a micrometer in several places and take a mean) into a cylinder of liquid. Measure the time it takes to fall between two elastic bands on the cylinder
• Stokes' law can now be used to calculate the viscosity of the liquid
• Multiple balls of different diameters could be used to improve the results as there will be a large uncertainty in time measurements
• This will be the most accurate if the ball is falling at terminal velocity and it doesn't touch the sides of the cylinder

## 53 Hooke's law:

• Hooke's law shows that the force needed to extend/compress a spring is proportional to the distance it is being extended/compressed by
• $\Delta F = k\Delta x$ where
- $\Delta F$ is the force ($\mathrm{N}$)
- $k$ is the stiffness constant for the spring ($\mathrm{N}\ \mathrm{m}^{-1}$)
- $\Delta x$ is the extension ($\mathrm{m}$)
• A stiff material (with a high value of $k$) does not easily change shape when a force is applied

## 54 Stress:

• Stress is a measurement of internal force in an object
• Stress is represented with the symbol $\sigma$ (sigma)
• $\sigma =$ $\frac{F}{A}$ where
- $F$ is the force ($\mathrm{N}$)
- $A$ is the cross-sectional area ($\mathrm{m}^{2}$)
• Therefore, if the cross-sectional area of an object is doubled, it will be able to withstand twice the force
• Tensile stress is a type of stress where the object has been pulled/stretched
• Compressive stress is for when the object is pushed or 'squashed'
• Stress, like pressure, is measured in pascals. $1\ \mathrm{Pa}$ $=$ $1\ \mathrm{N}\ \mathrm{m}^{-2}$
• Hooke's law can be applied for stress: $\Delta x \propto \sigma$

## Strain:

• Strain is the change in shape/size of an object due to an external force
• Strain is the ratio of initial length to extension, represented by the epsilon symbol ($\varepsilon$)
• $\varepsilon =$ $\frac{\Delta x}{x}$ where
- $\varepsilon$ is the strain (it has no units)
- $\Delta x$ is the extension ($\mathrm{m}$)
- $x$ is the initial length before a force has been applied ($\mathrm{m}$)
• Usually, a percentage is used to represent strain (so multiply it by 100)
• Like stress, strain can also be compressive or tensile. The same definitions apply

## Young's modulus:

• The gradient of a stress-strain graph is a measure of stiffness, Young's modulus ($E$), measured in pascals ($\mathrm{Pa}$)
• $E =$ $\frac{\sigma}{\varepsilon}$

## 55 Force-extension graphs:

• A force-extension graph shows the effect of extension on force required
• They are specific to a specific material and dimensions
• An example is shown below, for a wire: • Up to the limit of proportionality, P, it obeys Hooke's law
• After the elastic limit, E, it starts to behave plastically rather than elastically
(materials with plastic deformation have no tendency to return to their original size/shape after being deformed, unlike those following elastic deformation)
• This is why, with zero force acting on it, the unloaded wire never returned to its original extension of 0
• But both lines are still parallel because $k$ doesn't change
• Once a material starts to significantly plastically deform, it is said to have reached its yield point. This occurs just after the elastic limit is reached

## 56 Stress-strain graphs:

• A compressive/tensile stress-strain graph looks similar to a force-extension graph, but it tends to be more useful
• Stress-strain graphs are independent of the dimensions of the material
• The breaking stress is the maximum stress a material can take before breaking

## 57 Determining the Young Modulus of a material (CP 5): • Use a long copper wire (as long as possible to reduce uncertainty). Measure its diameter in multiple places, take the mean and calculate the cross-sectional area ($A = \pi r^2$)
• With a small weight, measure the distance between the marker and end of the wire. This is the base length, $l$ $\mathrm{m}$
• Increase the weight and record the marker distance. Repeat several times
• Plot a stress-strain graph. The gradient is the Young Modulus, $E$. The $y$-intercept should be 0

## 58 Elastic strain energy:

• Before the elastic limit, all work done in stretching a wire is stored as elastic strain energy ($E_\mathrm{el}$)
• Elastic strain energy is the area under a force-extension graph (before the elastic limit)
• This is because $\Delta E_\mathrm{el} = \frac{1}{2}F\Delta x$, which is similar to the formula for the area of a triangle

## 59 Waves:

• Amplitude, $A$ (measured in $\mathrm{m}$), is the distance from the equilibrium position to the maximum displacement from the equilibrium
• Wavelength, $\lambda$ ($\mathrm{m}$), is the distance from one point on a wave to the same point on the next cycle
• Frequency, $f$ ($\mathrm{Hz}$ or $\mathrm{s}^{-1}$), is the number of complete cycles (wavelengths) per second
• Period, $T$ ($\mathrm{s}$), is the time taken for one oscillation (complete cycle)
• Frequency and period are reciprocals of each other; i.e. $T =$ $\frac{1}{f}$ [not given in the exam]
• Wave speed, $v$ ($\mathrm{m}\ \mathrm{s}^{-1}$), is how fast the wave is moving

## 60 The wave equation:

• $v = f\lambda$
• This applies to all wave types

## 61 Longitudinal waves:

• Longitudinal waves (e.g. sound) consist of displaced air/medium molecules into compressions (areas of higher pressure) and rarefactions (areas of lower pressure)
• The oscillations move parallel to the direction of energy transfer
• In air, longitudinal sound waves travel at the speed of sound (this will be given in the exam if it's required)

## 62 Transverse waves:

• Transverse waves (e.g. light) are waves where the oscillations are perpendicular to the direction of energy transfer

## 63 Drawing transverse waves:

• A sinusoidal transverse wave is a smooth oscillation in the shape of a $\sin$ graph, with one complete wave being 360° or 2π radians. Most transverse waves are sinusoidal
• The equilibrium is indicated with 0 in the centre of the $y$-axis on a displacement-distance graph (displacement on $y$ axis, distance on $x$)
• The same graph can also be plotted for a standing transverse wave, except that there will be two curves instead of one

## Drawing longitudinal waves:

• Longitudinal waves can be plotted with dots representing air molecules across one axis (distance)
• This can be converted into a displacement-distance graph if there are a second set of dots showing the standard distribution of the recorded molecules (i.e. the molecule positions when there is no wave travelling through them), by measuring the displacement from the standard position and plotting this on a graph, as shown below: The numbers at the top and bottom indicate the displacement
• A pressure-distance graph can also be created for a longitudinal wave; crests are plotted where the air pressure is dense (dots close together) and troughs where the pressure is lower than normal

## 64 Determining the speed of sound in air (CP 6):

• Equipment: two-beam oscilloscope, signal generator, speaker, microphone
• Connect the signal generator to a speaker and an oscilloscope input
• Connect the microphone to a second oscilloscope input
• Move the microphone away from the speaker until the waves are in phase on the oscilloscope. Record this distance and repeat, further away. A mean can be found from the differences in distance; multiply this by the frequency to calculate the speed of sound (wave equation)
• It is recommended to use the oscilloscope display to calculate frequency rather than reading off the signal generator as it is more accurate

## 65 Interference and superposition:

• Wavefronts are lines in the path of a wave where every point on the line is in phase with every other point on the line • Two waves are coherent when they have the same frequency, wavelength and direction, which is required for effective interference
• Path difference is the difference in paths between two waves, measured in wavelengths

## Interference and superposition:

• Interfering waves must approximately have the same amplitude
• Superposition is the combination of two or more waves
• Interference is the effect caused by superposition, most noticeable if the waves are coherent
- Constructive interference is where two waves combine, reinforcing each other (causing amplitude to increase)
- Destructive interference is where two waves cancel (or partially cancel) for a decrease in amplitude
• Phase ($\phi$) is the current stage in the cycle of an oscillation, in degrees (up to 360°) or more commonly radians (360° $= 2\pi\ \mathrm{rad}$)

## 66 Phase difference and path difference:

• Path difference and phase difference can be expressed in the following relationship:
$\Delta \phi =$ $\frac{2 \pi \Delta x}{λ}$ [not given in the exam] where
- $\Delta x$ is the path difference in $\mathrm{m}$
- $\Delta \phi$ is the phase difference in $\mathrm{rad}$

## 67 Standing (stationary) waves:

• When a wave reflects, it undergoes a phase change of $\mathrm{\pi}\ \mathrm{rad}$ (180°)
• A standing wave is formed when two waves of the frequency and wavelength but opposite directions combine
• This causes a superposition where there are stationary points on the transverse resultant wave, nodes
• The points where the amplitude difference reaches a maximum are called antinodes

## 68 Calculating the speed of a transverse wave on a vibrating string:

• The following equation can be used to calculate the speed from tension ($T$) and mass per unit length ($\mu$):
$v =$ $\sqrt{\frac{T}{\mu}}$
• Mass per unit length is measured in $\mathrm{kg}\ \mathrm{m}^{-1}$
• Calculate mass per unit length with $\mu =$ $\frac{m}{l}$ [not given in the exam]

## 69 The effects of length, tension and mass per unit length on the frequency of a vibrating string/wire (CP 7):

• A signal generator and vibration generator can be used to vibrate a wire. By adjusting the frequency to form half of a wavelength of a standing wave, record the frequency for several different lengths of wire
• Because $\lambda = 2l$ for half of a wave, you can use the relationship
$f =$ $\frac{1}{2l}$ $\sqrt{\frac{T}{\mu}}$ [not given in the exam]

## Observations:

• From the above equation, you can determine that:
- Increasing the length will decrease the frequency
- Increasing the mass per unit length will decrease the frequency
- Increasing the tension will increase the frequency

• The received power of a beam of radiation can be measured with its intensity, $I$, with the units $\mathrm{W}\ \mathrm{m}^{-2}$ (watts per square metre)
• The equation $I =$ $\frac{P}{A}$ relates the received intensity, received power and surface area that the beam hits

## 71 Refraction:

• When light meets a boundary between two materials, it refracts (changes direction)
• The normal is a line perpendicular to the material boundary
• If a ray of light crosses a boundary into a medium and speed reduces, the ray will be refracted towards the normal
• If a ray of light crosses a boundary and speeds up, the ray will be refracted away from the normal

## Refractive index:

• The refractive index of a transparent material is the ratio of the speed of light in a vacuum ($3.00 \times 10^8$ $\mathrm{m}\ \mathrm{s}^{-1}$) to the speed of light in the material
• $n =$ $\frac{c}{v}$ where
- $n$ is the refractive index of a material
- $c$ is the speed of light in a vacuum
- $v$ is the speed of light in the material
• Since light travels fastest in a vacuum, a refractive index can never be less than 1
• In air, the refractive index is approximately 1

## Snell's law of refraction:

• Snell's law is $n_1 \sin\theta_1 = n_2 \sin\theta_2$ where
- $\theta_1$ is the angle of incidence (the angle between the normal and incoming light)
- $\theta_2$ is the angle of reflection/refraction (the angle between the normal and outgoing light)
- $n_1$ is the refractive index of material 1
- $n_2$ is the refractive index of material 2

## 72 Critical angle:

• When light travels from an optically more dense material to a less dense material, the angle of refraction can sometimes be 90°. The angle of incidence for this to happen is called the critical angle
• For light going from a material to air, $\sin C =$ $\frac{1}{n}$ where $n$ is the refractive index of the material

## 73 Total internal reflection:

• Total internal reflection is where light is reflected backwards when it hits a boundary with air instead of passing through and refracting
• If the angle of incidence is greater than the critical angle, total internal reflection will occur

## 74 Measuring the refractive index of a solid material:

• Trace the block onto paper with a pencil
• Trace the path of light from a ray box at different angles
• Use Snell's law (see 71) and the angles of incidence and refraction to calculate the refractive index of the material (remember that the refractive index of air is 1)

## 75 Focal length:

• A converging lens is thicker in the middle than the edges and causes parallel rays to converge (meet at a point)
• A diverging lens is the opposite; it is thicker at the edges and causes parallel rays to diverge (spread out)
• Any ray travelling through the centre of a lens will not change direction
• Focal length is a measure of how strongly a lens will converge or diverge light
• If parallel rays pass through a lens, they will converge at the principal focus (or focal point), F, which is one focal length away from the centre of the lens

## 76 Ray diagrams - converging lenses:

• For a converging lens: draw three lines emerging from the top of the object:
- The first should be parallel to the horizontal axis, converging through the lens and passing through the principal focus
- The second should pass straight through the lens
- The third should pass through the principal focus, with its divergence being reduced by the lens so that it leaves the lens parallel to the horizontal axis
• The intersection of these three lines is where the top of the image will be (it'll be upside down) • If the lines don't intersect, trace the second line back towards the left of the page and trace the reflection of the first line in the same direction. The point where they meet is the image location and height

## Diverging lenses:

• Draw the lines as shown below, with the first two being traced back
• The point where the three lines meet is where the top of the image will be ## 77 Lens power:

• The power of a lens is the reciprocal of this - $P =$ $\frac{1}{f}$
• The unit is the dioptre, $\mathrm{D}$ (or $\mathrm{m}^{-1}$ in SI units)

## 78 Lenses in series:

• Several lenses together will provide the same effect as one lens with the sum of the powers, $P = P_1 + P_2 + P_3 + ...$

## 79 Images:

• A real image is produced when a lens converges light which could be projected onto a screen
• In some situations, a virtual image is produced, like in a magnifying glass. Virtual images cannot be projected onto a screen or film because the image is formed on the same side of the lens as the object
• A diverging lens will always produce a virtual image
• If the object is between a converging lens and the principal focus, it will form a virtual image. Otherwise, a converging lens will form a real image

## 80 The thin lens equation:

• $\frac{1}{u}$ $+$ $\frac{1}{v}$ $=$ $\frac{1}{f}$ where
- $u$ is the distance from the object to the lens
- $v$ is the distance from the image to the lens
- $f$ is the focal length
• All three variables are measures of length, so you can easily use $\mathrm{mm}$, $\mathrm{cm}$, $\mathrm{m}$, etc as long as all variables are in this unit
• Hence the name, the equation only works for thin lenses

## Predicting image type:

• The thin lens equation can be used to predict the type of image (virtual or real)
• For this, the real is positive convention could be used:
- The focal lengths of converging lenses are positive
- The focal lengths of diverging lenses are negative
- Virtual distances are negative
- Real objects/images have positive distances

## 81 Magnification:

• magnification $m$ = image height $\div$ object height
• magnification $m$ $=$ distance from image to lens $v$ ($\mathrm{m}$) $\div$ distance from object to lens $u$ ($\mathrm{m}$)

## 82 Plane polarisation:

• If all of the vibrations/oscillations of a group of waves are in a single plane, they are plane-polarised
• Longitudinal waves cannot be plane-polarised
• A polarising filter can be used to polarise a wave. If two are used together, in certain orientations no light will pass through

## 83 Diffraction and Huygens' construction:

• Diffraction is what happens when a wave encounters a gap/slit. It will curve around the corners of the gap: • Huygens' principle states that each point on a wavefront is the source of a new spherical wavefront (secondary wavelet) which is a tangent to all of the other secondary wavelets. This is illustrated in two-dimensions in the following diagrams: • Therefore, when a wave passes through a slit, the path of the secondary wavelet curves it around the corners as shown in the first diagram

## 84 Diffraction gratings:

• A wave will diffract the most if its wavelength is similar to the diameter of the slit
• A diffraction grating is an array of narrow slits which create diffraction with visible light
• The distance, $d$, is the distance from the centre of one slit to the centre of the next
• Due to destructive interference, if a coloured laser is shone through a diffraction grating, a series of dots will be produced
• The diffraction order, $n$, is a way of numbering these dots, counting from the centre. The centre dot is order 0, the dots to the left and right of this are order 1, etc
• $\sin\theta_n$ is the sine of the angle from the between the path that the light has taken to reach the dot and the normal (perpendicular) to the grating
• The equation is $n\lambda = d \sin\theta_n$

## 85 Determining the wavelength of light from a laser using a diffraction grating (CP 8):

• Position a laser a set distance from a wall or screen. Place a diffraction grating directly in front of it
• Measure the distance between the zero order and first order dots on the wall/screen
• Use the diffraction grating equation and trigonometry to derive the following (or just memorise them) and calculate the wavelength:
- $\theta = \tan^{-1}($zero and first order dot distance $\div$ laser to wall distance$)$ [not given in the exam]
- $\lambda = d\sin\theta$ [not given in the exam]

## 86 Evidence for the wave nature of electrons:

• A beam of electrons can be diffracted, with the same properties as with light (see 87)
• This is evidence for the wave nature of electrons

## 87 The de Broglie equation:

• The 'wavelength' (de Broglie wavelength) of a particle is equal to Plank's constant divided by its momentum;
$\lambda =$ $\frac{h}{p}$

## 88 Reflection and transmission of waves at an interface:

• An interface is a boundary between two different mediums
• When a wave moves from one medium to another, some energy is reflected and some is transmitted straight ahead (with some refraction)
• If the densities of the mediums are very different, most energy is reflected. If they are similar, most energy is transmitted

## 89 The pulse-echo technique:

• The pulse-echo technique can be used to measure the distance to an object
• A signal pulse is sent towards the object and the time it takes to return is measured. By knowing the speed of that pulse, $\mathrm{velocity} =$ $\frac{\mathrm{displacement}}{\mathrm{time}}$ can be used to calculate the distance the pulse has travelled (remember to divide by 2 because this distance will be for the round-trip). It is used in applications like radar (microwaves) and sonar (ultrasound)
• A restriction is that only one pulse can be travelling at any one time. Therefore, the number of pulses per second is limited
• The pulse must have a short enough wavelength to not superpose with the reflection

## 90 Wave-particle duality:

• Since light is made up of packets of energy, each packet can be considered like a particle - a photon
• Many things can only be explained by the photon model, even though light is an electromagnetic wave
• Light acting like a wave and a particle is called wave-particle duality
• Until the discovery of the photoelectric effect (see 95), it was not thought that light could act as a particle as well as a wave

## 91 Photon energy and wave frequency:

• $E = hf$ where
- $E$ is the energy of a photon
- $h$ is the Planck constant
- $f$ is the frequency
• This can be used to calculate the energy of a photon from its frequency

## 92 Photoelectrons:

• Under certain conditions, when radiation (e.g. light from the sun) hits a metal, it may release photoelectrons
• For each photon, one photoelectron is released from the surface of the metal. This happens because the electrons in the surface absorb the energy from the photon and if one has enough energy to escape the electrostatic forces binding it to the metal, it will escape

## 93 Work function and threshold frequency:

• The amount of energy required to release an electron (as a photoelectron) is known as the work function ($\Phi$)
• Even with a very high intensity, if the frequency of the radiation is below the threshold frequency of a metal, no electrons are released from it
• If the frequency is above the threshold, photoelectrons will be released, no matter how low the intensity (but the intensity controls the rate of release). This is because intensity is simply the number of photons, not the energy of each
• Work function and threshold frequency are just different ways of expressing the same thing, one can be worked out from the other with $\Phi = hf$ (because $E = hf$, see 91)

## The photoelectric equation:

• If a photon has more energy than the work function, most of the remaining energy will be converted into the photoelectron's kinetic energy, as demonstrated in the photoelectric equation:
$hf = \Phi + \frac{1}{2}mv^2_{\mathrm{max}}$ where
- $h$ is Planck's constant
- $f$ is the frequency of the light
- $m$ is the mass of an electron (from the data sheet)
- $v_{\mathrm{max}}$ is the maximum velocity of the released electron
• In reality, some energy will always be lost due to the electron having to travel back through the metal to the surface, which is why we use $v_{\mathrm{max}}$ instead of $v$

## 94 Electronvolts:

• The electronvolt ($\mathrm{eV}$) is used to measure the energy of a single electron moving through a pd of $1\ \mathrm{V}$. This makes it more suitable than $\mathrm{J}$ for measuring energy on tiny scales
• $1\ \mathrm{eV}$ $= 1.60 \times 10^{-19}$ $\mathrm{J}$ (given in the data sheet)

## 95 Conclusions from the photoelectric effect:

• According to the wave theory of light, multiple electrons in the surface of the metal would gain energy at a near-constant rate and many would be released at once. Increasing intensity would increase the rate of energy gain, and increase the emission rate
• However, the above does not happen, which is why we need the particle model of electromagnetic radiation to explain the photoelectric effect

## 96 Atomic line spectra:

• When electrons are excited (gain energy), they are elevated to a higher energy level. They then lose this energy by emitting a single photon of light and moving to a lower energy level (not always their original level). The amount of energy emitted can be used to calculate the wavelength (using $v = f\lambda$ since transverse waves travel at the speed of light)
• These wavelengths can be plotted onto an emission spectrum, a black background with coloured lines indicating the wavelengths of emitted photons
• Spectra are unique for each element
• High energy levels are closer together because doubling the distance from the nucleus to an electron will reduce the attractive force by a factor of 4