Short Questions (Mixed Topics)

In this question you are asked to make reasoned estimates and assumptions. These must be clearly stated.
a)
(i) On a really dark moonless night the planet Venus is sufficiently bright to produce shadows of objects on the Earth. Explain why the shadows are different to those produced by the Sun and the Moon.
(ii) It is important not to under inflate car tyres - they could overheat and explode. Why is this?
(iii) It is difficult to open small plastic bags for vegetables in the supermarket. A sharp blow along the edge in the plane of the bag usually works. Why?
(iv) How long will it take to hard boil an ostrich egg? Explain how you reached your answer.
(b)

(i) Figure 1.1 shows an equilateral glass prism illuminated by a 100 W laser beam of wavelength $\lambda=600 \mathrm{~nm}$. The refractive index of the glass of the prism is 1.50 at $\lambda=600 \mathrm{~nm}$. The path of the light in the prism is parallel to the base of the prism. The momentum of a photon is $h / \lambda$. Calculate the change in weight of the prism when the beam is switched on.
(ii) Optical tweezers, which are composed of two lasers beams, are able to manipulate small transparent spheres. Explain clearly how this can be done and why at least two beams are needed.
(iii) Small smoke particles in air are seen under a low magnification microscope to move randomly at a speed of $0.10 \mathrm{~mm} \mathrm{~s}^{-1}$. The speed of sound in air is $330 \mathrm{~m} \mathrm{~s}^{-1}$. Estimate the mass of the smoke particles.
(c)
(i) Show that the de Broglie wavelength, $\lambda$, of a particle of mass $m$ moving at a velocity of $v$ where $v \ll c$ is related to the KE of the particle by:

$$\lambda=\frac{h}{\sqrt{2 m \cdot K E}} .$$

(ii) Consider the particle in a small rectangular box with sides of length $a, b$ and $c$. The particle is moving at right angles to the " $b-c$ " plane. Find an expression for the smallest possible energy of the particle.
(Think about the amplitude of the wave at the wall of the box)
(iii) The box now contains many particles, and one " $b-c$ " plane of the box is replaced by a piston. Show that as the length, $a$, is slowly decreased the resulting change in wavelength ensures that Boyle's Law is obeyed
d)

If an isotope of nitrogen is bombarded by alpha particles, protons are observed. The reaction that occurs is:

$${ }_{7}^{14} N+\alpha \rightarrow{ }_{8}^{17} O+p .$$

Calculate the minimum K E of the alpha particle if this reaction is to occur. Explain your answer.

Data

e) A man is fishing in a boat floating in a small pond. To stop him drifting in the wind he takes a large rock, to which a rope is attached from inside the boat, and lowers the rock to the bottom of the pond, fastening the rope to the boat. Explain what happens to the level of the water in the pond?
f) Figure 1.2

Figure 1.2 shows a circuit containing a resistor, coil, neon lamp, switch and 2 V cell. The bulb needs 80 V to light it and goes out if the PD across it drops below 60 V . The switch is closed and nothing is observed. The switch is opened and the Neon light flashes once. Explain.

In this question you are asked to make reasoned estimates and assumptions. These must be clearly stated.
a)
(i) Foolishly I turn on the shower tap when the showerhead is resting on the floor - the flexible pipe thrashes around like a berserk snake. Explain this behaviour.
(ii) The garden hose is 50 m long and connected to a tap. It is then turned on. I partially close the hole emitting the water. The water spurts out faster. Why?
(iii) How could I measure the increased speed of the water - I only have a metre rule?
b)

(i) Figure 1.1 shows a cross-section of the Earth and a typical magnetic field line. A proton from the Sun is shown approaching the Earth. Sketch the path of the proton. What effects may be observed when these protons reach the atmosphere of the Earth.
(ii) A charged particle, mass $m$, charge $q$, travelling in a circle radius $r$ in a magnetic field $\boldsymbol{B}$ at right angles to the plane of the circle emits electromagnetic radiation of frequency $f$, where $f$ is given by

$$f=\frac{B q}{2 \pi m} .$$

Show that this is equal to the rotational frequency of the charged particle rotating in a plane at right angles to the direction of motion.
(iii) The speed of the proton is $2.1 \times 10^{5} \mathrm{~m} \mathrm{~s}^{-1}$ and the magnetic field density, $\boldsymbol{B}$, is $1.0 \mu \mathrm{~T}$.

Calculate the frequency of electromagnetic waves given off.
c) Figure 1.2

Figure 1.2 shows the circuit diagram needed to test a three terminal device, D. It is found that within the working limits of the device that:

$$I_{c}=k I_{b},$$

where $k$ is a constant. The formula holds only if $I_{b}>0$.
(i) Write down a relationship between the three currents.
(ii) How would you modify the circuit so that a small changing $\mathrm{pd} \pm \delta v_{\text {in }}$, (where $\delta v_{\text {in }} \ll 5 V$ ) gave a much larger changing pd, $\delta v_{\text {out }}$, out?
d) According to the kinetic theory of gases for a perfect gas:

$$p=\frac{1}{3} \rho c^{2},$$

where $p$ is the pressure of the gas, $\rho$ is the density of the gas and $c$ is the root mean square velocity of the gas molecules.

Find an expression of the rate of leakage of a gas from a small hole, cross-sectional area $a$, in a cubic container, side $l, l \gg \sqrt{a}$.

A cylinder that contains helium has a diameter of 300 mm and a length of 1.5 m . The pressure of the gas in the cylinder is 10 atmospheres above the ambient air pressure. There is a small hole in the piston, diameter 2 mm . Calculate the rate of fall of pressure in the cylinder assuming that the expansion is isothermal.
e)

Figure 1.3 shows a diagram of a theoretical model of lightening striking damp ground. The constant potential of the small central hemisphere, radius $a$, is $V_{\text {strike. }}$ At a very long distance away, from the region where the lightening strikes the Earth, the potential of a point in and on the Earth is zero. The radius of the hemisphere in Figure 1.3 is $r$ and a thin hemispherical shell, radius $r$, is of thickness $\delta r$. You can assume that the specific resistivity of the damp earth is much less than that of the air and that the current flows radially through the hemisphere. Spherical symmetry is to be assumed on and below the ground. Sketch graphs (i) the current density $\sigma$, (ii) the potential gradient $E$, against $r$ the radial distance from the centre.
Show that for a small change in $r, \delta r$, there is a small change in potential $\delta V$, given by:

$$\delta V=-\frac{K \rho}{2 \pi r^{2}} \delta r$$

where $\rho$ is the specific resistivity of the earth ( $\Omega \mathrm{m}$ ) and $K$ is a constant. What is the physical significance of $K$ ?

Hence or otherwise find an expression for the potential with distance from the centre of the strike and sketch $V(r)$ against $\boldsymbol{r}$.

In one newspaper report it was stated that lightening struck a field in which there were a herd of cows, and most cows were killed. However in another incident lightening struck a rugby pitch whilst a game was in progress. All, but one, of the players were uninjured, though shocked. The injured man recovered. Suggest a physical reason for the different medical effects in the two incidents.

In this question you are asked to make reasoned estimates and assumptions. These must be clearly stated.
a)
(i) Large print heading in the National Geographic Magazine reads: "Olivine is sharper than steel". Comment and explain. Note olivine is a very hard glass-like volcanic rock.

(ii) A motorist gets his car stuck in a ditch Figure 1.1. He is advised to attach a rope to a tree trunk, tighten it, and push the rope with a force $F$ at right angles to the rope. When the car has moved, he repeats the above procedure. Explain why this procedure is more effective than simply pulling the rope without bothering to use the tree trunk
(iii) A razor, whether electric or manual, needs to be sharp in order to be comfortable when used. Why is this?
(iv) A stationary mass of 0.5 kg is hit by a 2 kg hammer that is moving at a speed of $1.3 \mathrm{~ms}^{-1}$. If the collision is perfectly elastic calculate the subsequent speed of the mass. Since the hardness of the hammer does not enter into the calculation one might as well use a rubber hammer to drive in a nail. Comment.
b) Table 1 shows the timetable of a passenger boat on the river Rhine.

Obtain a numerical estimate for the speed of the Rhine current and the speed of the boats in still water. N.B the position marks follow the river Rhine indicating the distance, along its course, from a zero mark.

It is desired to make an analogue electrical circuit that will replicate the behaviour of the mountain, lake, river system, Figure 1.2. Draw an electrical circuit, analogue, and explain how the circuit may be used, to indicate the rate of rise of river level at A due to sudden rainfall on the high ground.
d) The distance from the centre of the Earth to the poles is 21 km shorter than the radius of the equator. A one second pendulum is taken from the equator (at sea level) to the North Pole. Make a rough estimate of its change in period. You may assume for the purposes of your calculation that the Earth has a constant density.
e) When a perfect monatomic gas expands adiabatically (no heat interchange with its surroundings) the pressure and volume of the gas are related by the formula:

$$P_{1} V_{1}^{\gamma}=P_{2} V_{2}^{\gamma},$$

where $\gamma$ in this case is $5 / 3$ and $P_{1}, V_{1}, P_{2}, V_{2}$ refer to the initial and final states of the volume and pressure of the gas . A helium balloon at atmospheric pressure is allowed to rise rapidly to a height such that the external pressure is half atmospheric pressure. The envelope of the balloon is at all times loose, of negligible mass, and regarded as a perfect heat insulator.
(i) Calculate the ratio of the initial to final volume of the balloon.
(ii) If the initial temperature of the gas was 300 K , what will be its new temperature?
f) During the 1998 International Olympiad in Iceland competitors and leaders were surprised to see a cone of snow in the sea on the North West horizon. This was the mountain of Snaefell that is about 2000 m high. However only the top 800 m of the mountain could be seen. The mountain is about 120 km from Rekjavik. Use these values to estimate the radius of the Earth.
g) (i) How do tsunamis arise?
(ii) What is the order of magnitude of the wavelength, speed and amplitude of the wave in the deep ocean?
(iii) How do these change when they approach land?

In this question you are asked to make reasoned estimates and assumptions. These must be clearly stated.

a)

Figure 1.1 shows a photograph of the ice in a birdbath made of stone. Figure 1.2 shows a detailed picture of the pillar of ice shown in Figure 1.1. The birdbath was in the open air and no water fell from above. Suggest an explanation.
b) A recent item of the national news reported ``Scientists say that magnetic bracelets fail to cure aches and pains''. Outline the difficulties in performing a double blind test to settle this question.
c)

Figure 1.3 shows a demonstration of the dark shadow of a gas flame that has particles of salt in it. The lamp is a bright low pressure sodium lamp. Without the salt there is little shadow. Replacing the lamp with a tungsten filament lamp and a yellow filter destroys the black shadow. Explain.
d) On a clear day in summer the normal incident radiation on the surface of the Earth from the Sun is about $1000\ \mathrm{W\,m^{-2}}$. By making suitable simplifying assumptions, determine
(i) the power per square metre of radiation emitted from the surface of the Sun and
(ii) the pressure on the surface of the Sun due to this radiation. You may assume that the energy of a photon is $hf$, frequency $f$, and its momentum is $hf/c$. Calculate how many photons of wavelength 590 nm would be needed to accelerate a proton to a speed equivalent to a temperature of $10^{6}$ K.

a)

Figure1.1

Figure 1.1 shows a picture of the International Space Station (ISS). Figure 1.2 shows a photograph taken by an amateur photographer in Normandy of the ISS silhouetted against the Sun. The ISS measures about 200 m by 150 m .
(i) Use the data given about the Sun and the photograph to find a value for the height of the ISS above the surface of the Earth.
(ii) Comment on the accuracy of the answer.
(iii) Doubt has been cast on the validity of the photograph. Why?
(iv) The surface of the Sun appears to get darker towards the edge. Why?
b)

Fig 1.2

(i) The average P.D.
(ii) The RMS P.D.
(iii) The heat dissipated in a $200 \Omega$ resistor in 100 s .
c)

(i) Fig 1.3 Shows a gas (considered a perfect gas) in an enclosed cylinder. The thermal capacity of the solid components can be considered small and the cylinder and piston are perfect thermal insulators. The piston is pushed in slowly and then slowly allowed to return to its original position. Sketch a graph of the temperature of the gas against the displacement of the piston from its initial position.
(ii) It is important to have the correct pressure of the air in a tyre on a car. What is the problem if the air in the tyre is at too high a pressure? What is the problem if the air is at too low a pressure?
(iii) A sealed insulated cylinder of air full of a perfect gas is allowed to fall freely under gravity (with the long axis vertical). What happens to (a) the temperature of the air (b) the pressure of the air in the cylinder?
d) Furniture removal companies often charge by volume not weight. To reduce the volume blankets are placed in special sealed flexible bags. A vacuum cleaner is attached to the bag and reduces the pressure of the air inside the bag. What happens to the weight of the bag and contents?
e) There is a small danger of a small asteroid, discovered in 2004, hitting the earth in 2029 or 2036. One of the many proposals to avoid this is to paint the asteroid white hence increasing its albedo from $30 \%$ to $80 \%$. This effect is roughly constant over all significant wavelengths. The asteroid has a radius of 160 m and its density is roughly $3000 \mathrm{kgm}^{-3}$. Calculate its change in velocity due to this paint in the course of one day's illumination from the sun.

The momentum of a photon is $h f / c$, the energy of a photon is $h f$.
Albedo is the percentage of photons reflected off an illuminated surface.
The effective temperature of the surface of the Sun is $6400 K$.
Assume that the asteroid when the painting occurs is $2 R_{o}$, from the Sun.
$R_{o}$ is the radius of the orbit of the Earth round the Sun-assumed circular.
f)

20 kV

Figure 1.4 shows a photograph of a man hanging on a $20 \mathrm{kV}, 50 \mathrm{~Hz}$ power cable that supplies an electric train. A newspaper journalist wrote "he survived in spite of 20 kV going through him". Comment. A very simple model of the situation shows a sphere hanging from a wire with a resistance in series. Estimate the maximum current charging the sphere that can be regarded as remote from the ground. Does this model suggest that he will survive? Explain using numerical data where possible.

Data and formula.
(i) Electrical capacity $C$ of an isolated sphere is given by $C=4 \pi \varepsilon_{o} r$ where $r$ is the radius and $\varepsilon_{o}=8.9 \times 10^{-12} \mathrm{Fm}^{-1}$.
(ii) The current needed to electrocute a human being is probably a few mA . The maximum "safe" p.d. in dry conditions is considered to be 80 V .

(i) Normal broadcast receivers have a bandwidth of $\pm 25 \mathrm{kHz}$. Why has the radio telescope receiver a much wider bandwidth?
(ii) Initially the dishes are simply connected to the receiver by equal length cables and the receiver sums the two signals. Show that the signal amplitude has a very slow component that varies given by,

$$I_{\theta}=I_{o} \cos \omega t,$$

at time $t$, around midday. Find the value of $\omega$.
(iii) Sketch a graph showing how the maximum intensity of the signal varies with time between 1000 hrs and 1400 hrs on the $21^{\mathrm{st}}$ of September.
(iv) Sketch on the same graph how the maximum intensity of the signal varies with time between 1000 hrs and 1400 hrs on the $21^{\text {st }}$ of June.
(v) In a modified arrangement a length of cable equivalent to half a wavelength is inserted and removed at a frequency of 600 Hz . After the receiver there is a further amplifier that is so tuned that it amplifies the 600 Hz signal. How will the amplitude of the 600 Hz signal vary with $\theta$ ?
(vi) Why is the modified scheme better?

Note
Bandwidth is loosely defined as the difference between the highest frequency received by a receiver and the lowest frequency received when it is set to receive a certain frequency $f_{0}$. If $2 \Delta f$ is the bandwidth then the highest frequency received is $f_{0}+\Delta f$.

1a This question is about surface tension. You are unlikely to have studied it in depth, if at all; however sufficient help is given to enable you to solve the problems.
(i)

The surface tension $\tau$ is the force per unit length in the surface of a liquid when the liquid is in contact with a solid perpendicular to the line of contact of the surface with the solid. An example is given in Figure 1a1 where a clean glass microscope slide is hung on a balance and lowered into clean water such that the water just touches the base of the slide. The water wets the glass and the surface of the water adjusts so that it climbs and makes a very small angle $(\delta)$ with the surface of the glass. This angle is named the angle of contact and is practically zero for clean water and clean glass. There is tension between the molecules of water and it is that tension that pulls down the glass slide. The length of the slide "into the paper" is $l$ giving the total perimeter approximately $2 l$. Show that

$$\tau=\frac{W_{s t}-W}{2 l \operatorname{Cos} \delta},$$

where $W_{s t}$ is the weight of the slide after immersion and $W$ is the weight of the slide before immersion.

Figure 1 a2 shows a small water droplet, radius $R$. The dark blue is the circular water surface if the droplet is cut in half. The perimeter of the blue circle is the line along which the tension acts, illustrated by the orange arrows. By considering the stability of one hemisphere, or otherwise, show that:

$$p=\frac{2 \tau}{R} \quad \text { Equation 1a2 }$$

where $p$ is the excess pressure in the droplet above the surroundings.

1b

Bubble formation caused by sound waves.

The phenomenon of cavitation is caused by low pressure. This a very difficult problem affecting a ship's propellers, pumps etc. Parts of the liquid reach a lower pressure causing the liquid to superheat locally and cause bubbles to form. These grow and then may suffer catastrophic collapse. The temperature in a collapsing bubble may reach values greater than $10^{4} \mathrm{~K}$. Some have speculated that this might be used to initiate fusion. Sound waves may initiate the formation of bubbles and light may be emitted during their collapse, Figure 1b1. Equation 1a2 shows that the pressure in a very small bubble will be high so that a higher water temperature than normal boiling point may be needed to initiate the bubble formation. A charged bubble will reduce the temperature needed to start the bubble. This is used to show the tracks of charged particles through hydrogen liquid bubble chambers.

Figure 1b3

Figure 1b3 shows a cross section of two large metal plates with a charge of $+q$ and $-q$. The area of a plate is $A$. Ignoring edge effects show that the force of attraction between the plates is given by:

$$F=\frac{q^{2}}{2 \varepsilon_{o} A} .$$

(Hint. The theorem of Gauss states that if a surface A, is found such that the lines of electric field pierce that surface at right angles to the surface then for a simple symmetrical case,

$$E A \varepsilon_{o}=q,$$

where $E$ is the electric field, $\varepsilon_{o}$ is a constant, $q$ is the charge enclosed by the surface and $A$ is the surface area of the surface that surrounds the charge. Any surface that encloses a charge in this way is called a Gaussian surface. Figure $1 b 2$ shows a spherical Gaussian surface around a small isolated charge $Q$. The purple line in Figure $1 b 3$ shows the Gaussian surface enclosing the charge $q$. Consider the force on an elemental charge - $\delta q$ due to the $E$ field it is in.)

Hence or otherwise find the outward pressure on an isolated spherical water drop with a charge $q$, radius $r$. Find the outward pressure due to the electrostatic effect on a bubble in water with a radius of $20 \mu \mathrm{~m}$ whose charge is caused by $10^{9}$ excess electrons. How does this compare with normal atmospheric pressure?

A proton is constrained to move in a circle of radius $r$ at an angular velocity of $\omega$ in a non conducting ring, in the $x, y$ plane Figure 1c1. Note that $r \omega \ll c$ the velocity of light Consider an electron a large distance $x$ away from the centre of the ring, find an expression for the force on the stationary electron. Is the force the same if the electron moves? Explain your answer. Why will a charged particle moving in a circular track in a very good vacuum lose energy?

Figure 1.a.i shows the side on view and Figure.2.a.ii. the end on view of a new type of table tennis net. It consists of a thin piece of wood supported by two lead bars 100 mm long. The thickness of the 'net' is 5.0 mm , the width of the bars is 50.0 mm and the thickness of the lead is 3.0 mm . Whilst playing, a ball strikes the top of the net horizontally with its centre in line with the top of the 'net'. Assume the ball is not revolving when it hits the 'net'. The mass of a table tennis ball is $2.7 \times 10^{-3} \mathrm{~kg}$. The ball bounces back with $20 \%$ of its incident speed. Calculate the fastest speed it can hit the 'net' without the 'net' toppling over. Assume the 'net' and the bars do not slide on the surface of the table.

Q1.b

The Sun shines vertically down on to static seawater, Figure 1.b. The fraction of the power, $P$, absorbed by a thin layer of water thickness $\delta x$ at depth $x$, per square metre of surface is $\sigma P \delta x$. Write down an equation that connects the above variables. The thermal conductivity of water, $\kappa$, relates the rate of heat flow due to a temperature gradient. Write down an equation for this relation. Obtain a differential equation that relates the temperature of the water $\theta$ with the depth of the water $x$. Assume there are no convection currents. What is the condition for convection currents to start?

Q1.c
Figure 1.c.

Figure 1.c. shows a rotating glass cylinder containing a viscous liquid, density $\rho$. The electric motor rotates the cylinder at an angular velocity of $\omega$. Show that the liquid forms a parabolic surface and find the equation of the parabola. Some astronomical telescopes use rotating dishes filled with mercury to produce a reflecting parabolic surface. Show that a parabolic mirror does produce a focus and derive an expression relating the angular velocity of rotation to the focal length of the mirror. What type of advantages and disadvantages does this type of mirror have?

Q 1.d

Figure. 1.d.ii.

A tube of disc magnets was dropped through a coil. The computer recorded the EMF generated in the coil Figure 1.d.i. The coil is wound in a clockwise direction looking down, Figure 1.d.ii. and Figure 1.d.iii. The actual display is shown enlarged in Figure1.d.i. What quantitative and qualitative information can you obtain from the display? The physical length of the tube of magnets was 8.0 cm . The coil had 100 turns and an average diameter of 10 cm and was 1 cm thick.

Q 1e

Figure.1.e shows a photo of a laser beam incident at $90^{\circ}$ on a piece of CD Rom. The laser has a wavelength of 620 nm . The dotted blue lines indicated the directions of the reflected laser light that strikes the paper at the red oblongs shown in the photo. Find the effective reflection grating spacing and hence estimate the maximum number of bits ( 1 or 0 ) that might be stored on the CD Rom.
(You can take measurements from the photo).
Q.1.f. Show that the energy stored in a capacitor $C$ storing a charge of $Q$ is $\frac{1}{2} \frac{Q^{2}}{C}=\frac{1}{2} C V^{2}$ where $V$ is the potential difference between the plates of the capacitor.
Show that the energy stored in a coil is given by $\frac{1}{2} L I^{2}$ where $I$ is the current in the coil and $L$ is the inductance of the coil. Find the energy stored in the electromagnet of the CMS detector of the Large Hadron Collider. The inductance of the large magnet is $L=14 \mathrm{H}$, the normal current is 19.5 kA , giving a magnetic field intensity at this current of 4.0 T . Calculate the energy stored and the average power output if the circuit was broken and the current reduced to zero in 2 ms .

In this question you are asked to make reasoned estimates, assumptions and explanations. These must be clearly stated.
(a)

(i) The planet Venus orbits the Sun and is nearer the Sun than the Earth is. Viewed through weak binoculars Venus can often seem to be crescent shaped. Its size and shape appear to change some time later. Why is this?
(ii) The planet Venus does not appear to change in this way viewed by the naked eye. However, its brightness does vary throughout its orbit. How and why?
(iii) It was realised in the $18^{\text {th }}$ century that if two telescopes on different parts of the Earth measured the position of Venus at the same time, the distance of the Earth to Venus could be determined and hence the scale of the Solar System. The radius of the Earth was quite well known. With the backing of the Royal Society, Captain Cooke performed the experiment when an eclipse of the Sun by Venus occurred (appearing as a black dot crossing the surface of the Sun). Why did he need the eclipse of the Sun? Why did he go to the South Pacific? Explain.
(iv) A long hose is connected to a tap. If the end of the hose is partially covered by a thumb, the water seems to squirt out much faster. Explain.
(v) Alpha radiation is observed and photographed digitally in a cloud chamber. The density of the track, ('light' pixels/per unit length $p_{x}$ ), is plotted against distance from the source ( $x$ ). Explain the shape of the graph in Figure 1.1.
(b)

A beam of electrons is accelerated through a p.d. of 100 V in a high vacuum tube, Figure 1.2. There is no external magnetic field. The beam current is $1 \mu \mathrm{~A}$. The diameter of the cross section of the circular beam is 0.5 mm . Assume the electrons are constant density per unit area across the beam. Calculate:
(i) The maximum velocity of the electrons.
(ii) The velocity of electrons after they have travelled half the distance in the accelerating field.
(iii) The magnetic field a distance of 5 mm from the centre of the beam.
(iv) What happens if the accelerating p.d. is increased to 100 kV ?
(c)

(i) Figure1.3 shows an active four terminal device (it has internal power supplies). $I_{\text {in }}$ is very small and the output behaves as if it is of p.d. $V_{\text {out }}$ and internal resistance $r$, where $r$ is a constant. Draw a very simple circuit that will behave in this way.
(ii) Show that if the resistance $r$ is large compared to the resistance of the subsequent resistors in a complete circuit then the current $I_{\text {out }}$ will be constant and the device will act as a constant current source. What is the condition for a constant voltage source? Suggest a circuit for a constant voltage source.

Explain or comment on the following. More marks will be awarded for numerate or algebraic solutions where possible. Estimate quantities that you may need if these are not given.
i) Light hitting a single small hole will spread out.
ii) Why do you think that stars appear to twinkle to the naked eye when planets do not?
iii) Figures 1.1 and 1.2 show a comet with its tail. On the original photo the tail is so transparent that stars can be seen through the tail. Why does the comet's tail always point away from the sun? Why are there two tails, as seen in Figure 1.2? Explain what orbit the comet might follow.
iv) It seems odd that the planets' orbits round the Sun are nearly circular. Why do you think this is?

In this question you are asked to make reasoned estimates, assumptions and explanations. These assumptions and estimations must be clearly stated.
(a)
(i) The planet Venus orbits the Sun. Viewed through weak binoculars it appears to be crescent shaped. Its size and shape appear to change a few days later. Why is this?
(ii) The planet does not appear to change in size and shape to the naked eye. However, its brightness does vary through its orbit. Why?
(iii) It was realised in the $18{ }^{\text {th }}$ century that if two telescopes on different parts of the Earth measured the position of Venus at the same time, the distance from Earth to Venus could be measured and hence the scale of the Solar system (at that time, the radius of the Earth was well known). Why did they need to do this when Venus was in transit across the Sun?
(b) If a long hose is connected to a tap and the end of the hose is partially covered by a thumb, the water squirts out faster. Explain why.
(c) Alpha radiation from a source is observed in a cloud chamber and the tracks are photographed with a digital camera. The density of the track (number of dark pixels/per unit length, $\frac{\mathrm{d} N_{\text {dark pixels }}}{\mathrm{d} x}$ ) is plotted against the distance from the source, $x$. Explain the graph in Figure 1a.

(d) A beam of electrons is accelerated from rest by a pd of 1 kV . The electric field is uniform. The beam current is $1 \mu \mathrm{~A}$ and diameter of the cross section of the circular beam is 5 mm .
Calculate:
(i) the maximum velocity of the electrons.
(ii) the velocity of the electrons after they had traversed half the distance in the field.
(iii) the magnetic field a distance of 5 mm from the centre of the beam.
(iv) how does the magnetic field vary across the beam?
(v) how does this field effect the electrons?

Answer the following questions briefly, but including physics ideas and explanations.
(a) A container of water in which a rat is swimming is in equilibrium on a balance (as in a ruler balanced on a centre pivot). A string is lowered from a rod attached to the other side of the balance so that it just touches the water. The rat begins to climb up the string. What happens to the balance?
(b) It is possible to observe a bright star (not the Sun) in the sky in the daytime using an ordinary telescope. Explain why the magnification of the eyepiece is crucial to being able to observe such a star, rather more than the objective lens or mirror of the telescope.
(c) The human eye is used to observe a distant point source of light. If the intensity of the light is $I_{0}$ at the eye's pupil, by what factor is the intensity increased on the retina of the eye?
(d) There are six push-button switches. When none of them is pressed, a light bulb remains unlit. When buttons 1, 2 and 5 are simultaneously pressed, the bulb lights. No other combination of buttons can be pressed so as to light the bulb. Draw a diagram of the simplest electric circuit having this behaviour.
(e) The image of the living plant in Figure 1 has ice wings. Suggest how these might have formed.

(f) Neutrons pass readily through a lead block, but are stopped in the same distance in paraffin, water or other compound containing hydrogen atoms. Why might this be?
(g) An ordinary car is driven along a straight road, but it has been fitted with square wheels instead of the usual round ones. What is an estimate of the extra power converted by the engine in order to drive the car at 30 mph (miles per hour)?
(h) A washer, shown in Figure 2, is made of a conducting material of resistivity, $\rho$. It has a square cross-section of width $a$ and thickness $t$, and its outer radius is $2 a$. There is a small slit cut in the washer, and wires of negligible resistance are connected to the exposed faces at the slit. What is the resistance of the washer measured between the pair of connecting wires?
(i) A child stores his crayons in a plastic water bottle which has a narrow neck. When he shakes the bottle to get the crayons out, they jam in the neck. However, when several of the crayons have come out, the rest can be shaken out much more easily. Comment on this effect from a physics viewpoint.

(a). In 1927 a famous conference on Photons and Electrons was held in Bruxelles, the $5{ }^{\text {th }}$ Solvay Conference. The leading physicists of the day, in the rapidly developing field of Quantum Mechanics, attended. A photo of the conference delegates is shown in Figure 1 below. Write down as many names of the physicists from that era who you think might be in the photo. You may include relevant names, even if you may not be able to recognise them in the photo.

(b). In Figs 2a and 2b are two pictures taken of the same lens resting on a sheet of newsprint. The camera used was held many focal lengths above the lens. What can you deduce about the lens from these observations. No calculations are required, but diagrams may be helpful.

(c). Estimate the mass of a patch of mist which forms on a cold window when you breathe on it.
(d). An interesting paradoxical two terminal circuit formed from passive components can be seen in the following example. Two $1.0 \Omega$ resistors and two 1.0 V zener diodes, $Z_{1}$ and $Z_{2}$, are connected in a manner known as a Wheatstone bridge, in Figure 3a. (A typical diode will not conduct in the reverse direction, but a zener diode will conduct in the reverse direction if the potential difference across it is greater than a specified voltage; e.g. an ideal 1.4 V zener diode will conduct perfectly in the forward direction, and will also conduct perfectly in reverse if the reverse the potential across it is greater than 1.4 V .)
(i) In Figure 3a, a +2 V potential from a power supply is connected to point A . What current flows through each branch of the circuit down to earth?
(ii) In Figure 3b, a $\frac{3}{8} \mathrm{~V}$ zener diode, $Z_{3}$, is connected as shown. The same +2 V potential from a power supply connected to point A . What current now flows through the left and right hand arms of the circuit (i.e. paths AB0 and AC0)?
(iii) In Figure 3a, a power supply connected to the arrangement shown. A current of $\frac{1}{2} \mathrm{~A}$ flows into the circuit at A and flows down to earth. What are the potentials at points $\mathrm{A}, \mathrm{B}$ and C ?
(iv) In Figure 3b, the $\frac{3}{8} \mathrm{~V}$ zener diode $Z_{3}$ is connected as shown. The current from the supply is maintained at $\frac{1}{2} \mathrm{~A}$. What are the potentials at points $\mathrm{A}, \mathrm{B}$ and C now?
(v) Comment on what you might see as the paradox in this circuit.

(e). A small bead of mass $m$ can slide along a loop of wire bent in the form of two $90^{\circ}$ curves joined by a straight section, shown in Fig. 4. Gravity does not apply here. As the bead slides, it is only slowed down by a contact frictional force given by $F=\mu N$, where $N$ is the normal force as it goes round the curve. If the bead has an initial speed $v_{0}=1.4 \mathrm{~m} \mathrm{~s}^{-1}$ and $\mu=0.25$, what would be its speed $v$ after it exits the second bend in the wire? The radius of the bead is much smaller than the radius of a bend.

(a). A powerboat instructor has mass 60 kg . With the bow (front) of her boat just touching the harbour, she removes the outboard motor from the stern (back) and intends to carry it along the length of the boat to shore. Her boat is 5 m long with a mass of 180 kg . The outboard motor has a mass of 40 kg . As she walks she realises that she has forgotten to affix the boat to shore. Determine how far from the shore she will be when she reaches the bow of the boat in the cases that:
(i) Frictional forces between the boat and water can be ignored;
(ii) the frictional force between the boat and water is proportional to the velocity of the boat.
(b). The BepiColumbo mission to Mercury launched in October 2018 is only the second satellite destined to enter an orbit around Mercury, 6 years after launch in 2024. Given that, at their closest, Mercury and Mars are approximately equidistant from the Earth, explain why a mission to put a satellite in orbit around Mercury requires significantly more energy expenditure than a similar mission to Mars.
(c). A paper sphere is constructed from thin spherical lunes of paper glued together, as illustrated in Fig. 1, with a photo of the constructed paper sphere shown in Fig. 2 (a). There is a small hole shown in the top end of the sphere through which air can enter or leave. The paper is very thin, but krinkly and stiff. It supports itself as a sphere when resting on the table,

as in Fig. 2 (c).

When a crumpled sphere is bounced gently from one hand to another as in Fig. 2 (b), it gradually takes on a spherical shape, filling with air. Suggest a possible explanation for why it might become filled with more air rather than crumple even more.
(d). A class II 1 mW laser, suitable for use in the classroom, is deemed safe enough in the unfortunate case that it is shone directly in the eye, due to the blink reflex. The diameter of such a laser beam is 6 mm when it reaches the eye, and we can assume that the intensity is uniform across the beam. For an eye focusing the beam on the retina, explain why the beam does not focus to a point, and estimate the factor by which the intensity of the beam is increased on the retina. Make clear any numerical values that you estimate.
(e). Estimate the number of protons that you could pour into an empty teacup before it overflowed. The protons and the cup do not interact electrostatically, but the protons cannot leak through the cup.
(f). The composition of dry air by volume is $78 \%$ nitrogen, $21 \%$ oxygen and $1 \%$ argon, with atomic masses $14.0,16.0$ and 39.9 respectively. The density of dry air is $1.23 \mathrm{~kg} \mathrm{~m}^{-3}$ at $15^{\circ} \mathrm{C}$. Calculate the total kinetic energy of the molecules of $1 \mathrm{~m}^{3}$ of air under these conditions.

Write you answer starting on a new page for this question.
(a) For a modest temperature range, a sample of a metal in the form of a cube of side $\ell_{0}$ and volume $V_{0}$ at room temperature is heated by $\Delta T$. For small temperature changes, its volume expands by a temperature dependent factor such that $V=V_{0}(1+\gamma \Delta T)$, whilst its length expands linearly by a factor given by $\ell=\ell_{0}(1+\alpha \Delta T)$. If the expansion factors are both small, $\gamma=k \alpha$, where $k$ is a numerical factor. Derive a value for $k$.
(b) Two cells have e.m.f.s $\varepsilon_{1}$ and $\varepsilon_{2}$ and internal resistances $r_{1}$ and $r_{2}$.
(i) What are the effective e.m.f.s and internal resistances when these cells are arranged:
i. in series?
ii. in parallel? (assume that the e.m.f.s and internal resistances are now the same)
(ii) A cell has an e.m.f. of 1.0 V and an internal resistance of $50 \Omega$. How many of these cells, and in what arrangement, would be needed to power a $5.0 \mathrm{~W}, 6.0 \mathrm{~V}$ filament lamp?
(c) Shown in Figure 1 below is an image of an electrified fence with ice crystals on it. Give a physics explanation of this effect that you can see.

(d) An electric car carries an accelerometer which also records the distance travelled. A graph of the recording when the car starts from rest is shown in Figure 2. Estimate the velocity of the car after five seconds, and the power per unit mass which is used to accelerate the car at that moment.

(a) The bottle below is filled to the brim with salad dressing made from oil and vinegar. The bottle is shaken before being left to stand, whereupon it is observed that the oil slowly separates and rises to the top. Is the pressure on the bottom of the bottle now the same, greater or less than before? Explain your reasoning carefully.

(b) Table 1 shows the timetable of a passenger boat on the River Rhine.

Obtain a numerical estimate of the speed of the Rhine current and the speed of the boats in still water. N.B. the position markers follow the river Rhine, indicating the distance along its course from a zero marker.
(c) Estimate the work done in inflating a bicycle tyre. Provide your own approximate data and work in SI units.
(d) A sodium salt is heated in a flame at a temperature of 1500 K and the sodium line of wavelength 590 nm is observed. Calculate the spread of wavelengths assuming that all the molecules move with the same speed.
Mass number of sodium is 23 .
(e) A thick walled flask has an outer surface which is a sphere of radius 10 cm and an inner surface which is also spherical and concentric with the outer. When it is filled with an opaque liquid and held up against a diffuse source of light, the flask looks black all over. Is the flask able to contain 1 litre of liquid?

\begin{equation*} 1 \text { litre }=1000 \mathrm{~cm}^{3} \tag{5} \end{equation*}

You are expected to write clear physics, to suggest a model, and to suggest ideas for these effects in Q1. You may find it helpful to bullet point your answers.
(a) In 1998 the McLaren F1 became the world's fastest production car, achieving a top speed of 240 miles per hour. If the maximum useful power developed by the engine is 461 kW , estimate the drag coefficient of the McLaren F1. The drag force is proportional to $v^{2}$, the cross-sectional area of the car, and the density of air. The drag coefficient is then the constant of proportionality.

(b) A resistance $R$ is placed in series with an alternating voltage source, of negligible internal resistance, which has an amplitude $V_{0}$ and frequency $f$. Determine an expression for the average power dissipated by the resistor. If the source can supply a total energy $E$, calculate how long the source lasts before being drained.
(c) A bag of dried sage and onion stuffing is unpacked from its box. It consists of dried organic particles of a range of sizes. On examination it is seen that the larger particles in the mixture are on top. Suggest, based on your knowledge of physics, why this might be so.

(d) A car is left outside overnight in the cold when there is a frost (below zero). Explain why the glass front windscreen of the car can form a thicker layer of ice, compared to that on the side windows of the car.

(a). Air bubbles in water are observed to rise towards the surface at different speeds according to their initial size. Use your knowledge of physics to suggest an explanation.
(b). Walking out of the front door on a cold, frosty morning, two effects are observed outside.
(i) A nail, hammered in to the wooden gate, has a splay of ice crystals growing from it, shown in Fig. 1. With a few brief statements, suggest a possible explanation of this effect.

(ii) The windscreen of a car has a pattern on it, as in Fig. 2, which is not scratches on the glass, but ice. Again, with a few brief statements, suggest a possible explanation of this effect.

(c). Estimate the number of peas (the small, green edible vegetable) that are consumed in the UK in a year.
(d). A voltage amplifier of the type known as an op-amp consists of an integrated circuit constructed from many elements (transistors, resistors, etc.). Its internal workings are unimportant in this question. It is symbolised by a triangle with two inputs and an output, as shown in Fig. 3 . It has a dc power supply to enable it to work, but that is not shown. The signal input terminals are labelled + and -, which is a notation and not the polarity of voltage supplied. They are respectively the

non-inverting input and the inverting input. We will assume a perfectly adjusted op-amp in order to make the following simplified statements.

• If the two inputs are connected together, the output is 0 V .
• If the inverting input ( - ) is set to 0 V then a small voltage, $V_{\text {in }}$, applied to the non-inverting input ( + ) will produce an output voltage $A_{0} V_{\text {in }}$ where $A_{0}$ is the open-loop gain and $A_{0} \gg 1$ ( $A_{0}$ is typically of order $10^{5}-10^{6}$ ). If $V_{\text {in }}$ is not small, then the output will saturate at the supply voltage and not at $A_{0} V_{\text {in }}$.
• If the non-inverting input (+) is set to 0 V , and a small voltage $V_{\text {in }}$ is applied to the inverting input ( - ) then the output voltage will be $-A_{0} V_{\text {in }}$.

Three examples are shown in Fig. 4 . The op-amp is rarely used in this simple manner as the $A_{0}$ is

large but not known precisely. So usually, a small fraction of the output signal is fed back to the input and, as we shall see, this controls the gain of $A_{g}$ of the circuit ( $A_{0}$ is for the op-amp alone).
(i) In the circuit shown in Fig. 5, we can assume that negligible current enters the inverting terminal.
i. Copy the circuit and sketch the path of the current $i$ that flows into the circuit at $V_{\text {in }}$ through the pair of resistors to $V_{\text {out }}$.
ii. Given that the non-inverting input is at 0 V , and that the (small) potential at the inverting input is given by $v=-\frac{V_{\text {out }}}{A_{0}}$, obtain equations for the current $i$ flowing through $R_{\text {in }}$ in terms of $V_{\text {out }}, V_{\text {in }}$ and $A_{0}$, and the same current $i$ through $R_{\mathrm{f}}$ in terms of these quantities.
iii. Eliminating the current $i$ between these two equations, obtain an expression for $A_{g}=\frac{V_{\text {out }}}{V_{\text {in }}}$.
iv. For $A_{0} \gg 1$, obtain an expression for $A_{g}$ in terms of $R_{\mathrm{f}}$ and $R_{\text {in }}$.

(ii) For the non-inverting amplifier of Fig. 6 the non-inverting input is now at $V_{\text {in }}$ whilst the inverting input is again at potential $v$ given by $V_{\text {out }}=A_{0}\left(V_{\text {in }}-v\right)$. A current $i$ flows from the output down to 0 V with no current entering the inverting input.
i. Write down equations for the current $i$ flowing through $R_{1}$ and $R_{2}$ in terms of $V_{\text {out }}$, and the same current through $R_{2}$ in terms of $v$.
ii. Eliminating the current $i$ between these two equations, obtain an expression for $A_{g}=\frac{V_{\text {out }}}{V_{\text {in }}}$.
iii. For $A_{0} \gg 1$, obtain an expression for

(iii) In the following circuit, we assume that $A_{0} \gg 1$ and so $v$ is small enough that the two inputs can be taken to be the same value and $v \approx 0$, and also that no current enters the op-amp inputs (these are the two "Golden Rules" for op-amps).

For the circuit of Fig. 7 a current $i$ flows through $R_{\text {in }}$ and onto the capacitor when switch S is opened at time $t=0$. No current enters the inverting input.
i. Obtain equations for the current $i$ in terms of $R_{\text {in }}$ and $V_{\text {in }}$, and then the charge $Q$ on the capacitor in terms of $C$ and $V_{\text {out }}$.
ii. Obtain a differential equation $\frac{d V_{\text {out }}}{d t}$ which can then be expressed in terms of $V_{\text {in }}, R_{\text {in }}$ and $C$. Solve this for $V_{\text {out }}$ in terms $V_{\text {in }}$.
iii. Sketch a graph of $V_{\text {out }}$ against time, $t$.

(a). An aircraft is moving around the Earth at $900 \mathrm{~km} / \mathrm{h}$ in clear daytime conditions. The pilot maintains a course of constant latitude and contrary to the Earth's rotation direction. Determine the latitude at which the pilot observes the Sun to remain in a fixed position in the sky. The equator is at zero latitude and the North pole at $90^{\circ}$.
(b). A bucket of water is tipped out quickly from the top of a tall tower. It arrives at the ground as droplets in the form of rain. Give an explanation for this effect.
(c). Capacitor $C_{1}$ has a capacitance $6.0 \mu \mathrm{~F}$ and capacitor $C_{2}$ has a capacitance $12.0 \mu \mathrm{~F} . C_{1}$ is charged in a circuit with an emf of 12.0 V , whilst $C_{2}$ is initially charged with an emf of 6.0 V . The two capacitors are then put into a single loop circuit with an open switch, with the same polarities connected together. This is shown in Fig. 1. The switch is then closed, and an equilibrium state is reached.

(i) What is the final pd across $C_{1}$ ?
(ii) What fraction of the original energy stored in the capacitors has been dissipated as heat in the wires?
(d). Three particles of equal mass $m$ are free to move under their mutual gravitational interactions. The particles occupy the same fixed circular orbit of radius $r$, and remain equally spaced throughout the motion. Calculate the orbital period of the system.
(e). The current $I$ through a semiconductor thermistor is measured at a variety of temperatures. The current is given by $I \propto e^{-E_{g} / k_{\mathrm{B}} T}$ where $T$ is in kelvin, $k_{\mathrm{B}}$ is the Boltzmann constant, and $E_{g}$ is the band gap energy in the semiconductor. At $90^{\circ} \mathrm{C}$ the current is 10.90 mA and at $60^{\circ} \mathrm{C}$ it is 3.97 mA .
(i) Find the value of $E_{g}$ (in eV )
(ii) Find the current when $T$ is very large

(a) Half the length of a 1.0 m rope is stretched out in a straight line on a frictionless table. The other half of the rope hangs vertically from the edge of the table. The rope is released from rest. Calculate the speed of the rope as it leaves the table.
(b) Stoodley Pike Monument in West Yorkshire stands 37 m tall at high elevation above exposed moorland (Fig. 1). As such, in winter, it is subject to ferocious wind speeds $v$.

Roughly speaking, the tower is cylindrical in shape and solid throughout with an average diameter of 7.5 m . The local sandstone, which used as construction material, has a density of $3.0 \mathrm{~g} \mathrm{~cm}^{-3}$. A horizontal wind exerts an inertial force $F=\rho A v^{2}$, where $A$ is the projected area of the tower and $\rho=1.3 \mathrm{~kg} \mathrm{~m}^{-3}$ is the density of air.

Assuming that it has negligible foundations (below-ground construction), determine the wind speed that would topple the monument. Suggest the significance of the actual shape of the monument.

(c) The angular momentum $L$ of a particle about a specified axis $A$ is defined as

\begin{equation*} L=r p, \tag{1} \end{equation*}

where $r$ is the perpendicular distance from $A$ to the line of motion of the particle and $p$ is the linear momentum of the particle.
(i) Calculate the angular momentum of a 25 g rubber bung on a 50 cm light string being whirled in a circle 4.0 times per second. Give your answer in the standard (S.I.) unit for angular momentum.
(ii) A cyclist is travelling at $12 \mathrm{~m} \mathrm{~s}^{-1}$. Her front wheel has a mass of 1.3 kg and a diameter of 62 cm . If the wheel does not slip with respect to the ground, find its angular momentum.
(iii) We can crudely model a uniform rod of mass $M$ and length $\ell$ as a system of $N$ equally spaced particles, each of mass $M / N$. If the rod is pivoted at one end and rotates with angular velocity $\omega$, determine an expression for its angular momentum in terms of $N$.
Hint: the sum of the first $n$ square numbers is given by

$$\sum_{i=1}^{n} i^{2}=\frac{1}{6} n(n+1)(2 n+1)$$

(iv) By taking an appropriate limit, find an exact expression for the angular momentum of the rotating rod from part (iii) in terms of $M, \ell$ and $\omega$ only.
(v) Determine the kinetic energy of the rotating rod from part (iii) in terms of $M, \ell$ and $\omega$ only.