BPHO paper2 Section1
catalogue
2009.........................................................................................1
2010.........................................................................................5
2011.........................................................................................11
2012.........................................................................................17
2013.........................................................................................23
2014.........................................................................................30
2015.........................................................................................36
2016.........................................................................................42
2017.........................................................................................48
2018.........................................................................................54
2019.........................................................................................61
2020.........................................................................................68
BRITISH PHYSICS OLYMPIAD
British Physics Olympiad 2010
Paper 2
13th November 2009
Section 1
Important Constants
Speed of light
c
3.00 x 108
ms-1
Planck constant
h
6.63 x 10-34
Js
Electronic charge
e
1.60 x 10-19
C
Mass of electron
me
9.11 x 10-31
kg
Permittivity of a vacuum
ε0
8.85 x 10-12
Fm-1
Gravitational constant
G
6.67 x 10-11
Nm2kg-2
Acceleration due to free fall
g
9.81
ms-2
Mass of Earth
ME
5.9700 x 1024
kg
Mass of Moon
MM
7.35 x 1022
kg
Radius of Earth
RE
6.38 x 103
km
Radius of the Moon
RM
1.74 x 103
km
Earth – Moon distance
REM
3.84 x 105
km
1
Q1
(a)
Molten lead, mass 3.0 kg and melting point 600 K, is allowed to cool down until it has solidified. It is found that the temperature of the lead falls from 605 K to 600 K in 10 s, remains
constant at 600 K for 300 s, and then falls to 595 K in a further 8.4 s.
Assuming that the rate of loss of energy remains constant, and the specific heat of solid lead is
140 Jkg -1K-1, calculate:
(i)
(ii)
(iii)
the rate of loss of energy from the lead
the specific latent heat of fusion
the specific heat capacity of liquid lead
[4]
(b)
One end of a rope is fixed to a vertical wall, making an angle of 300 with the wall, and the other
end is pulled by a horizontal force of 20N.. What is the mass of the rope?
[3]
(c)
A compact disc, CD, contains 650 MB (megabyte) of information. Estimate the area of one bit
on the CD (8 bits =1 byte).The inner diameter of a CD is 4.4 cm and the outer diameter is
11.0 cm.
[3]
(d)
A camera, which has a lens of diameter of 2.0 cm, takes a photograph of a 100 W filament lamp
from 100 m away. If 1.0 % of the energy is emitted as light and the exposure lasts 0.015 s,
estimate the number of photons that strike the film, assuming all have a wavelength of 600
nm.
[5]
(e)
A tungsten filament rated at 250 W, 230 V, has a resistance of 20 Ω at 273 K. Its mean
temperature coefficient of resistance is 5.0 x 10-3 K-1.
What is its working temperature?
[4]
(f)
A monochromatic light wave of amplitude a is incident normally on a Polaroid sheet at an
angle θ to the plane of polarization. What is the amplitude of the transmitted wave? The
intensity of an unpolarized light beam incident normally on the Polaroid sheet is I.
What is the intensity of the transmitted light?
[3]
(g)
Determine an expression for the escape velocity of a body of mass m from a planet of mass M
and radius R. Why do some planets possess an atmosphere and others do not?
[4]
(h)
A plane mirror rotates about a vertical axis in its plane at 35 revs s-1 and reflects a narrow beam
of light to a stationary mirror 200 m away. This mirror reflects the light normally so that it is
again reflected from the rotating mirror. The light now makes an angle of 2.0 minutes with the
path it would travel if both mirrors were stationary. Calculate the velocity of light.
[3]
2
2
(i)
A battery, internal resistance r and emf E, drives a current of 3.0 A round a circuit consisting of
two 2.0 Ω resistors connected in parallel. When these resistors are connected in series the
current is 1.2 A. Calculate:
(i)
(ii)
(iii)
(j)
(l)
(m)
Show that the rocket does not leave the pad immediately.
Calculate the time interval between ignition and lift off.
[4]
What properties do molecules of an ideal gas possess? How do they differ from those of a real
gas? What macroscopic properties distinguish an ideal gas from a real gas?
[3]
-1
A boat can travel at a speed of 3.0 ms on a pond. A boatman wants to cross a river by the
shortest path. In what direction should he row, with respect to the bank, if the speed of the
water is 2.0 ms-1.
Explain, using a diagram, which path he should take if the water speed is 4.0 ms-1.
[7]
A small positively charged ball B, mass m, is suspended by an insulating thread of negligible
mass. Another identical ball, with the same charge, is moved slowly, from a great distance, to
the original position of B. B rises by a distance h.
(i)
(ii)
(n)
[4]
A rocket stands vertically on its launch pad. Prior to ignition the mass of the rocket and its fuel
is 4.1 x 103 kg. On ignition gas is ejected from the rocket at a speed of 2.5 x 103 ms-1 relative to
the rocket and the fuel is consumed at a constant rate of 16 kgs-1.
(i)
(ii)
(k)
the emf of the battery E
the internal resistance of the battery r
the power dissipated in a resistor in each case, Pp and Ps respectively
What is the final tension in the thread?
Obtain an expression for the work done, W, and show that it is independent of
the charges. (Hint: identify similar triangles)
[8]
Two identical parallel insulating plates, each of area A and having a charge +Q, are separated
by a distance x. Sketch the field lines between the plates:
(i)
for the system
(ii) for a capacitor with identical dimensions having charges Q and -Q
(iii) Deduce, giving an appropriate explanation, the ratio of the forces between the
plates in (i) and (ii).
(iv) Obtain an expression for the energy, E, of the capacitor using an appropriate
expression for its capacitance as a function of x.
(v) Determine the force between the plates of the capacitor either directly or by
comparison with the energy of a physical system with the same dependence on
the parameter ‘x’.
[7]
3
3
(o)
A student rotates a whistle, of frequency 256 Hz, at the end of a 1.2 m length of string, at 3.0
revs. per sec. Derive the extreme values of the frequency experienced by an observer at some
distance from the student. The velocity of sound is 340 ms-1.
[7]
(p)
Electrons with speed v, much less than c, are injected into a uniform magnetic field of flux
density B at an angle θ to the field.
(i) Show the motion is periodic.
(ii) Determine the time for one period, T.
(iii) Determine the distance travelled, L, along the direction of the field in time T .
Hint: Consider the motion along and perpendicular too the field .
[8]
End of Section 1
4
4
BRITISH PHYSICS OLYMPIAD
British Physics Olympiad 2011
12th November 2010
Paper 2
Section 1
Instructions
Questions: Any or all parts of Section 1 can be attempted. However students are not expected
to complete all parts of Section 1 as only 40 marks are available.
Time: It is recommended that students spend 1 hour 15 minutes on this section.
Marks: There are 78 marks available; however only a maximum total mark of 40 will be
awarded. Therefore students need to plan which questions they will attempt in the time
recommended.
Question answers
Answers can be written on loose paper or examination booklets. Graph paper and a formula
sheet should be available.
Students should ensure their name and school is clearly written on their answer sheets.
Sittings
Section 1 and Section 2 of Paper 2 may be sat in one session of three hours. Alternatively, the
paper may be sat in two sessions, 1 hour 15 minutes for Section 1 and 1 hour 45 minutes for
Section 2. If the paper is taken in two sessions, students should not receive Section 2 until the
start of the second session, and should not be allowed to return to their answers to Section 1.
1
5
BRITISH PHYSICS OLYMPIAD
British Physics Olympiad 2011
12th November 2010
Paper 2
Section 1
Important Constants
Speed of light
c
3.00 x 108
ms-1
Planck constant
h
6.63 x 10-34
Js
Electronic charge
e
1.60 x 10-19
C
Mass of electron
me
9.11 x 10-31
kg
Permittivity of a vacuum
ε0
8.85 x 10-12
Fm-1
Acceleration due to free fall
g
9.81
ms-2
Gravitational constant
G
6.67 x 10-11
Nm2kg-2
Avogadro’s number
N
6.02 x 1023
Mol
Mass of Earth
ME
5.9700 x 1024
kg
Mass of Moon
MM
7.35 x 1022
kg
Radius of Earth
RE
6.38 x 103
km
Radius of the Moon
RM
1.74 x 106
m
Earth – Moon distance
REM
3.84 x 108
m
3
6
Q1
a) Gas is contained in a tank at a pressure of 10 atm and a temperature of 15oC. If one half
of the gas is withdrawn and the temperature is raised to 65oC, what is the new pressure
in the tank?
[3]
b) In Figure 1.b, what is the value of the resistor R3, in terms of the resistances R1 and R2,
expressed in its simplest form, if the total resistance across AB is equal to R1 ?
[2]
Figure 1.b
c) What is an electric field line? Sketch the field lines due to two charges 3Q and (-Q).
[5]
d) A uniform cable has a mass of 100 kg and is suspended between two fixed points A and
B, at the same horizontal level, (Figure 1.d). At the support points the cable makes
angles of 30o.
Figure 1.d
What is :
(i) The force exerted on each support?
(ii) The tension in the cable at its lowest point?
[5]
4
7
e) When the Sun is directly overhead a narrow shaft of light enters an ancient temple
through a small hole in the ceiling and produces a light spot, 10m below, on the floor.
(i) At what speed does the spot move across the floor?
(ii) If a mirror is placed on the floor to reflect the beam, at what speed will the
reflected spot move across the ceiling?
[4]
f) The potassium isotope 42K19 disintegrates into 42Ca20.
(i) What are the likely source/s of radiation produced?
(ii) How many protons, neutrons and electrons are present in an atom of the
daughter nucleus 42Ca20?
[3]
g) A thin film of glass, refractive index 1.52, and thickness 0.42 µm is viewed by reflection
with white light at normal incidence. What visible wavelength is most strongly reflected?
[6]
h) A 50 kg ball is attached to one end of a 1.2 m chord that has a mass of 0.13 kg and
initially hangs vertically in equilibrium. The other end of the chord is attached to a ring
that can slide on a smooth horizontal rod, (Figure 1.h). A horizontal blow is delivered
to the chord which excites its fundamental mode. Assume the ball remains stationary as
the chord vibrates.
Figure 1.h
(i) What is the frequency, f , and period, T , of the fundamental mode?
(ii) What is the amplitude, A, of the ring if its maximum velocity is 15 ms-1 ?
(iii)If, initially, the ball is not stationary, determine its natural period, T0.
(iv) Determine the ratio ( T / T0). Is the original assumption justified?
[11]
5
8
i) A sphere, mass M and speed u, collides elastically head-on with an identical sphere of
mass m which is initially at rest. After the collision the masses M and m have
respectively speeds, in the direction of u, of v and w .
(i) Prove or verify that
u+v=w.
(ii) If R = (u-v)/ u, prove R = mw/(Mu) .
(iii)Express R in terms of M and m only.
[8]
j) Two identical plastic balls of mass 5.00 g are charged to +1.00 µC and suspended
from a fixed point by massless non-conducting threads, each of length 1.00 m. Verify
that the angle between the threads is 41.00 .
[7]
k) The height of mercury, density 1.35 x 104 kg m-3, in a barometer, (Figure 1.k), is 75.9
cm, at 15 0C. The height of the evacuated space in the barometer is 8.0 cm. The internal
diameter of the barometer is 6.5 mm. A small amount of nitrogen is introduced into the
this space and the mercury level drops to 62.2 cm. Determine the mass, m , of nitrogen
present.
[7]
Figure 1.k
6
9
l) A battery consisting of two cells, in series, each of emf E, is used to charge a capacitor,
capacitance C.
(i) What is the energy of the charged capacitor?
(ii) How much energy has been lost?
(iii) If the capacitor is charged in two stages, first with one cell and then with
two cells, determine the energy lost. Comment on the result.
[8]
m) Two masses of 0.90 kg and 1.10 kg are hung vertically from identical springs on a
common support each with force constant 39.48 Nm-1. Both are released
simultaneously from a position of maximum extension to describe simple harmonic
motion. Calculate:
(i) The frequencies of the two masses.
(ii) The beat period and frequency.
[4]
n) The tangential frictional force produced by a band break on a rotating metal drum of
circumference 0.25 m is 20 N. The mass of the drum is 0.40 kg and its specific heat
capacity is 0.35 kJ kg-1K-1. Calculate the number of complete revolutions required to
increase its temperature by 5.0 K.
[3]
o) If the atmosphere is assumed to be composed of a layer of air of uniform density, 1.23
kg m-3, calculate its height if it produces a pressure of 1.01 x 105 Pa at the Earth’s
surface.
[2]
End of Section 1
7
10
BRITISH PHYSICS OLYMPIAD
British Physics Olympiad 2012
11th November 2011
Round 1
Section 1
Instructions
Questions: Any or all parts of Section 1 can be attempted. However students are not expected
to complete all parts of Section 1 as only 40 marks are available.
Time: It is recommended that students spend 1 hour 15 minutes on this section.
Marks: There are 76 marks available; however only a maximum total mark of 40 will be
awarded. Therefore students need to plan which questions they will attempt in the time
recommended.
Question answers
Answers can be written on loose paper or examination booklets. Graph paper and a formula
sheet should be available.
Students should ensure their name and school is clearly written on their answer sheets.
Sittings
Section 1 and Section 2 of Paper 2 may be sat in one session of three hours. Alternatively, the
paper may be sat in two sessions, 1 hour 15 minutes for Section 1 and 1 hour 45 minutes for
Section 2. If the paper is taken in two sessions, students should not receive Section 2 until the
start of the second session, and should not be allowed to return to their answers to Section 1.
1
11
Important Constants
Speed of light
c
3.00 x 108
m s-1
Planck constant
h
6.63 x 10-34
Js
Electronic charge
e
1.60 x 10-19
C
Mass of electron
me
9.11 x 10-31
kg
Mass of proton
mp
1.67 x 10-27
kg
Acceleration due to free fall
g
9.81
m s-2
Gravitational constant
G
6.67 x 10-11
N m2 kg-2
Radius of Earth
RE
6.38 x 106
m
Mass of Earth
ME
5.97 x 1024
kg
Mass of the Sun
MS
1.99 x 1030
kg
Mass of Moon
MM
7.35 x 1022
kg
Radius of the Moon
RM
1.74 x 106
m
Density of water
ρ
1.00 x 103
kg m -3
2
12
Q1
In a circuit the following resistor combination is found.
Figure 1.a
All the resistors in Figure 1.a have resistance R ohms.
What is the total resistance across (i) AC and (ii) AB?
[4]
(b) The energy levels, En , of the hydrogen atom are given by
(i)
(ii)
. J, where n is a positive integer.
What is the ionization energy of the atom?
What is the wavelength of the Hα line, which is due to transitions from the n = 3
to n = 2 level?
[4]
(c) You are challenged to construct a bridge using two identical uniform rectangular blocks,
length 24 cm, which overhang a table as indicated in Figure 1.c. The lower block
overhangs the table by x cm and the upper block overhangs the lower block by 6.0 cm.
Under what condition will one or both blocks collapse?
[5]
Figure 1.c
(d) A proton travelling with a velocity of 3.00 x 107 m s-1 collides with an oxygen nucleus,
of mass 2.56 x 10-26 kg that is at rest, and is scattered through an angle of 90°. Calculate
the velocity and direction of the oxygen nucleus using Newtonian mechanics.
[12]
(e) A submerged wreck, mass 104 kg and mean density 8 x 103 kg m-3, is lifted out of the
water by a crane with a steel cable 10 m long, cross-sectional area 5 cm2 and Young’s
modulus 5 x 1010 N m-2. Determine the change in the extension of the cable as the wreck
is lifted clear of the water.
[5]
3
13
(f)
MM', Figure 1.f, is a plane mirror. A and B are points in front of the mirror and O is a
variable point on the mirror. B' is the image of B in the mirror. Prove geometrically
that:
(i) the paths AOB and AOB' are equal.
(ii) the path length of the ray reflected in the mirror has the minimum possible value
of AOB.
[6]
(g)
An exoplanet is discovered by the Kepler mission. It has a mass M with angular
velocity ω. A small moon of mass m and radius a orbits the planet at a centre to centre
distance of r. What is the condition for this circular orbit?
If R is the reaction force on a loose rock on the moon’s surface, write down the
equation for the ‘equilibrium’ of the rock on the moon’s surface. Assume that the
moon orbits the planet always keeping the same face towards the planet. Deduce the
condition, independent of ω, to be satisfied by M/m, for the rock to be lifted off the
moon by the planet’s gravitational attraction.
[8]
(h)
A beaker is fitted with a heating coil and stirrer and contains 40.0 cm3 of liquid A.
When the power dissipated in the heating coil is 4.80 W, the temperature of the
contents rises from 15.0°C to 35.0°C in 400 s. The experiment is repeated using 20.0
cm3 of liquid A mixed with 20.0 cm3 of liquid B. It is found that, with a heater power
of 4.90 W, the temperature again rises from 15.0°C to 35.0°C in 400 s.
Determine
(i)
the specific heat capacity, s, of B and
(ii)
the heat lost, H, in both experiments.
Density of A is 1.60 x 103 kg m-3,
Density of B is 2.00 x 103 kg m-3
Specific heat capacity of A is 8.60 x 102 J kg-1 K[6]
(i)
The electron gun of a cathode ray tube consists of a small hot filament F which is
located at x = 0, Figure 1.i, and which produces electrons in the x-y plane of the page
with a very small range of velocities. A typical electron has velocity components vx
and vy. Between x = 0 and x = d there is a horizontal uniform electric field, E, which
accelerates the electrons produced at the filament to velocities which are much greater
than vx and v y. The electrons emerge from the field, beyond x = d, travelling in straight
lines. Show that the paths of the emerging electrons, when projected back, appear to
have come from a point along the axis at approximately x = −d.
[6]
4
14
Figure 1.f
Figure 1.i
(j) Determine the binding energy per nucleon, in MeV, of an alpha particle.
Mass of proton
Mass of neutron
Mass of alpha particle
1u
= 1.0080 u
= 1.0087 u
= 4.0026 u
= 930 MeV/c2
[3]
(k) An organ pipe has one end closed and at the other end is a vibrating diaphragm, which is
a displacement antinode. When the frequency of the diaphragm is 2,000 Hz a stationary
or standing wave pattern is set up in the tube. The distance between adjacent nodes is
8.0 cm. As the frequency is slowly reduced the stationary wave pattern disappears, but
another stationary wave pattern reappears at frequency 1,600 Hz.
Calculate:
(i) the speed of sound in air
(ii) the distance between adjacent nodes at 1,600 Hz
(iii) the length of the tube
(iv) the next frequency below 1,600 Hz at which a stationary pattern occurs
[6]
(l) A rescue helicopter of mass 810 kg, supports itself in a stationary position by imparting
a downward velocity, v, to the air in a circle of radius 4.0 m. The density of the air is
1.20 kg m-3.
Calculate:
(i) the value of v
(ii) the power, P, required to support the helicopter
[5]
(m) A radioactive substance, with a half-life of T, contains a particular nucleus that has NOT
decayed over an observational period of 5T. What is the probability that it will decay
over a further period of (i) T and (ii) 3T ?
[4]
5
15
(n) A rectangular block has a mass of 1.5 kg with an uncertainty of magnitude 0.03 kg, and
a volume of 80 mm x 50 mm x 30 mm, with uncertainties of magnitude 1 mm in each
dimension. Determine the magnitude of the fractional uncertainty in the density of the
block.
[2]
End of Section 1
6
16
BRITISH PHYSICS OLYMPIAD 2012-13
BPhO Round 1
Section 1
16th November 2012
Instructions
Time: 1 hour 20 minutes.
Questions: Students may attempt any parts of Section 1. Students are not expected to
complete all parts.
Marks: A maximum mark of 40 is awarded for Section 1. There are a total of 83 marks
allocated to the problems of Question 1 which makes up Section 1.
Solutions: Answers and calculations are to be written on loose paper or examination
booklets. Graph paper and formula sheets should also be made available. Students should
ensure their name and school is clearly written on all answer sheets.
Setting the paper: There are two options for setting BPhO Round 1:
 Section 1 and Section 2 may be sat in one session of 2 hour 40 minutes.
 Section 1 and Section 2 may be sat in two sessions on separate occasions; with
1 hour 20 minutes allocated for each section. If the paper is taken in two sessions on
separate occasions, Section 1 must be collected in after the first session and
Section 2 handed out at the beginning of the second session.
1
17
Important Constants
Speed of light
c
3.00 x 108
ms-1
Planck constant
h
6.63 x 10-34
Js
Electronic charge
e
1.60 x 10-19
C
Mass of electron
me
9.11 x 10-31
kg
Gravitational constant
G
6.67 x 10-11
Nm2kg-2
Acceleration of free fall
g
9.81
ms-2
Permittivity of a vacuum
ε0
8.85 x 10-12
Fm-1
2
18
Q1.
(a) In an experiment to measure the Planck constant, h, the following results were
obtained, in units of 10-34 Js: 6.65, 6.67, 6.57, 6.61, 6.72, 6.60, 3.66, 6.71, 6.66, 6.64,
6.67. What would you give as:
(i)
The best estimate of h?
(ii)
The uncertainly of h from your reading?
[3]
(b) The pulley system in Figure 1.b consists of two pulleys of radii a and b rigidly fixed
together, but free to rotate about a common horizontal axis. The weight W hangs
from the axle of a freely suspended pulley P, which can rotate about its axle. If
section A of a rough rope is pulled down with velocity V :
(i)
Explain which way W will move.
(ii)
With what speed will it move?
[5]
Figure 1.b
(c) Estimate the mean density of the Earth, in g cm-3, assuming that the radius of the
Earth RE = 6.38 x 103 km and the value of g = 9.81 ms-2. State any assumptions
made.
[5]
(d) Determine the half-life, T, of a substance with activity that decreases by 1.00 % in
108 s (approximately 3 years).
[5]
3
19
(e) A horizontal bar, 8.0 m in length, has a 2.0 m rope attached at each end, with a small
metal sphere at each end of each rope hanging under gravity, Figure 1.e. When the
bar rotates about a vertical axis through its centre, the ropes are inclined at 30 o to
the vertical. Determine the period, T, of rotation of the system.
[5]
Figure 1.e
(f) A lithium surface, with work function energy W = 3.7 x 10-19 J, is irradiated with
photons of frequency f = 6.3 x 1014 Hz. The loss of photoelectrons from the surface
causes the metal to acquire a positive potential, V. What will this potential be when
the metal prevents the loss of further electrons?
[4]
(g) To determine the specific heat capacity, s, of a liquid flowing at a constant rate of
0.060 kg min-1 down a pipe, heat from an electrical supply is maintained at the rate
of 12 W. It produces a temperature rise of 2.0 C along the flow. Calculate s.
[2]
(h) A thundercloud has a horizontal lower surface, area 25.0 km2, 750 m above the
surface of the Earth. Using a capacitor as a model, calculate the electrical energy, E1,
stored when its potential is 1.00 x 105 V above the earth potential.
If the cloud rises to 1250 m:
(i) explain why the energy, E2, has increased or decreased
(ii) What is the change in electrical energy, ΔE?
[10]
(i) A square metal plate of uniform density has dimensions 3R x 3R. Its centre is the
centre of a Cartesian coordinate system, with axes parallel to the edges of the plate.
A complete circular region, radius R, is removed from the plate. Determine the
position of the centre of gravity of the remaining square plate when the centre of
the circular hole is at (i) (x,0) and (ii) (x,y).
[6]
4
20
(j) A battery of emf E and internal resistance r drives 3.0 A round a circuit consisting of
two 2.0 ohm resistors in parallel. When these resistors are connected in series the
current is 1.2 A. Calculate E, r and the power dissipated, W, in each resistor.
[5]
(k) Sketch, on the same diagram, the paths of three alpha particles, of the same energy,
which are directed towards a fixed nucleus so they are deflected through (i) 10o, (ii)
90o, and (iii) 180o. If the nucleus in (iii) is not fixed, what is the relative velocity of the
two particles at the distance of closest possible approach?
[5]
(l) A tank contains water to a depth of 1.0 m. Water emerges from a small hole in the
vertical side of the tank at 20 cm below the surface. Determine:
(i) the speed at which the water emerges from the hole
(ii) the distance from the base of the tank at which the water strikes the floor on
which the tank is standing.
[5]
(m) A smooth flat horizontal turntable 4.0 m in diameter is rotating at 0.050 revs per
second. A student at the centre of the turntable, and rotating with it, releases a
smooth flat puck on the turntable 0.50 m from the edge. Describe the motion of the
puck as seen by a stationary observer who is standing at the side of the turntable.
(i) How long does the puck remain on the turntable?
(ii) At what relative velocity would the student, at the centre of the turntable,
see the puck leave the turntable?
[10]
(n) A closed wire loop in the form of a square of side 4.0 cm is placed with its plane
perpendicular to a uniform magnetic field, which is increasing at the rate of
0.030.Ts-1.The loop has a resistance of 2.0 x 10-3 ohms. Calculate the current induced
in the loop, and explain, with the aid of a diagram, the relation between the
direction of the induced current and the direction of the magnetic field.
[4]
(o) The value of a device for storing energy can be assessed by the height to which it
would rise if all the stored energy were used to propel it upwards under gravity.
Calculate the height of the following:
(i) a 12 V car battery of capacity 60 Ah and mass 20 kg
(ii) a quantity of petrol of calorific value 4 x 107 J/kg
[5]
5
21
(p) A soap bubble, diameter 3.0 cm, containing helium floats in the air. Determine the
thickness of the bubble, as a multiple of the wavelength, 650 nm, of red light,
assuming soap solution has a density of 1.00 x 103 kg m-3. The density of the air and
helium are, respectively, 1.28 kg m-3 and 0.18 kg m-3.
[4]
End of Questions
6
22
BRITISH PHYSICS OLYMPIAD 2013-14
BPhO Round 1
Section 1
15th November 2013
Instructions
Time: 1 hour 20 minutes on Section 1.
Questions: Students may attempt any parts of Section 1 but are not expected to complete
all parts.
Marks: A maximum of 40 marks can be awarded for Section 1, out of the total of 78 marks
allocated to the problems in Section 1.
Solutions: Answers and calculations are to be written on loose paper or in examination
booklets. Graph paper and formula sheets should also be made available. Students should
ensure their name and school is clearly written on each page of their answer sheets.
Setting the paper: There are two options for setting BPhO Round 1:
 Section 1 and Section 2 may be sat in one session of 2 hours 40 minutes.
 Section 1 and Section 2 may be sat in two sessions on separate occasions; with
1 hour 20 minutes allocated for each section. If the paper is taken in two sessions on
separate occasions, Section 1 must be collected in after the first session and
Section 2 handed out at the beginning of the second session.
23
Important Constants
Speed of light
c
3.00 x 108
m s-1
Planck constant
h
6.63 x 10-34
Js
Electronic charge
e
1.60 x 10-19
C
Mass of electron
me
9.11 x 10-31
kg
Gravitational constant
G
6.67 x 10-11
N m2 kg-2
Acceleration of free fall
g
9.81
m s-2
Permittivity of a vacuum
ε0
8.85 x 10-12
F m-1
Avogadro constant
NA
6.02 x 1023
mol-1
2
24
Section 1
(a) A chain of resistors, Figure 1.a.i, is composed of n units, each consisting of three
resistors, each resistor of resistance R, Figure 1.a.ii. A unit is attached to the left hand
end of the chain in order to increase the number of units from n to (n+1).
A
B
Figure 1.a.i
Figure 1.a.ii
(i)
Calculate the resistance (between A and B) across a chain with two units, R2 ,
and the resistance R3, across a chain with three units .
(ii)
A unit is attached to a long chain. The resistance of the chain, RT , is not
altered by this addition. Determine the resistance of the chain.
[6]
(b) .
(i)
A satellite is in orbit just above a spherical planet of radius R and uniform
density ρ. If the periodic time for each orbit is T, find an expression for ρT2.
Comment on the result.
(ii)
A man with a mass of 75 kg stands at the end of a diving board, depressing it
by 0.30 m. What would be the period of his motion if he was to jump lightly
in rhythm with the harmonic motion of the diving board?
[6]
(c)
(i)
A uniform vertical tube, open at the lower end and sealed at the upper end,
is lowered into sea water, trapping air in the tube. When the tube is
submerged to a depth of 10.0 m, sea water has exactly filled the lower half of
the tube. To what depth must the tube be lowered so that sea water fills 90%
of the tube?
(ii)
A mercury barometer has some air above the mercury, Figure 1.c.i. The top of
the barometer is 1.000 m above the level of the mercury in the reservoir.
When the tube is vertical the height of the mercury column is 0.700 m. When
the tube is inclined at 60o to the vertical, Figure 1.c.ii, the reading of the
mercury level is 0.950 m. What is the atmospheric pressure in mm of
mercury?
[6]
3
25
Figure 1.c.i
(d)
Figure 1.c.ii
A car travels along a horizontal road starting at time t = 0 , and finishing at t = π /10.
At time t it has travelled a distance x, has a speed v and an acceleration f given by
x = A sin (5t), v = 5A cos(5t) and
f = -25A sin(5t),
where A is a constant.
Determine the average speed, vAV , and average acceleration, f AV.
[3]
(e)
One gram of hydrogen atoms is separated into electrons and protons. The electrons
are deposited on the Moon, the protons remaining on the Earth. What, numerically,
is the force that results? The Earth – Moon distance is REM = 3.84 x 108 m.
[5]
(f)
Determine the half life of uranium given that 3.23 x 10-7g of radium is found per
gram of uranium in ancient minerals. The half life of radium is 1,600 years. The
atomic weights of uranium and radium are 238 and 226 respectively. All the radium
arises from the uranium.
[5]
(g)
A mixed beam of deuterons (an isotope of hydrogen, 2H+) and protons, which have
been accelerated through 1.00 x 104 V, enter a uniform magnetic field of 0.500 T in a
direction at right angles to the field. Calculate the separation of the proton beam
from the deuteron beam when each has described a semicircle in the field.
[6]
4
26
(h)
A sound source, frequency f and velocity u , is moving along a straight line towards
an observer who is approaching the sound source with velocity v. Determine the
frequency fO heard by the observer if the speed of sound is c.
A moving sound source, S, has velocity of 15.0 m s-1 and frequency 200 Hz. An
observer P, speed 18.0 ms-1, and S are approaching a point Q along paths inclined at
30o to each other, Figure 1.h. What frequency is heard by the observer when S and P
are equidistant from Q? The speed of sound is 331 ms-1.
[8]
Q
Figure 1.h
(i)
A beaker, containing some water, has a total mass of 0.300 kg. The beaker
rests on a weighing scale. A 250 g copper sphere, density 8.93 x 103 kg m-3, is
suspended so that it is completely immersed in the water, but does not touch
the bottom of the beaker. What is the reading on the weighing scale in
newtons? The density of water is 1.00 x 103 kg m-3.
[3]
(j)
A capacitor, capacitance C1 , with a charge Q0 , is connected in a closed (loop)
series circuit with an uncharged capacitor, capacitance C2 and a switch,
which is initially open. Compare the energy stored in the capacitors before
and after the switch is closed by considering the potential across each
capacitor. What can one conclude?
[8]
5
27
(k) ……
…determine the specific heat
Figure 1.k
A uniform sphere, radius R and mass M = 5.00 kg, is pulled up an inclined plane,
inclination 33.0o to the horizontal, by a string of tension T, which is attached to a
point P on its surface, making an angle θ with the line joining the centre of the
sphere, O, and its contact point with the plane, C. The string is parallel to the plane.
The coefficient of friction between the sphere and the plane μ = 0.420. The sphere is
about to slide up the plane. The frictional force is F and the normal reaction is N,
Figure 1.k.
Determine, numerically:
(i)
T, by resolving the forces along and perpendicular to the slope
(ii)
θ.
[10]
(l)
Three boats start at time t = 0 from the corners of an equilateral triangle, of side 50
km, and maintain constant speeds of 30 km hr-1 during the subsequent motion. They
each maintain a heading, clockwise, towards the neighbouring boat. They all
eventually meet at P.
Determine:
(i)
qualitatively, the evolution of the triangle formed by the three boats
(ii)
the velocity components of the three boats in the direction of P ,as a function
of time, t, and in the perpendicular directions
(iii)
the time, tM , at which they all meet
(iv)
the distance travelled by each boat, D.
[8]
6
28
(m) A bicycle tyre has a volume of 1.20 x 10-3 m3 when fully inflated. The barrel of the
bicycle pump has a working volume of 9.0 x 10-5 m3. How many strokes of the pump
are needed to completely inflate the flat tyre to a total pressure of 3.0 x 10 5 Pa? The
atmospheric pressure is 1.00 x 105 Pa. Assume the air is pumped in slowly, so that
the temperature remains constant.
[4]
End of Questions
7
29
BRITISH PHYSICS OLYMPIAD 2014-15
BPhO Round 1
Section 1
14th November 2014
Instructions
Time: 1 hour 20 minutes.
Questions: Students may attempt any parts of Section 1. Students are not expected to
complete all parts.
Marks: A maximum of 40 marks can be awarded for Section 1. There are a total of 67 marks
allocated to the problems of Question 1 which makes up the whole of Section 1.
Solutions: Answers and calculations are to be written on loose paper or examination
booklets. Graph paper and formula sheets should also be made available. Students should
ensure their name and school is clearly written on all answer sheets.
Setting the paper: There are two options for setting BPhO Round 1:
• Section 1 and Section 2 may be sat in one session of 2 hours 40 minutes.
• Section 1 and Section 2 may be sat in two sessions on separate occasions; with
1 hour 20 minutes allocated for each section. If the paper is taken in two sessions on
separate occasions, Section 1 must be collected in after the first session and
Section 2 handed out at the beginning of the second session.
30
Important Constants
Speed of light
c
3.00 x 108
ms-1
Planck constant
h
6.63 x 10-34
Js
Electronic charge
e
1.60 x 10-19
C
Mass of electron
me
9.11 x 10-31
kg
Gravitational constant
G
6.67 x 10-11
Nm2kg-2
Acceleration of free fall
g
9.81
ms-2
2
31
Q1.
(a) The circuit in Figure 1.(a) contains a cell of emf E , a known variable resistance R0 ,
an unknown resistance R and an ammeter. When X and Y are short circuited E = I0R0
When R is inserted the current is αI0 , where α is a constant.
Figure 1.(a)
(i)
Express R in terms of R0 and α, giving the range of validity of R and α.
(ii)
In order to extend the range of α, modify the circuit by putting R in parallel
with R0. Determine the ranges of R and α for the modified circuit.
[4]
(b) A man, on an open wagon of a train travelling along a straight horizontal track at a
constant speed of 10 ms-1 , throws a ball into the air in line with the track, that he
judges to be at 60o to the horizontal. A woman standing on the ground observes the
ball rise vertically.
How high does the ball rise relative to
(i)
the man and;
(ii)
the woman?
[5]
(c) A glass block of refractive index μ = 1.5 has an ‘L’ cross-section, Figure 1.(c), and is of
constant width and thickness.
Figure 1.(c)
(i)
A laser beam enters the block from the left, as indicated in Figure 1.(c), at an
incident angle of θ = 45o. If the block was absent the beam would pass
through the point P. Determine the angle at which the beam will emerge
from the bottom face after refraction through the block.
(ii)
If this beam enters the block below the horizontal through P, determine its
possible subsequent path(s).
[6]
3
32
(d) The largest moon of Jupiter, Ganymede, revolves around the planet in a circular orbit
of radius 1.07 x 106 km and period 7.16 days. Determine the mass of Jupiter, MJ, in
terms of the mass of the Earth, ME.
The radius of the Earth RE = 6.38 x 106 m
[5]
(e) Two 1.00 m lengths of wire, one copper and one tungsten, are joined vertically end
to end. The copper wire has a diameter of 0.500 mm. When a 100 kg block is
suspended from one end, the combined length of wire stretches by 6.00 cm. What is
the diameter, d, of the tungsten wire if the Young’s modulus for copper is 12.4 x 1010
Pa, and that for tungsten is 35.5 x 1010 Pa?
[6]
(f) Wood from the coffin of an Egyptian mummy showed a specific activity of 1.2 x 102
s-1kg-1. Comparable living wood has a value of 2.0 x 102 s-1kg-1. The half life of
carbon-14 is 5.70 x 103 years. What is the time interval, TB , in years, since the burial?
[5]
(g) Explain why the centre of gravity of a triangular plate lies along a median; the line
joining a vertex to the midpoint of the opposite side. An equilateral triangular plate,
sides of length b, has a triangle, formed by two corners and the centre of gravity of
the original plate, removed. Determine the centre of gravity of the remaining plate.
The centre of gravity of a triangular plate is at a point two thirds along the length of
a median measured from the vertex.
[7]
4
33
(h) A vertical U-tube, partially filled with liquid, is accelerated vertically upwards in a lift,
acceleration α. What is the effective value of 'g', gv? If the U-tube is mounted in a
vehicle accelerating in a horizontal straight line, acceleration a, Figure 1.(h), what is
the effective 'g' , gh ? Express a in terms of the distance between the arms of the Utube, L , and the difference in heights, h, of the liquid in the arms.
[7]
Figure 1.(h)
(i) In Figure 1.(i) a fixed mirror, a light source and a light receiver are all 0.30 km from a
rotating mirror, with angular frequency ω. The distance between the light source
and the receiver is 0.60 m. What is the lowest value of ω required for detection of
the reflected light?
[4]
Figure 1.(i)
(j) A car travelling at 90 km/hr in a straight line sounds its horn continuously, frequency
400 Hz, as it passes a stationary observer. At the closest point, A, to the observer the
car is at a distance D = 100 m from the observer. Determine the frequency heard by
the observer when the car is:
(i)
at A;
(ii)
at a distance x from A, after passing A
The velocity of sound is vS = 343 ms-1.
[8]
5
34
(k) A ball of mass m and velocity u collides elastically with a larger ball of mass M,
initially at rest. The ball of mass m rebounds along its original line of motion with
speed v1 and the ball of mass M has velocity v2 in the direction of u.
(i)
Write down the conservation equations for the system.
(ii)
Deduce the result that u – v1 = v2.
[5]
(l) A velocity selector, Figure 1.(l) , consists of two slotted discs mounted on a common
axis a distance d apart. The slots are displaced relative to each other by an angle θ .
The axis is driven at an angular velocity ω . Particles in a horizontal beam, with all
possible velocities, will get through the first slit, in the first disc, for a short time
interval. To subsequently get through the second slit, particles must travel a distance
d in the times it takes the second slot to line up with the beam. This will occur, for
rotations of the second slit of θ, 2π+θ, 4π+θ ,.. etc.
If d =1.00 m, ω = 24,000 rpm and θ = 60o, what are the speeds of those particles that
pass through the velocity selector?
[5]
Figure 1.(l)
End of Questions
6
35
BRITISH PHYSICS OLYMPIAD 2015-16
BPhO Round 1
Section 1
13th November 2015
Instructions
Time: 1 hour 20 minutes on this section.
Questions: Students may attempt any parts of Section 1. Students are not expected to
complete all parts.
Working: Working, calculations and explanations, properly laid out, must be shown for full
credit. The final answer alone is not sufficient.
Marks: A maximum of 40 marks can be awarded for Section 1. There are a total of 71 marks
allocated to the problems of Question 1 which makes up the whole of Section 1.
Solutions: answers and calculations are to be written on loose paper or examination
booklets. Graph paper and formula sheets should also be made available. Students should
ensure their name and their school is clearly written on all answer sheets.
Setting the paper: There are two options for setting BPhO Round 1:
• Section 1 and Section 2 may be sat in one session of 2 hours 40 minutes.
• Section 1 and Section 2 may be sat in two sessions on separate occasions, with
1 hour 20 minutes allocated for each section. If the paper is taken in two sessions on
separate occasions, Section 1 must be collected in after the first session and
Section 2 handed out at the beginning of the second session.
36
Important Constants
Speed of light
c
3.00 x 108
m s-1
Planck constant
h
6.63 x 10-34
Js
Electronic charge
e
1.60 x 10-19
C
Mass of electron
me
9.11 x 10-31
kg
Gravitational constant
G
6.67 x 10-11
N m2 kg-2
Acceleration of free fall
g
9.81
m s-2
Permittivity of a vacuum
ε0
8.85 x 10-12
F m-1
Avogadro constant
NA
6.02 x 1023
mol-1
2
37
Q1.
(a) A measurement is carried out to check the speed of a camera shutter of 1/15 s. The
camera is focused symmetrically on a rotating turntable which revolves at 33.3 ± 0.1
revolutions per minute and has a spot at its centre and at its circumference. A
photograph shows the arc produced by the spot on the circumference subtends an
angle of 12.4 ± 0.1° at the centre of rotation. What is the correct exposure time?
[3]
(b) The temperature coefficients of resistance, α, of certain alloys are positive and
others are negative. They have resistance per unit length of r. This makes it possible
to produce a resistor, using the two wires in series, which does not vary with
temperature. The values of r, at 0 ℃, and are given in Table 1.b for constantan and
manganin. These wire have lengths Lc and Lm respectively at 0 oC. What values of Lc
and Lm are required to produce a 5.0 Ω resistor?
Wire
r /Ωm-1
/ oC-1
Constantan
6.3
-3.0 x 10-5
Manganin
5.3
+1.4 x 10-5
Table 1.b
[5]
(c) A monochromatic sodium lamp, wavelength λ = 6.0 x 10-7 m, radiates 100 W of
radiation uniformly in all directions. At what distance from the lamp will the photons
have an average density of 106 m-3 ?
[5]
(d) Protons are accelerated from rest through a p.d. of 2.0 x 106 V and fired at a gold
( Au) foil. What is the distance of closest approach of a proton to the gold
nucleus?
[4]
3
38
(e) Figure 1.e is a section through a smooth parabolic metal bowl, which can be rotated
about its vertical axis of symmetry, the - axis. Its equation, in Cartesian
coordinates, is =
. The gradient at the point ( , ) is 2 .
There is one angular speed of rotation, , of the bowl about the - axis for which a
small metal sphere remains at rest relative to the rotating bowl, wherever it is placed
on the inner surface. Determine in terms of and .
Figure 1.e
[4]
(f) A ray of light is incident on a 60°glass prism of refractive index 1.500 at an angle of
incidence of 48.59°. Determine:
(i) the angle of emergence, , from the prism; i.e. the angle between the emergent
ray and the normal to the prism face.
(ii) the angle of deviation of the ray, .
[5]
(g) A man blowing a whistle of frequency moves away from a stationary observer at
speed . Derive the formula for the frequency heard by the observer. The velocity
of sound is c.
If a man blowing a whistle of frequency 500 Hz moves away from a stationary
observer towards a fixed wall, in a direction perpendicular to the wall at 2.00 m s-1,
determine the beat frequency heard by the observer if c = 330 m s-1.
[6]
4
39
(h) Determine, in Figure 1.h, the total resistances, RTBC , across BC, RTBD across BD and
RTBA across AB.
Figure 1.h
[6]
(i) What mass of radium, mass number 226, half-life of 1620 years, and an α emitter, is
required to produce an average of 10 α particles per second?
[4]
(j) An a.c. voltmeter displays the rms value of the voltage, , for a.c. signals and also for
periodic signals that are not sinusoidal. What reading will it display if connected to a
periodic voltage, period 4, that changes instantaneously from +10 V to -2 V to
+4V repeatedly, the voltages lasting, respectively, for , and 2?
Determine the following;
(i) the mean voltage i.e. .
(ii) the rms voltage i.e. ! .
(iii) the rms value of deviation from the mean, ( − ) which is ! .
[4]
5
40
(k) Assuming the Earth is a homogeneous sphere, calculate the fractional difference
between the acceleration due to free fall at the Earth’s equator and at the poles,
indicating which is the greater. You may not use a value of g given in the constants
table as that is an average value, neither correct at the poles nor at the equator.
Mass of the ME = 5.98 x 1024 kg
Radius of the Earth RE = 6.38 x 106 m
[8]
(l) A horizontal square wire loop of side 4.00 cm has a resistance, #, of 2.00 x 10-3 Ω.
The loop is situated in a vertical downward magnetic field of 0.700 T. When the field
is switched off, it decreases to zero, at a uniform rate, in 0.800 s.
Determine:
(i) the induced current, $, and its direction in the loop.
(ii) the energy dissipated in the loop.
[7]
(m) Two well separated identical conducting spheres of radius 10.0 cm are charged to
+200 V and +400 V. If they are joined by a long wire, how much heat is generated?
[10]
End of Questions
6
41
BRITISH PHYSICS OLYMPIAD 2016-17
BPhO Round 1
Section 1
18th November 2016
This question paper must not be taken out of the exam room.
Instructions
Time: 1 hour 20 minutes on this section.
Questions: students may attempt any parts of Section 1. Students are not expected to
complete all parts.
Working: working, calculations and explanations, properly laid out, must be shown for full
credit. The final answer alone is not sufficient. Writing must be clear.
Marks: a maximum of 40 marks can be awarded for Section 1. There is a total of 70 marks
allocated to the problems of Question 1 which makes up the whole of Section 1.
Solutions: answers and calculations are to be written on loose paper or in examination
booklets. Graph paper and formula sheets should also be made available. Students should
ensure that their name and their school are clearly written on each and every answer sheet.
Setting the paper: There are two options for setting BPhO Round 1:
• Section 1 and Section 2 may be sat in one session of 2 hours 40 minutes.
• Section 1 and Section 2 may be sat in two sessions on separate occasions, with
1 hour 20 minutes allocated for each section. If the paper is taken in two sessions on
separate occasions, Section 1 must be collected in after the first session and
Section 2 handed out at the beginning of the second session.
42
Important Constants
Speed of light
Planck constant
3.00 × 10
m s-1
6.63 × 10
Js
Electronic charge
1.60 × 10
C
Mass of electron
9.11 × 10
kg
Gravitational constant
6.67 × 10
N m2 kg-2
ℎ
Acceleration of free fall
Permittivity of a vacuum
Avogadro constant
9.81
m s-2
8.85 × 10
F m-1
6.02 × 10
mol-1
2
43
Question 1.
(a) Sketch the electric field lines due to two point charges, of magnitudes +
at A and B, separated by a distance !.
and +2 ,
(i)
Determine the location of the neutral point, P, where the electric field is zero.
(ii)
Why does the magnitude of the electric field vary along a field line?
[5]
(b) A charge of 0.5 x 106 C passes through a 12 V battery when the battery discharges.
Assuming that the p.d. across the terminals remains constant, calculate the time for
which it can supply 0.45 kW.
[2]
(c) Draw a general resistive network diagram with:
(i)
two resistors in series which are, in turn, in series with three resistors in
parallel.
(ii)
five resistors that are not in series or parallel, or in a combination of series
and parallel arrangements.
Calculate the resistance in (i) and (ii) if all the resistors have resistance ".
[5]
(d) Two spheres, of uniform density, one of mass
and radius # and the other of
mass
and radius # , attract each other gravitationally . What is their relative
speed at the instant of collision if they are released from rest when a great distance
apart?
[7]
4
44
(e) A bicycle tyre has a volume of 1.2 x 10-3 m3 when fully inflated. A bicycle pump has a
working volume of 9.0 x 10-5 m3. How many strokes, $, of the pump are needed to
inflate the completely flat tyre, containing no air, to a pressure of 3.0 x 105 Pa? The
atmospheric pressure is 1.0 x 105 Pa. Assume the air is pumped in slowly so that there
is no change in temperature.
[5]
(f) A van, travelling at constant speed of 80 km hr-1 (km/hour), passes a car. The car
immediately begins to accelerate at a constant rate of 1.2 m s-2 and passes the van
0.50 km further down the road. What is the speed, %, of the car when it passes the
van?
[4]
(g) A calorimeter contains 0.800 kg of water at a temperature of 15.0 oC. The heat
capacity of the calorimeter is 42.8 J oC-1. 0.400 kg of molten lead is poured into the
calorimeter. The final equilibrium temperature is 25.0 oC. What was the initial
temperature of the lead?
The specific heat of molten lead is 158 J kg-1 oC-1, the specific heat of solid lead is 137 J
kg-1 oC-1 and the specific latent heat is 2.323 x 104 J kg-1. Lead freezes at 327 oC. The
specific heat of water is 4200 J kg-1 oC-1.
[5]
(h) A small object of mass rests on a scale-pan which is supported by a spring. The
period of vertical oscillations is 0.50 s. When the amplitude of the oscillations
exceeds the value, &, the mass leaves the scale-pan. Determine &.
[3]
5
45
(i) Uncharged metallic spheres of radii 6", 3" and 2" are mounted on insulated stands.
The spheres of radii 2" and 6" are charged to a potential ' above earth potential.
All three spheres are then briefly joined by a copper wire. What, in terms of ' , is the
subsequent potential of the sphere of radius 3" ?
What fraction of the original total charge is held by the sphere of radius 3" ?
[5]
(j) The maximum kinetic energy of photoelectrons ejected from a tungsten surface by
monochromatic light of wavelength 248 nm is 8.60 x 10-20 J.
What is the value of the work function, (, of tungsten?
[3]
(k) A ladder of length ) and mass , with a uniform density, rests against a frictionless
vertical wall at an angle of 60o to the wall. The lower end rests on a flat surface with
a coefficient of static friction of *+ = 0.40. A student with a mass - = 2 attempts
to climb the ladder. What fraction of the distance up the ladder will the student have
reached when the ladder begins to slip?
[5]
(l) A smooth ball of radius 10.0 cm, mass 0.600 kg, hangs by a weightless string from a
support. What is the speed of a horizontal wind necessary to keep the string inclined
at 39o to the vertical? Make the assumption that the wind speed drops to zero on
collision with the ball. The density of the air is 1.293 kg m-3.
[4]
(m) The activity of polonium, Po, fell to one eighth of its initial value in 420 days. Calculate
the half-life, ./ , of polonium.
Give the numerical values of a, b, c, d, e, and f in the nuclear equation
:
9Po
→
>
=α
+
?
Pb +
B
Aγ
[4]
6
46
(n) Four masses of 1 kg, 4 kg, 3 kg, and 4 kg are arranged cyclically at the corners of a
square of side 2C and centre O. A thin circular metal ring has radius D, mass 8 kg, and
with the same centre O lies in the same plane as the square. Determine the position
of the centre of mass of the system from O.
[3]
(o) A trumpeter travelling in an open car sounds a note at 440 Hz. A stationary
pedestrian directly in the path of the car hears a note at frequency 466 Hz. What is
the speed of the car? The velocity of sound is 331 ms-1.
[3]
(p) A beam of protons is accelerated from rest through a potential difference of 2000 V
and enters a uniform magnetic field which is perpendicular to the direction of the
proton beam. If the flux density is 0.2000 T, calculate the radius of the path of the
beam.
How is the result modified for deuterons?
[4]
(q) A particle, mass m, slides down the smooth track, Figure 1(q), from a height E under
gravity. It is to complete a circular trajectory of radius " when reaching its lowest
point. Determine the smallest value of E.
[3]
Figure 1(q).
End of Section 1
7
47
British Physics Olympiad 2017-18
BPhO Round 1
Section 1
17th November 2017
This question paper must not be taken out of the exam room
Instructions
Time: 5 minutes reading time (NO writing) and 1 hour 20 minutes for writing on this section.
Questions: Students may attempt any parts of Section 1, but are not expected to complete all parts.
Working: Working, calculations, explanations and diagrams, properly laid out, must be shown for full
credit. The final answer alone is not sufficient. Writing must be clear.
Marks: A maximum of 50 marks can be awarded for Section 1. There is a total of 94 marks allocated
to the problems of Question 1 which makes up the whole of Section 1.
Instructions: You are allowed any standard exam board data/formula sheet.
Calculators: Any standard calculator may be used.
Solutions: Answers and calculations are to be written on loose paper or in examination booklets.
Graph paper and formula sheets should also be made available. Students should ensure that their name
and their school are clearly written on each and every answer sheet. Number each question clearly.
Setting the paper: There are two options for sitting BPhO Round 1:
a. Section 1 and Section 2 may be sat in one session of 2 hours 40 minutes plus 10 minutes reading time.
b. Section 1 and Section 2 may be sat in two sessions on separate occasions, with 1 hour 20 minutes
plus 5 minutes reading time allocated for each section. If the paper is taken in two sessions on
separate occasions, Section 1 must be collected in after the first session and Section 2 handed out at
the beginning of the second session.
48
Important Constants
Constant
Symbol
Value
Speed of light in free space
c
3.00 × 108 m s−1
Elementary charge
e
1.60 × 10−19 C
Planck’s constant
h
6.63 × 10−34 J s
Mass of electron
me
9.11 × 10−31 kg
Mass of proton
mp
1.67 × 10−27 kg
Gravitational constant
G
6.67 × 10−11 m3 kg−1 s−2
Acceleration of free fall at Earth’s surface
g
9.81 m s−2
Permittivity of free space
ε0
8.85 × 10−12 F m−1
Permeability of free space
µ0
4π × 10−7 H m−1
Avogadro constant
NA
6.02 × 1023 mol−1
Mass of Sun
MS
1.99 × 1030 kg
Mass of Earth
ME
5.97 × 1024 kg
Radius of Earth
RE
6.37 × 106 m
49
Question 1
a) Physicists sometimes use the approximation that light travels in a vacuum at a speed of 1 foot in
1 ns. What is the percentage error in using this value?
(1.000 m = 1.094 yards and 1.000 yard = 3.000 feet)
[3]
b) A window cleaner’s ladder shown in Figure 1 is narrower at the top than the bottom. It has a
weight of 350 N and a length of 5.0 m. When it lies flat on the ground, a force of 80 N is needed
to lift the narrow end off the ground.
(i) How far is the centre of mass from the narrow end?
(ii) What force is required to lift the wide end of the ladder off the ground?
Figure 1
[5]
c) A particle moves in a straight line with an intial acceleration of 10 m s−2 . The acceleration
decreases uniformly with time until, after ten seconds, the acceleration is 5 m s−2 , and from then
on the acceleration remains constant. If the intial velocity is 100 m s−1 ,
(i) find when the velocity has doubled;
(ii) sketch a graph of the velocity against time.
[7]
d) A student standing at a distance a from a vertical wall kicks a ball from ground level with velocity
V at an angle α to the horizontal in a plane perpendicular to that of the wall. The ball strikes the
wall and rebounds. The coefficient of restitution for the collision is e = 2/3. The ball first strikes
the ground at a distance 2a from the wall. e is the ratio of the components of velocity at normal
vafter
incidence to the wall, before and after collision; e =
≤ 1.
vbefore
Find a in terms of V, α and g, the gravitational field strength.
[6]
1
50
e) A helicopter of total mass 1000 kg is able to remain in a stationary position by imparting a uniform
downward velocity to a cylinder of air below it of effective diameter 6 m. Assuming the density
of air to be 1.2 kg m−3 , calculate the downward velocity of the air.
[5]
f) In this question, distances are measured in nautical miles and speeds in nautical miles per hour.
A motor boat sets out at 2 p.m. from a point with position vector −4î − 5ĵ relative to a marker
buoy (where
√ î and ĵ are two fixed perpendicular unit vectors) and travels at a steady speed of
magnitude 41 in a straight line to intercept a ship S. The ship S maintains a steady velocity
vector î + 4ĵ and at 3 p.m. is at a position 3î − ĵ relative to the buoy. Find
(i) the position vector of the ship at 2 p.m.,
(ii) the velocity vector of the motor boat,
(iii) the time of interception.
[7]
g) In a factory heating system, water enters the radiators at 60 ◦ C and leaves at 38 ◦ C. The system
is replaced by one in which steam at 100 ◦ C is condensed in the radiators, the condensed steam
leaving at 82 ◦ C. What mass of steam will supply the same heat energy as 1.00 kg of hot water
described in the first instance? (The latent heat of vaporisation of water is 2.260 × 106 J kg−1 at
100 ◦ C. The specific heat capacity of water is 4200 J kg−1 ◦ C−1 .)
[4]
h) A cell, a resistor and an ammeter of negligible resistance are connected in series and a current of
0.80 A is observed to flow when the resistor has a value of 2.00 Ω. When a resistor of 5.00 Ω is
connected in parallel with the 2.00 Ω resistor, the ammeter reading is 1.00 A.
Calculate the emf and the internal resistance of the cell.
[5]
i) A battery with an emf of 6 V can produce a maximum current of 3 A. A resistor is connected
to the terminals whose value is such that the power dissipated in it is a maximum. Calculate the
maximum energy which can be dissipated in the external resistor in one minute.
[4]
2
51
j) Calculate the number of photons emitted in a one nanosecond (10−9 s) pulse of light from a
0.5 mW laser of wavelength 639 nm.
[3]
k) A lead ball is attached to the end of a light metal rod of length l, the other end being attached to
a horizontal axle of negligble friction. The rod is given an intial impulse and swings round in a
vertical circle. When it is at the top of the circle, the tension in the rod is zero. What is the tension
in the rod at the lowest point of its swing?
[6]
l) Some sand is sprinkled onto a loudspeaker cone which is pointing vertically upwards. The
louspeaker is driven in simple harmonic motion when attached to a signal generator and the
frequency is gradually raised. At a particular frequency, when the amplitiude of oscillation is
0.20 mm, the sand begins to lose contact with the cone. At what frequency does this occur?
[3]
m) Two radio stations on the equator, diametrically opposite each other, communicate by sending and
receiving radio signals that are tangential to the Earth’s surface via two geostationary satellites
in circular orbits at 3.59 × 104 km above the Earth’s surface. Calculate the time delay between
sending and receiving a signal.
[6]
n) A thin film of transparent material of refractive index 1.52 and thickness 0.42 µm forms a thin
coating on glass of refractive index 1.60. It is viewed by reflection with white light at normal
incidence. What visible wavelength in vacuo is most strongly reflected?
[5]
o) Monochromatic light of wavelength 600 nm is incident on two vertical slits hence producing two
coherent sources. Before the light leaving these slits overlaps and interferes, each beam passes
through a tube 5.0 cm long. One of the tubes is now gradually evacuated and it is noted that the
fringe pattern shifts 25 fringes. Calculate the refractive index of air.
[5]
3
52
p) A submerged wreck is lifted from a dock basin by means of a crane to which is attached a steel
cable 10 m long of cross-sectional area 5.0 cm2 and Young modulus 5.0 × 1010 Pa. The material
being lifted has a mass of 1.0 × 104 kg and mean density 8000 kg m−3 . Find the change in
extension of the cable as the load is lifted clear of the water. Assume that at all times the tension
in the cable is the same throughout its length. (Density of water is 1000 kg m−3 .)
[5]
q) A uniform beam AOB, O being the midpoint of AB, mass M , rests on three vertical springs with
stiffness constants k1 , k2 , k3 at A, O and B respectively. The bases of the springs are fixed to a
horizontal platform. Determine the compression of the springs and their compressional forces in
the following two instances:
(i) k1 = k3 = k and k2 = 2k
(ii) k1 = k, k2 = 2k and k3 = 3k
[8]
r) A pond is covered by a layer of ice 5 cm thick. How long will it be before the ice is 10 cm thick
if the air temperature stays constant at −10 ◦ C?
Assume the density of ice = 900 kg m−3 ; the latent heat of fusion of ice = 330 kJ kg−1 ; the
thermal conductivity of ice = 2.1 W m−1 K−1 .
The power flowing perpendicular to the faces through a uniform material is given by power flow
(TH − TC )
P = λA
, in which λ is the thermal conductivity of the material, TH is the hotter
x
temperature at one face of the material, TC is the colder temperature on the other face, A is the
area of a face, and x is the thickness of the material.
[7]
END OF SECTION 1
4
53
British Physics Olympiad 2018-19
BPhO Round 1
Section 1
16th November 2018
This question paper must not be taken out of the exam room
Instructions
Time: 1 hour 20 minutes for this section.
Questions: Students may attempt any parts of Section 1, but are not expected to complete all parts.
Working: Working, calculations, explanations and diagrams, properly laid out, must be shown for full
credit. The final answer alone is not sufficient. Writing must be clear.
Marks: A maximum of 50 marks can be awarded for Section 1. There is a total of 87 marks allocated
to the problems of Question 1 which makes up the whole of Section 1.
Instructions: You are allowed any standard exam board data/formula sheet.
Calculators: Any standard calculator may be used, but calculators cannot be programmable and must
not have symbolic algebra capability.
Solutions: Answers and calculations are to be written on loose paper or in examination booklets.
Graph paper and formula sheets should also be made available. Students should ensure that their name
and their school/college are clearly written on each and every answer sheet. Number each question
clearly.
Setting the paper: There are two options for sitting BPhO Round 1:
a. Section 1 and Section 2 may be sat in one session of 2 hours 40 minutes plus 5 minutes reading time
(for Section 2). Section 1 should be collected in after 1 hour 20 minutes and then Section 2 given out.
b. Section 1 and Section 2 may be sat in two sessions on separate occasions, with 1 hour 20 minutes
plus 5 minutes reading time allocated for Section 2. If the paper is taken in two sessions on separate
occasions, Section 1 must be collected in after the first session and Section 2 handed out at the
beginning of the second session.
54
Important Constants
Constant
Symbol
Value
Speed of light in free space
c
3.00 × 108 m s−1
Elementary charge
e
1.60 × 10−19 C
Planck constant
h
6.63 × 10−34 J s
Mass of electron
me
9.11 × 10−31 kg
Mass of proton
mp
1.67 × 10−27 kg
u
1.661 × 1027 kg
Gravitational constant
G
6.67 × 10−11 m3 kg−1 s−2
Acceleration of free fall at Earth’s surface
g
9.81 m s−2
Permittivity of free space
ε0
8.85 × 10−12 F m−1
Permeability of free space
µ0
4π × 10−7 H m−1
Avogadro constant
NA
6.02 × 1023 mol−1
Mass of Sun
MS
1.99 × 1030 kg
Radius of Earth
RE
6.37 × 106 m
atomic mass unit
(1u is equivalent to 931.5 MeV)
T(K) = T(◦ C) + 273
Volume of a sphere = 34 πr3
For small angles, sin θ ≈ tan θ ≈ θ
For x 1,
(1 + x)n ≈ 1 + nx
55
Question 1
a) The Milky Way galaxy has a period of rotation of 240 × 106 years. The Sun is 26 light years from
the centre of the galaxy. How fast is the Sun moving with respect to the centre of the galaxy, given
in units of m s−1 ?
A light year is the distance that light travels in one year of 365.25 days.
[3]
b) A smooth sphere of radius 6.0 cm is suspended from a thread of length 9.0 cm attached to a smooth
wall as shown in shown in Figure 1. If the mass of the sphere is 0.5 kg, calculate the tension, T ,
in the thread.
Figure 1
[3]
c) The displacement of an object is determined by the following function:
s = 2t3 − 9t2 + 12t + 4
where s is the displacement in metres, and t the time elapsed in seconds. Determine
(i) the times when the object comes to rest,
(ii) the time when the acceleration is zero,
(iii) the object’s velocity when its acceleration is zero,
(iv) the object’s accelerations when its velocity is zero.
[4]
d) The distance in which a train can be stopped is given by:
s = av + bv 2
where s is the stopping distance, v the initial velocity, and a and b are constants. When moving at
40 km hr−1 , the train can be stopped in 100 m, and at 80 km hr−1 it can be stopped in 280 m.
Find the greatest speed such that the train can be stopped in 500 m.
[4]
1
56
e) Two planes set out at the same time from an aerodrome. The first flies north at 360 km h−1 , the
second south-east at 300 km h−1 . After 40 minutes they both turn and fly towards each other.
Calculate
(i) the bearing, and
(ii) the distance
of the meeting point from the aerodrome.
[7]
f) A neutron moving through heavy water strikes an isolated and stationary deuteron (the nucleus of
an isotope of hydrogen) head-on in an elastic collision.
(i) Assuming the mass of the neutron is equal to half that of the deuteron, find the ratio of the
final speed of the deuteron to the initial speed of the neutron.
(ii) What percentage of the initial kinetic energy is transferred to the deuteron?
(iii) How many such collisions would be needed to slow the neutron down from 10 MeV to
0.01 eV?
[6]
g) A uniform chain of mass per unit length, µ, is suspended from one end above a table, with the
lower end just touching the surface. The chain is released, falls and comes to rest on the table
without bouncing.
(i) Determine an expression, in terms of µ and the gravitational field strength g, for the reaction
force exerted by the table on the chain as a function of time, t.
Hint: you might consider F in the form F = ∆m
∆t v.
(ii) In terms of the total weight W of the chain, what is the maximum reaction force exerted by
the table, and at what time during the fall does this occur?
[6]
h) A small particle of mass m can slide without friction round the inside of a cylindrical hole of
radius r, in a rectangular shaped object of mass M . The rectangular object is held between rigid
walls by small wheels so that it can slide up and down without friction, as shown in Figure 2. If
the small particle m is initially at rest at the bottom of the cylindrical hole, and is then given an
impulse to give it a speed v, what is the minimum speed v needed to just lift the rectangular mass
M off the ground?
M
m
Figure 2
[5]
2
57
i) Two resistors and two cells are connected in the circuit shown in Figure 3. One cell has an
e.m.f. of 2.0 V and an internal resistance of 1.0 Ω, the other an e.m.f. of 1.5 V and an internal
resistance of 0.5 Ω. The resistors are connected in series and the point between them is at earth,
i.e. zero potential. Calculate
(i) the current through the cells,
(ii) the potential difference across each cell, and
(iii) the potential, relative to earth, at points A and B.
Figure 3
[4]
j) A thick-bottomed, cylindrical glass beaker is placed on a bench. Water and oil are poured into the
beaker and form discrete layers, as shown in Figure 4. The bottom of the beaker is 1.8cm thick,
the water is 1.2 cm deep, and the oil layer is 0.8 cm deep.
(i) Draw a diagram showing the path of a ray at a small angle to the normal, travelling from the
underside of the beaker and being refracted through the layers.
(ii) Assuming the angles of deviation of the ray are small, calculate the apparent vertical
displacement of the lab bench when viewed from above.
The refractive indices are 1.5, 1.3 and 1.1 for the glass, water and oil respectively.
observer's
eye
oil
water
glass
Figure 4
[7]
k) A person might reasonably expect to jump a height of 1 m on Earth. On a planet with a density
two thirds that of Earth, and radius twice that of the Earth, to what height might the person jump?
Assume that they supply the same energy to make the jump
[4]
3
58
l) A pond containing water of density ρ is covered to a depth b by oil of density 23 ρ. A long wood
block of square cross section 4b × 4b, with the same density as the oil, floats in the pond, as shown
in Figure 5. What fraction of the wood block is immersed below the surface oil level?
4b
4b
Figure 5
[4]
m) A volume of 80 cm3 of water in a copper calorimeter of mass 150 g takes 12 minutes to cool from
40 ◦ C to 15 ◦ C in a cold room. The same volume of ethanol of density 0.8 g cm−3 takes 8 minutes
to cool also from 40 ◦ C to 15 ◦ C in the same calorimeter in the same circumstances. Calculate the
specific heat capacity of ethanol.
The specific heat capacity of copper = 400 J kg−1 ◦ C−1 and of water = 4200 J kg−1 ◦ C−1 .
The density of water, ρw = 1.0 g cm−3 .
[5]
n) A steel girder is planted securely between two sides of a ravine in order to provide a bridge. The
total cross-sectional area of the girder is 30 cm2 , and the length of the girder is 4.0 m. Installed
at a temperature of 5 ◦ C, the temperature now rises to 20 ◦ C. Calculate the force exerted by the
girder due to the change in temperature, assuming the ends do not move.
Young modulus of steel = 2.0 × 1011 Pa
Linear expansivity of steel (fractional expansion per unit temperature rise) = 1.2 × 10−7 ◦ C−1 at
5 ◦ C.
[4]
o) A narrow beam of monochromatic light falls on a diffraction grating of 1200 lines mm−1 , and
two diffracted beams of successive orders are observed at 14◦ and 73◦ to the normal, both of them
on the same side of the normal. The incident beam of light is not along the normal to the grating.
(i) Sketch a diagram to show the path difference between rays passing through adjacent slits, for
a ray incident on the diffraction grating at angle θ1 , and for the corresponding ray emerging
from the grating at angle θ2 , with respect to the normal.
(ii) Derive an equation relating the angles θ1 and θ2 to the order of diffraction, n, and the
wavelength, λ.
Determine:
(iii) The wavelength of the light used.
(iv) The angle of incidence of the beam on the grating.
(v) The angle of diffraction of a third transmitted beam.
4
[6]
59
p) Two identical spherical glass containers are joined by a narrow tube, whose volume is negligible
compared to the spheres. The spheres contain air at 100 ◦ C. One of the spheres is then heated by
50 ◦ C whilst the other is cooled by 50 ◦ C. This produces a small change in pressure, from Pinitial
to Pfinal , of the air in the system. What common temperature of the two spheres could produce
the same final pressure Pfinal ?
[4]
q) A simple pendulum consists of a small mass on the end of a light, inextensible string, as shown in
Figure 6. It swings from an initial angle θ = 14◦ , for which it would have a period T0 , but it hits
a wall elastically, which is at angle φ = 7◦ to the vertical. What is the new period of oscillation in
terms of T0 ?
(θ, φ are small angles such that sin θ ≈ θ and sin φ ≈ φ).
wall
Figure 6
[4]
r) Four charges are placed at the corners of a square of side 10 cm, as shown in Figure 7.
A = +10 × 10−9 C
B = +8 × 10−9 C
C = −12 × 10−9 C
D = −6 × 10−9 C
(i) Calculate the magnitude and direction of the electric field strength at the centre of the square.
(ii) Calculate the work done taking an electron from the centre to the mid-point of side CD.
A
B
D
C
Figure 7
[7]
END OF SECTION 1
5
60
BPhO Round 1
Section 1
15th November 2019
This question paper must not be taken out of the exam room
Instructions
Time: 1 hour 20 minutes for this section.
Questions: Students may attempt any parts of Section 1, but are not expected to complete all
parts.
Working: Working, calculations, explanations and diagrams, properly laid out, must be
shown for full credit. The final answer alone is not sufficient. Writing must be clear.
Marks: A maximum of 50 marks can be awarded for Section 1. There is a total of 88 marks
allocated to the problems of Question 1 which makes up the whole of Section 1.
Instructions: You are allowed any standard exam board data/formula sheet.
Calculators: Any standard calculator may be used, but calculators cannot be programmable
and must not have symbolic algebra capability.
Solutions: Answers and calculations are to be written on loose paper or in examination
booklets. Graph paper and formula sheets should also be made available. Students should
ensure that their name and their school/college are clearly written on each and every answer
sheet. Number each question clearly and number the pages.
Sitting the paper: There are two options for sitting BPhO Round 1:
a. Section 1 and Section 2 may be sat in one session of 2 hours 40 minutes plus 5 minutes
reading time (for Section 2 only). Section 1 should be collected in after 1 hour 20 minutes
and then Section 2 given out.
b. Section 1 and Section 2 may be sat in two sessions on separate occasions, with 1 hour 20
minutes plus 5 minutes reading time allocated for Section 2. If the paper is taken in two
sessions on separate occasions, Section 1 must be collected in after the first session and
Section 2 handed out at the beginning of the second session.
61
Important Constants
Constant
Symbol
Value
Speed of light in free space
c
3.00 × 108 m s−1
Elementary charge
e
1.60 × 10−19 C
Planck constant
h
6.63 × 10−34 J s
Mass of electron
me
9.11 × 10−31 kg
Mass of proton
mp
1.67 × 10−27 kg
u
1.661 × 10−27 kg
Gravitational constant
G
6.67 × 10−11 m3 kg−1 s−2
Acceleration of free fall at Earth’s surface
g
9.81 m s−2
Permittivity of free space
ε0
8.85 × 10−12 F m−1
Avogadro constant
NA
6.02 × 1023 mol−1
Mass of Sun
MS
1.99 × 1030 kg
Radius of Earth
RE
6.37 × 106 m
atomic mass unit
(1u is equivalent to 931.5 MeV)
T(K) = T(◦ C) + 273
Volume of a sphere = 34 πr3
ex ≈ 1 + x + . . .
x1
(1 + x)n ≈ 1 + nx
x1
1
(1+x)n
x1
≈ 1 − nx
sin θ ≈ θ
for θ 1
tan θ ≈ θ
for θ 1
cos θ ≈ 1 − θ2 /2
for θ 1
62
Question 1
a) A golf ball is struck and begins to move at an initial velocity of 60 m s−1 at an angle 40◦
above the horizontal. Determine at time t = 3 s after the strike
(i) the velocity of the ball, and
(ii) the position of the ball relative to the origin.
[4]
b) A drone flies horizontally. The displacement of the drone is given by s = 2tî + 6tĵ,
where î and ĵ are unit vectors to the East and North respectively. Determine at t = 2 s:
(i) the speed of the drone,
(ii) its bearing in degrees,
(iii) its acceleration.
Note: all bearings are measured clockwise from North.
[3]
c) Estimate the mass of a piece of paper the size of a pinhead (the blunt end of a sewing
pin). Show your calculation.
[2]
d) The speed of surface waves of wavelength λ on a liquid of density ρ is given by
1
aλ 2πb 2
+
v=
2π
ρλ
where a and b are constants. Determine the units of a and b.
[2]
e) Figure 1 shows the cross section of a high voltage overhead electrical transmission cable.
The central strand is of steel and the six outer strands are of aluminium. The resistivity
of steel is 2.0 × 10−7 Ω m, and that of aluminium 3.2 × 10−8 Ωm. The cross-sectional
area of each strand is 5.0 × 10−4 m2 . The steel is present to give mechanical strength to
the cable and only reduces the resistance of a length ` of cable by 1.4 × 10−4 Ω when it
is included. Calculate the length of the cable.
Figure 1
[5]
1
63
f) Platinum (symbol Pt) and potassium (symbol K) have densities of 21.5 g cm−3 and
0.89 g cm−3 respectively. How many cubic centimetres (cm3 ) of platinum could be
attached to 10.0 cm3 of potassium before the combination sinks in mercury of density
13.6 g cm−3 ? Ignore any chemical reactions.
[4]
g) One kilogram of ice at 0 ◦ C is placed in a thermally insulated bucket of volume 5 litres.
Water at 15 ◦ C is added until the bucket is completely filled. Calculate the temperature
of the water when half of the ice has melted.
1 litre = 1000 cm3
Latent heat of fusion of ice, Lice = 3.34 × 105 J kg−1
Specific thermal capacity of water, cice = 4180 J kg−1 K−1
Densities: ρice = 920 kg m−3 and ρwater = 1000 kg m−3
[5]
h) This question concerns three vessels at sea: a ferry (F), a container ship (C), and a pilot
boat (P). The ferry is sailing on a bearing of 090◦ at 5 m s−1 . Relative to the ferry, the
container ship is sailing on a bearing of 160◦ . The pilot boat is sailing on a bearing of
270◦ at 7.5 m s−1 , and the pilot boat observes the container ship moving on a bearing of
120◦ .
Determine the speed and direction of the container ship relative to the water.
Note: all bearings are measured clockwise from North.
[7]
i) A car accelerates from a standing start. If the mass of the car is m, and the car is driven
at constant driving power P , find an expression for the velocity of the car v as a function
of distance travelled from a standing start, x. Ignore resistive effects and inefficiencies in
power transmission.
[4]
j) An experiment is proposed which involves submerging a ball of mass m and radius r to
a depth of d r in a swimming pool. The ball is then released, and emerges from the
water and rises to a height h r above the surface. The quantities d and h are measured
from the centre of the ball to the water surface. An initial model is proposed, which
ignores any resistive effects and the inertia of the water. Determine the prediction this
initial model makes for the ratio h/d in terms of m, r and the density of water, ρ.
[5]
2
64
k) A sand timer is a sealed glass vessel with a narrow section acting as a constraint, so that
sand can flow through at a steady rate. A fifteen minute sand timer is shown in Figure 2
below. Unlike a liquid, the rate of flow of sand grains through the constrained section is
independent of the height of the sand above.
Thus the rate of flow of sand through the time can be expressed as a product of powers
of the remaining relevant variables:
dm
= kρα × Aβ × g γ
dt
where k is a dimensionless constant, ρ is the density of the sand, A is the cross sectional
area at the narrowest point, and g is the gravitational field strength, and α, β, γ are
numbers.
(i) By considering the units of the variables on each side of the equation, find the values
of α, β and γ.
(ii) On the Moon, the gravitational field strength is gM = 1.6 N kg−1 . How long would
the sand timer last on the Moon if it runs for 15 minutes on Earth?
[4]
Figure 2: Sand timer. (image credit: John Lewis Partnership,
https://www.johnlewis.com/the-school-of-life-15-minutes-timer/p3827395)
l) An excited neon-20 isotope travels with a velocity of 3.0 × 106 m s−1 into a detector
and disintegrates into an alpha particle and oxygen-16. The event produces an additional
6.25 MeV of kinetic energy. The oxygen nucleus leaves the event at right angles to the
path of the original neon nucleus.
Determine the velocity of the alpha particle. Relativistic effects may be neglected.
1 eV = 1.6 × 10−19 J
[5]
3
65
m) An aeroplane flies due East along the equator at a constant low altitude and constant
speed relative to the ground. On the aeroplane, a one kilogram mass is suspended on a
spring balance and records a weight W1 . The aeroplane then flies due West along the
equator, at the same altitude and speed, and measures a balance reading of W2 .
If the speed of the plane relative to the ground is 250 m s−1 , calculate the difference in
apparent weights.
[5]
n) Two glass bulbs are connected by a thin tube. One glass bulb has a volume of 75 cm3 ,
the other 150 cm3 , and gas can move freely between them. Initially the system contains
nitrogen at −12 ◦ C and 0.91 × 105 Pa. The smaller bulb is then warmed to 24 ◦ C, whilst
the larger bulb is maintained at −12 ◦ C.
Calculate the new pressure in the system. Assume the thermal expansion of the bulbs and
the volume of the connecting tube are negligible.
[5]
o) Determine the current in the 6.0 Ω resistor shown in the Fig 3. The cells have no internal
resistance.
9.0
6.0
4.0
6.0 V
9.0 V
Figure 3
[5]
p) An electrically isolated copper sphere of radius 2 mm is illuminated by light of
wavelength 150 nm. Determine
(i) the maximum electric potential that the copper sphere can reach
(ii) the number of electrons lost reaching the maximum potential
(Work function of copper = 4.5 eV)
[4]
4
66
q) Three conducting spheres of radii 13 R, 21 R and R are mounted on insulating rods, and are
well separated from each other. The 13 R and R spheres are each charged to a potential V ,
whilst the 21 R sphere is uncharged. Then a thin copper wire is used to briefly connect all
three spheres. What fraction of the original charge on the two spheres is now on the 12 R
sphere?
[5]
r) The thermal power flowing by conduction through a surface is proportional to the
temperature difference across the surface, ∆θ, the area of the surface, A and inversely
proportional to the thickness ∆x. The constant of proportionality is known as the
thermal conductivity.
A 60 cm composite rod, of constant cross section, is made of 20 cm lengths of steel,
copper and aluminium joined together. The rod is well insulated. The tip of the steel end
of the rod is maintained at 100 ◦ C and the tip of the aluminium end, at 0 ◦ C. What are the
temperatures at each of the two junctions of dissimilar metals?
Thermal conductivities are as follows: steel 60 W m−1 K−1 ; copper 400 W m−1 K−1 ;
aluminium 240 W m−1 K−1 .
[5]
s) A bicycle pump of cross-sectional area 4.0 cm2 has one end sawn off and a cork is fitted
into the end. The piston is pushed slowly inwards and the cork is fired out with a popping
sound which has a frequency of 512 Hz. The initial distance between the cork and the
piston is 25 cm, with atmospheric pressure equal to 1.0 × 105 Pa and the speed of sound
in air being 330 m s−1 .
Calculate the force required to eject the cork.
[4]
t) A cup of tea cools from 30.2 ◦ C to 29.7 ◦ C in 1 minute, in an ambient temperature of
20.0 ◦ C. Assuming the tea cools at a rate directly proportional to the temperature
difference between the tea and the surroundings, calculate how long it will take for the
tea to cool from 24.0 ◦ C to 23.0 ◦ C.
[5]
END OF SECTION 1
5
67
BPhO Round 1
Section 1
17th November 2020
This question paper must not be taken out of the exam room
Instructions
Time: 1 hour 20 minutes for this section.
Questions: Students may attempt any parts of Section 1, but are not expected to complete all
parts.
Working: Working, calculations, explanations and diagrams, properly laid out, must be
shown for full credit. The final answer alone is not sufficient. Writing must be clear.
Marks: A maximum of 50 marks can be awarded for Section 1. There is a total of 73 marks
allocated to the problems of Question 1 which makes up the whole of Section 1.
Instructions: You are allowed any standard exam board data/formula sheet.
Calculators: Any standard calculator may be used, but calculators cannot be programmable
and must not have symbolic algebra capability.
Solutions: Answers and calculations are to be written on loose paper ON ONE SIDE ONLY
(pages will be scanned). Students should ensure that their name and their school/college are
clearly written on each and every answer sheet. Number each question clearly and number the
pages.
Sitting the paper: There are two options for sitting BPhO Round 1:
a. Section 1 and Section 2 may be sat in one session of 2 hours 40 minutes plus 5 minutes
reading time (for Section 2 only). Section 1 should be collected in after 1 hour 20 minutes
and then Section 2 given out.
b. Section 1 and Section 2 may be sat in two sessions on separate occasions, with 1 hour 20
minutes plus 5 minutes reading time allocated for Section 2. If the paper is taken in two
sessions on separate occasions, Section 1 must be collected in after the first session and
Section 2 handed out at the beginning of the second session.
68
Important Constants
Constant
Symbol
Value
Speed of light in free space
c
3.00 × 108 m s−1
Elementary charge
e
1.602 × 10−19 C
Planck constant
h
6.63 × 10−34 J s
Mass of electron
me
9.110 × 10−31 kg
Mass of proton
mp
1.673 × 10−27 kg
Mass of neutron
mp
1.675 × 10−27 kg
atomic mass unit
u
1.661 × 10−27 kg = 931.5 MeV c−2
Gravitational constant
G
6.67 × 10−11 m3 kg−1 s−2
Earth’s gravitational field strength
g
9.81 N kg−1
Permittivity of free space
ε0
8.85 × 10−12 F m−1
Avogadro constant
NA
6.02 × 1023 mol−1
Gas constant
R
8.3145 J K mol−1
Mass of Sun
MS
1.99 × 1030 kg
Radius of Earth
RE
6.37 × 106 m
T(K) = T(◦ C) + 273
Volume of a sphere = 34 πr3
ex
≈ 1 + x + ...
x1
(1 + x)n
≈ 1 + nx
x1
1
(1 + x)n
≈ 1 − nx
x1
tan θ
≈ sin θ ≈ θ
for θ 1
cos θ
θ2
≈1−
2
for θ 1
69
Section 1 — 50 marks maximum
Question 1
a) Estimate from what height, under free-fall conditions, a heavy stone would need to be
dropped if it were to reach the surface of the Earth at the speed of sound of 330 m s−1 .
[2]
b) A motorcycle rider is propelled up the left side of a symmetric ramp shown in Figure 1.
O
ℓ
P
θ
θ
Figure 1
The rider reaches the apex of the ramp at speed of u, and falls to a point P on the
descending ramp. In terms of u, θ and g, obtain expressions for,
(i) The time ta for which the rider is airborne.
(ii) The distance OP (= `) along the descending ramp.
[4]
c) Two buckets hang from a rope over a frictionless pulley as in Figure 2. The bucket on
the right has a mass m2 , which is greater than the mass of the bucket on the left m1
(m2 > m1 ). Bucket 2 starts at height h above the ground. If the buckets are released
from rest, determine:
(i) the speed with which bucket 2 hits the ground in terms of m1 , m2 , h, and the
acceleration due to gravity g, and
(ii) the further increase in height of bucket 1 after bucket 2 hits the ground and stops.
Ignore resistive effects and assume the rope is long compared to the height above the
ground.
2
1
ℎ
Figure 2
[4]
3
70
d) A rugby pitch lies in a north-south direction. In this question î represents a unit vector due
east, and ĵ represents a unit vector due north. Rugby player Y collects the ball and runs
with a velocity (3î + 4ĵ) m s−1 . Player X, starting 20 m due east of player Y, immediately
gives chase at a speed of 8 m s−1 . She is an expert player and runs in a straight line to
intercept player Y. Calculate
(i) the velocity of player Y,
(ii) the time taken for the players to meet, and
(iii) the displacement of Y from her original position at their point of contact.
[4]
e) A long wire of uniform diameter 1.40 mm has a resistance of 0.478 Ω. It is wrapped into
a ball in order to find its weight, which is 4.60 N. When weighed in water it is 4.08 N.
Calculate the resistivity of the wire.
Density of water = 1000 kg m−3
[3]
f) A train travels at a constant speed for one hour but is then delayed on the line for half
an hour. When it restarts, its speed is reduced to 75% of its previous speed. It arrives
at its destination 1 21 hours later than if it had travelled at its initial speed throughout. If
the delay had occurred 45 km further on, then the train would only have been 1 hour late.
Determine,
(i) the distance travelled, and
(ii) the initial speed of the train.
[4]
g) Emmy walks into a lift with a set of (bathroom) scales. She stands on the scales, presses
the button for the 30th floor, and starts a timer as the lift begins to move. She notices that
the reading on the scales varies with time, according to the equation below:
t
t2
m(t) = 60 1 +
−
10 100
(i) Write down an expression for the acceleration of the lift as a function of time.
(ii) How fast is the lift moving after 10 seconds?
(iii) After the initial 10 seconds, the lift decelerates at a constant rate until it arrives
at the 30th floor. Given that the 30th floor is 100 m above the ground, calculate
the minimum value of the mass reading (in kg) shown on the scales during this
deceleration.
[4]
4
71
h) Two transparent miscible liquids of refractive indices na = 1.15 and nb = 1.52 can be
mixed together to produce a liquid of refractive index n by mixing volumes Va and Vb of
the liquids. The refractive index of the mixture varies linearly with the volumes of the
two liquids. The refractive index of powdered glass, ng , poured into the mixture can be
found by adjusting the liquid mixture until the powdered glass cannot be observed in the
liquid.
(i) Obtain an expression for ng in terms of na , Va , nb and Vb .
(ii) If the powdered glass is poured into 100 ml of liquid A and is seen to disappear
when 64 ml of liquid B is added, what is the refractive index of the glass?
[3]
i)
(i) Five resistors, R1 . . . R5 are connected in a circuit between points A and B, as in
Figure 3. Resistors R2 and R4 can be changed in value to be connecting wires with
R = 0, finite values, or open circuit with R = ∞. Write down the values of R2 and
R4 so that the network between A and B is equivalent to
i. three resistors in series,
ii. three resistors in parallel, and
iii. two identical resistors in parallel.
R1
R2
A
B
R5
R3
R4
Figure 3
(ii) Figure 4 shows a simple circuit with two cells and three resistors. If the current
in the ammeter shown in the figure is 2.0 A determine the unknown e.m.f. ε, of the
battery, assuming the batteries and ammeter have no internal resistance.
5
4
3
12 V
Figure 4
[4]
5
72
j)
(i) In Figure 5a, resistors R1 , R2 and R3 are connected between A and B. Derive an
expression for R3 in terms of R1 and R2 , if the equivalent resistance, RAB is equal
to R1 .
(ii) A different arrangement is shown in Figure 5b for resistors R1 , R2 and R3 . Again
R3
= 6. Determine the value of
the equivalent resistance, RAB = R1 , and the ratio
R2
R1
the ratio of
.
R2
R1
R1
A
A
B
R3
R2
R2
B
R3
(b)
(a)
Figure 5
[4]
k) The stiffness, S, of a beam of rectangular cross-section, with width w, and thickness t,
is directly proportional to its width and the cube of its thickness, (t3 ); that is, stiffness,
S ∝ wt3 . Determine the cross-sectional dimensions of the stiffest rectangular crosssection wooden beam that can be cut from a log of diameter 20 cm.
[3]
l) A narrow beam of monochromatic light is passed normally through a diffraction grating
of 6×105 lines per metre. Three spots of light are observed on a screen placed 80 cm away
from the grating, and the outer spots are 30 cm away from the central spot. Determine
the wavelength of the light used.
[2]
m) Large craters can be produced on the Earth by meteorites. The size of a crater with
diameter d is dependent on the kinetic energy of the meteorite E, the density of the rock
removed from the crater ρ, and the field strength g, since the rock must be lifted out of
the crater. We can express this as
d = kE α ρβ g γ
where k is a numeric constant and k ≈ 1.
(i) By considering the dimensions (or units) of the quantities above, obtain an
expression relating the diameter of the crater to E, ρ and g.
(ii) The Barringer Crater in Arizona was made by a meteorite that landed there 30 000
years ago. It has a diameter of 1200 m and is in rock of typical density 3000 kg m−3 .
If the impact speed was 15 km s−1 , estimate the mass of the meteorite.
(iii) If the spherical meteorite was made of iron of density 8000 kg m−3 what was its
diameter?
[6]
6
73
n) The petrol engine of a car consumes 5.3 litres of petrol for every 100 km travelled at a
speed of 100 km h−1 with the outside temperature being 16 ◦ C. The heat of combustion
of the petrol is 30 MJ per litre, and 23 % of this energy finds its way to the water cooling
system. Calculate the mass rate of flow of cooling water such that the temperature rise of
the cooling water is limited to 40 ◦ C.
Specific thermal capacity of water = 4180 J kg−1 ◦ C−1 .
[4]
o) Stefan’s Law states that for a given perfectly radiating surface at an absolute
temperature T , the radiated power Φ is directly proportional to T 4 . The radiated energy
is distributed over a range of wavelengths of electromagnetic radiation, with the peak in
emission occurring at λmax , as shown in Figure 6. The value of λmax is determined by
Wien’s Law which states λmax is inversely proportional to absolute temperature.
Determine:
(i) The ratio of the radiative powers of the surface at 500 ◦ C and 1000 ◦ C, i.e. evaluate
Φ1000 /Φ500 ,
(ii) the wavelength of maximum emission at 1000 ◦ C if the wavelength of maximum
emission at 500 ◦ C is 3750 nm.
[4]
Figure 6
p) Electron beam lithography is used for etching microscopic patterns on surfaces.
Typically, a 5 nm layer of aluminium deposited on an insulating substrate will remove
the incident charge. If the beam current is 1.0 nA, and the electron beam of width 15 nm
is incident centrally on a circular aluminium covered surface of diameter 5.0 cm,
calculate the electrical resistance and potential difference between the edge of the spot
and the edge of the surface.
Resistivity of aluminium = 2.8 × 10−8 Ω m.
[5]
7
74
q) A leaky capacitor is one which does not have a perfectly insulating dielectric layer
between the plates upon which the charge is stored. Such a capacitor contains a
dielectric material filling the space between the plates, and with an effective resistivity
of 1.5 × 1012 Ω m. The area of the plates is A = 0.603 m2 , and the dielectric film
thickness d = 0.82 µm. The capacitor is charged from a 24 V supply connected in series
with a 4.7 MΩ resistor.
(i) Calculate the maximum charge that can be accumulated on one of the capacitor
0 A
]
plates. [For a parallel plate capacitor, C =
d
(ii) If the capacitor is disconnected from the circuit, calculate the time taken for the
capacitor to lose half its maximum charge.
[5]
r) A fixed mass of gas expands isothermally and the relationship between the pressure p,
and the volume of the gas V , is pV = 380 Pa m3 . The volume increases at a rate of
0.005 m3 s−1 when the volume is 0.17 m3 . At what rate does the pressure decrease at this
point?
[3]
s) A hot air balloon uses a gas flame to heat the air it contains to a temperature required to
enable it to hover at a small distance above ground level. The mass of the balloon, ropes,
basket and riders is 240 kg and the volume of the balloon is 1100 m3 . The temperature of
the surrounding air is 15 ◦ C and its density is 1.23 kg m−3 .
To what temperature does the air in the balloon need to be heated?
[5]
END OF SECTION 1
8
75