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Oxford Teachers Meeting – 2nd

March 2013

Problems and ideas to stretch students are often accumulated over many years and form a

valuable treasure trove to interest and intrigue our students. They can be seen as very personal,

although one frequently finds that good problems simply evolve as fashions change.

References:

1. BPhO papers at www.BPhO.org.uk

2. STEP papers

3. The Physics Teacher (AAPT)

There are numerous problem books that have been published. Currently available at a high level

is Riley: Problems for Physics Students (CUP) (tough)

WG Rees: Physics by Example (CUP)

TE Barrett: Introductory Physics with Calculus (Mastering Problem Solving) (Wiley)

P. Gnädig, G. Honyek, K. F. Riley 200 Puzzling Physics Problems (CUP) (tough)

Assorted Problems

1. Show how one fairly simple mathematical formula is used in some one branch of natural

science. Now translate it into English.

2. A person sitting in a rowing boat drops a large stone from the boat into the lake. Does the

level of the water in the lake rise or fall?

3. How would you attempt to measure the quantity of heat from the sun striking unit area of

the earth’s surface per second?

4. Explain what is meant by stable, unstable, neutral, rotational, translational, dynamic and

static equilibrium. Suggest an example of each.

5. Explain what is meant by thermal equilibrium.

6. Explain why the molecules of the air do not fall to the ground and stay there.

7. A man in a lift holds a cork below the surface of a bucket filled with water. The lift cable

breaks and at the same instant the cork is released. What happens to the cork?

8. A man holds a cork underwater and then releases it. It floats to the surface of the water.

Why?

9. When Galileo dropped a light body and a heavy body simultaneously from the leaning

tower of Pisa they reached the ground at nearly the same time. What is the significance of

this experiment?

10. What determines the speed with which smells travel in still air.

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11. Seen from the moon, the earth has 3.6 times the angular diameter of the sun. What is the

ratio of the densities of the sun and the earth?

12. It has been said that every breath you take contains several molecules from the dying

breath of Julius Caesar. Verify the basis for this statement.

13. Two solid copper spheres of radii 1 and 2 cm are released from rest in free space, their

centres being 20 cm apart. Estimate the velocity with which they eventually collide.

14. What is the source of heat on Io, a moon of Jupiter, which enables it to exhibit volcanic

activity?

15. “What goes up must come down.”

16. Comment on: most substances expand when heated.

17. What is the advantage of using a telescope to look at the stars, if no advantage is gained

over the naked eye?

18. Why can the brightest stars, remote from the sun, be observed through a telescope even

by day?

19. How does a nut and bolt work? i.e. why does it hold things together tightly?

20. Sketch the graphs of voltage, current, resistance and power against time for a light bulb as

it is switched on from cold. Explain your graphs.

21. Why does a tuning fork have 2 prongs not 1?

Estimates 1. Estimate the consumption of electricity in lifting ten people to the top of a ten storey

building.

2. Estimate the difference in pressure between the ground and roof of a bungalow.

3. Estimate the total energy input to a saucepan when it is used to boil an egg.

4. Estimate the minimum diameter of a hot air balloon which is designed to carry one man.

5. Estimate the mass of ice that would have to reach the ocean in order to raise the level by

100 mm.

Fermi Questions 1. How much energy do you radiate in a lifetime?

2. How does the number of grains of sand on a beach compare with the atoms in a beaker of

water?

3. How many cells are there in the human body?

4. How many pizzas will be ordered in London this year?

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Dr Anson Cheung

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St Johns College Admissions Paper

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Density of Nitrogen Nature 46, 512 (1892) [from Scientific Papers, vol IV (Cambridge, UK: Cambridge Press, 1903)]

Lord Rayleigh

I am much puzzled by some recent results as to the density of nitrogen, and shall be obliged if any of your chemical readers can offer suggestions as to the cause. According to two methods of preparation I obtain quite distinct values. The relative difference, amounting to about 1/1000 part, is small in itself, but it lies entirely outside the errors of experiment, and can only be attributed to a variation in the character of the gas.

In the first method the oxygen of atmospheric air is removed in the ordinary way by metallic copper, itself reduced by hydrogen from the oxide. The air, freed from CO2 by potash, gives up its oxygen to copper heated in hard glass over a large Bunsen, and then passes over about a foot of red-hot copper in a furnace. This tube was used merely as an indicator, and the copper in it remained bright throughout. The gas then passed through a wash-bottle containing sulphuric acid, thence again through the furnace over copper oxide, and finally over sulphuric acid, potash and phosphoric anhydride.

In the second method of preparation, suggested to me by Prof. Ramsay, everything remained unchanged, except that the first tube of hot copper was replaced by a wash-bottle containing liquid ammonia, through which air was allowed to bubble. The ammonia method is very convenient, but the nitrogen obtained by means of it was 1/1000 part lighter than the nitrogen of the first method. The question is, to what is the discrepancy due?

The first nitrogen would be too heavy, if it contained residual oxygen. But on this hypothesis, something like 1 per cent would be required. I could detect none whatever by means of alkaline pyrogallate. It may be remarked that the density of the nitrogen agrees closely with that recently obtained by Leduc using the same method of preparation.

On the other hand, can the ammonia-made nitrogen be too light from the presence of impurity? There are not many gases lighter than nitrogen, and the absence of hydrogen, ammonia, and water seems to be fully secured. On the whole it seemed the more probable supposition that the impurity was hydrogen, which in this degree of dilution escaped the action of the copper oxide. But a special experiment seems to preclude this explanation.

Into nitrogen prepared by the first method, but before its passage into the furnace tubes, one or two thousandths by volumes of hydrogen were introduced. To effect this in a uniform manner the gas was made to bubble through a small hydrogen generator, which would be set in action under its own electro-motive force by closing an external contact. The rate of hydrogen production was determined by a suitable galvanometer enclosed in the circuit. But the introduction of hydrogen had not the smallest effect upon the density, showing that the copper oxide was capable of performing the part desired of it.

Is it possible that the difference is independent of impurity, the nitrogen itself being to some extent in a different (dissociated) state? I ought to have mentioned that during the fillings of the globe, the rate of passage of gas was very uniform, and about 2/3 litre per hour.

http://web.lemoyne.edu/~giunta/rayleigh0.html

Find out why there was this discrepancy between the results obtained through these two

different techniques.

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Question 1 (j) (from BPhO Paper 1 2010)

A star with a diameter larger than that of the Sun can collapse to form a neutron star which has a

diameter of only a few kilometres. As the core collapses to form a neutron star, its electrical

conductivity becomes very high. This results in the magnetic field lines being trapped in the collapsing

matter so that the field lines become denser and the field strength increases.

If the magnetic field pattern is similar to that of a bar magnet, as shown in Fig. 1 below, then as the star

radius r decreases, its cross sectional area decreases, and the density of the field lines at the equator

changes as 1/��.

If the radius decreases from 1.4 x 106 km to 10 km, and the initial magnetic field strength of the star at

the equator is 10-2 T, calculate the magnetic field strength of the neutron star at the equator.

Fig. 1

(4 marks)

Question 5 (from BPhO Paper 1 2011)

Johannes Kepler used Tycho Brahe’s detailed observations on planetary motion, made without the

use of telescopes, to determine the elliptical orbits of the planets. He also ascertained that the square

of the period of orbit T is proportional to the cube of R (Kepler’s Third Law). The radial distance R

for a planet is the simple arithmetic average of the closest distance of approach to the sun, Rmin and the

furthest distance from the sun, Rmax.

(a) Sketch a diagram of a planetary orbit, marking on it Rmin and Rmax. [1]

(b) From the statement above, write down two equations, the first one relating T and R with a constant of proportionality k, and a second equation relating R, Rmin and Rmax.

[2]

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(c) The average distance of the earth from the sun is defined as 1 Astronomical Unit (1 AU). Determine the value of k for part (a), including units. (The period T can be measured in years).

[1]

(d) Halley’s Comet also orbits the sun and so the value of k is the same as in (b). Its period of orbit is 75.3 years. Determine the value of R for its orbit about the sun.

[1]

(e) The closest distance of approach to the sun for the comet is 0.585 AU, when it is visible to the naked eye. Calculate the furthest distance of the comet from the sun.

[1]

(f) Its speed is 70.6 km s-1 at closest approach to the sun. Is the speed greater or smaller than this at the comet’s furthest distance from the sun? Give a reason for your answer.

[3]

(g) As a man made satellite orbits the earth, there is always a point on the earth directly below it. This point follows the path of a satellite’s orbit and is plotted on a map of the earth, as shown below in figure 4. Describe or sketch the satellite’s orbit i.e. how it is oriented about the earth, and its shape.

[3]

(h) This orbit is known as a Molnya orbit and is used for some spy satellites. Apart from the obvious feature that it covers Russia and the USA, what is its advantage?

[1]

Figure 1. Path of a satellite in a Molnya orbit of around the earth.

(13 marks)

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Question 4 BPhO A2 Challenge 2012

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Question 2 A2 Challenge 2010 (Sept/Oct 2009)

Q2. An insight into the solution of a problem can often be made by looking at the dimensions

of the relevant physical quantities.

An example is the simple pendulum, in which a mass at the end of a light inextensible

string swings from side to side. The period of the swing, T, could be determined by

resolving the forces acting on the mass.

Alternatively, if we suggest that the relevant factors affecting the period are the length

of the string, l, the mass, m, and the strength of the gravitational field, g, then T must

depend upon the product of powers of the quantities l, m and g.

i.e cba gmlconstT ×××= (1)

The dimensions of l, m, g are given by

[l] = L, [m] = M, [g] = LT -2

and [T] = T. (2)

So then we can write the equation in terms of dimensions as

T = La × M

b × (LT

-2)c (3)

The powers of T, L, M on each side of the equation must be the same.

For T: T1 = T

-2c so that c = -½ .

For M: M0 = M

b so b = 0.

For L: L0 = L

a + c, so that a = ½ .

This results in the equation g

lconstT ×= . A full analysis of the forces will enable you

to deduce that const = 2p.

Now solve the following example in the same manner: when a river floods, large

boulders can be left behind on the riverbed, and yet the speed of the river does not

change very much (the slope remains the same). Assume that the mass of boulders

swept along by the river, m, depends upon the speed of the river, v, the gravitational

field strength, g, and the density of the boulder, ρ.

Write down:

i. The form of the equation relating m to v, g, ρ as exemplified in equation (1)

(1 mark)

ii. The dimensions of each of the quantities m, v, g, ρ, as in the set of

equations in (2).

(4 marks)

iii. The dimensional equation for these quantities, as in (3) and solve to obtain

the powers a, b, c.

(4 marks)

iv. The final equation for m in terms of the variables.

(2 marks)

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v. From your solution, explain why a flooding river which has a small change

of speed has a significant effect on the size of the boulders that are swept

along.

(1 mark)

[Q2: 12 marks]

Question 3 A2 Challenge 2009 (Sept /Oct 2008)

Q1. A typical man has a mass of 70 kg and the minimum cross sectional area of the bones in

each leg is approximately 5.0 x 10-4

m2. The compressive breaking stress of bone is

approximately 1.0 x 107 Nm

-2. If the man stands with his weight equally supported by

each leg, calculate the following:

(a) The maximum stress in his leg bones

(2 marks)

(b) The ratio of the maximum stress to the breaking stress

(1 marks)

If a giant grew to such a size that each of the linear dimensions of his body were a

factor of nine larger than those of a typical human being, calculate:

(c) The mass of the typical giant

(2 marks)

(d) The new ratio of the maximum stress to the breaking stress

(2 marks)

(e) Briefly explain whether the giant would be able to stand on one leg.

(1 marks)

[8]

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Trinity Atomic Weapon Test

The radius R of the fireball of an atomic bomb depends on the density, ρ, of the atmosphere, the

energy, E, of the bomb, and the time, t after the explosion. Derive a dimensionally homogeneous

relationship between these variables.

Images from the Trinity Atomic explosion are printed below. Plot a suitable graph using data from the

images in order to estimate the energy of the explosion. We assume that the constant of

proportionality ≈ 1 Density of air = 1.2 kg m-3

1 tonne TNT ≈ 4 x 109 J

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y = 0.3665x + 2.688R² = 0.9965

1.800

1.900

2.000

2.100

2.200

2.300

2.400

-2.400 -2.200 -2.000 -1.800 -1.600 -1.400 -1.200 -1.000

Lo

g(R

/m)

Log(t/s)

Trinity Explosion

R5 = 4.2 x 1013 t2 (+ 6 x 109)

R2 = 0.996

-5.0E+100.0E+005.0E+101.0E+111.5E+112.0E+112.5E+113.0E+113.5E+114.0E+11

0 0.002 0.004 0.006 0.008 0.01

R5

/ m5

t2 / s2

Trinity Explosion

t/ms picture

diameter/cm 100m

scale/cm R/m t/s t^2 /s^2 R^5 /m^5 log(t/s) log(R/m) 6 4.6 3 76.7 0.006 0.000036 2.65E+09 -2.222 1.885

16 10.3 5 103.0 0.016 0.000256 1.16E+10 -1.796 2.013 25 12.3 4.9 125.5 0.025 0.000625 3.11E+10 -1.602 2.099 53 18 5.4 166.7 0.053 0.002809 1.29E+11 -1.276 2.222 62 17.5 4.9 178.6 0.062 0.003844 1.82E+11 -1.208 2.252 90 19 4.7 202.1 0.090 0.0081 3.37E+11 -1.046 2.306

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Star Formation

• Dimensional analysis has applications throughout physics. Physicists like to range over their

territory, flexing their physics muscles. Here is a question, continuing our problem solving

approach, which brings in at least five traditional topic areas

• A star of uniform density is formed from a very large cloud of gas, which is spread out over a

very large radius compared to the radius of the star which is eventually formed. The loss of

gravitational potential energy appears as thermal energy of the star (random k.e. of the

particles. Average stars radiate due to fusion processes going on internally. But how does this

start?

• Our model looks at the gravitational potential energy lost in forming the star and estimates

the thermal energy of the protons (hydrogen atoms) in the star as a result of this energy

change. Do the “hot” protons get close enough to fuse, and then start the exothermic

(nuclear) reaction?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

• The force of attraction between two point masses is given by Newton's Universal Law of

Gravitation. This is expressed as2

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r

mGmF = . So in any expression involving gravitational

attraction, we expect the dimensioned constant G to be present.

• By dimensional analysis, determine the gravitational potential energy, U, lost by the cloud of

dust collapsing (from a large distance which we will assume to be almost infinite) to form a

star of mass M and radius R. (you need to know the dimensions of G).

The constant of proportionality for a uniform density star is ⅗, but we will just use 1.

• If the sun is an average star, how many hydrogen atoms (protons) does it contain?

What is the average k.e. of a proton in the star, Uav?

• The temperature of a gas is proportional to the average k.e. of the molecules.

i.e T α uav

or Uav = const. T

For air, 1 mole of air is 30 g, average speed of a molecule is 500 m/s (a little over the speed of

sound, the temperature is 300 k and NA = 6 x 1023

mol-1

.

Calculate the value of the constant and use the value to determine the temperature of the star.

• Two protons approach each other at high speed. As they approach closer and closer, the

electrostatic p.e. gain reduces their k.e. until they stop and then they move apart again (if

there paths are directly in line). The electrostatic p.e. is given by closesto r

eU

2

4

1

πε=

• Calculate the closest distance of approach of two hydrogen nuclei which are coming towards

each other at the average k.e. of the hydrogen atoms.

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• The range of the strong nuclear force which holds nucleons together is similar to the

diameter of the nucleus. What is the range?

• Do the protons approach each other close enough for them to fuse (and hence release

energy)?

• Calculate the de Broglie wavelength of an average k.e proton in the star.

• How has the sun managed to ignite and release energy in a fusion reaction? Give two factors

that we have not taken into account.

1. ..…….……………………………………………

2. ….……………………………………………….

Mass of sun = 2.0 x 1030

kg

G = 6.7 x 10-11

N m2 kg

-2

Radius of the sun = 6.7 x 108 m

Mass of a hydrogen atom = 1.67 x 10-27

kg

charge on an electron = 1.6 x 10-19

C

h = 6.6 x 10-34

Js

229109.84

1 −= CmNxoπε

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STEP 1996

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Solutions Question 1 (j) (from BPhO Paper 1 2010)

Field increases as 1/r2. So the calculation is

T100.2

1096.11010

)104.1(

)10(10

82

12222

26

2

2

2

21

22

2

1

×=

×××=×

=

=

−−

−

B

B

B

R

R

B

B

Right idea �, numbers substituted ��, answer �

(4 marks)

Question 5 (from BPhO Paper 1 2011)

(a)

�

(b) T2 = k R

3 �

2)minmax RR

R+= �

(c) 12 = k 1

3

k = 1 year2 AU

-3 �

(d) 75.32 = 1 x R

3 giving R = 17.8 AU �

(e) 17.8 = ½ (0.585 + Rmax)

Rmax = 35.1 AU �

(f) The speed is slower at the distant point �

sun

Rmin Rmax

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Total mechanical energy is constant �

Increase of gravitational pe accompanied by loss of ke. �

(g) The satellite is in a highly elliptical orbit �

(with the centre of the earth at one focus) not centred on the earth �

plane of the ellipse tilted at a large angle �

with respect to the plane of the equator.

(h) At the furthest point of the orbit, when the satellite is moving slowest / spends more time

(�), the region below the satellite is Russia or the USA

[13 marks]

Question 5 (from BPhO Paper 1 2011)

(a) 2 x 2 = 4 �

[1]

(b) Beginning of 1935 1 cm

1936 4 cm

1937 42

1938 43

1939 44

1940 45 cm answer; �

clear working – table/calculation; �

[2]

(c) 1 x 103 cm or 1 x 10

1 m �

[1]

(d) Beginning of 1941 40 m = 4 x 10 m

1942 160 m = 42 x 10 m

1943 640 m = 43 x 10 m

Molnya satellite orbit plane

equatorial plane

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After n years from 1941 the volume thickness will be 4n x 10 m

The velocity of the front page will be 4n x 10 ÷ 6 months

Year when this is equal to the speed of light is when

3 x 108 =

4nx10

364x3600x24/2 �

4.73 x 1014

= 4n

Taking logs to base 10

14.67 = n log 4 �

n = 24.4

So the year will be 1964 �

[4]

[Q4: 10 marks]

Question 2 A2 Challenge 2010 (Sept/Oct 2009)

i) cba gvconstm ρ×××= �

(1 mark)

ii) The dimensions are given by

[v] = LT -1

, [g] = L T -2

, [r] = ML-3

���

So then we can write

M = (LT -1

)a × (LT

-2)b × (ML

-3)c �

(4 marks)

iii) The powers of M, L, T on each side of the equation must be the same.

For M: M1 = M

c so that c = 1 �

For L: L0 = L

a + b -3c so that a + b -3c = 0 �

For T: To = T

-a – 2b so that -a – 2b = 0 �

b = -3 and a = 6 �

(4 marks)

iv)

3

6

36

g

vconstm

or

gvconstm

ρ

ρ

=

×××= −

all correct: ��

(2 marks)

v) The high power of v means that for a small increase in v there will be a relatively large

increase in v6 . owtte �

(1 mark)

[12]

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Question 3 A2 Challenge 2009 (Sept /Oct 2008)

Star Formation

R

GMU

2

=

N = 2 x 1030 / 1.67 x 10-27 kg = 1.2 x 1057 hydrogen atoms (protons)

Uav = (6.7 x 10-11 x 4 x 1060 /6.7 x 108 ) π 1.2 x 1057

= 3.3 x 10-16 (≈ 2 keV)

constant = ½ x (0.03 / 6 x 1023) x (5002 / 300) = 2 x 10-23 J k-1

Temperature of star ≈ 3.3 x 10-16 / 2 x 10-23 ≈ 1.6 x 107 k

Rclosest = 9 x 109 x (1.6 x 10-19)2 / (2 x 3.3 x 10-16) = 3.5 x 10-13 m

λde Broglie = h/p and mUp 2=

λde Broglie = 6.6 x 10-34 / (2 x 1.67 x 10-27 x 3.3 x 10-16)½ = 6.3 x 10-13 m

Sausages question: 50% overcharged!

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Entrance Paper June 1983 (age 13)

O-level 1986

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Nuffield O-level June 1986

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O&C A level June 1988

O&C Special Paper 1988

Oxford Entrance Paper November 1987