British Physics Olympiad Competition
Sections 1 & 2
12th November 2022
This question paper must not be taken out of the exam room
Instructions
Time: 2 hours 40 minutes.
Questions: There are two sections in the paper and you should spend about 1 hour 20 minutes
on each section.
Section 1 - Students may attempt any parts but are not expected to complete all parts.
Section 2 - Only two questions out of the four questions should be attempted.
Each question contains independent parts so that later parts should be
attempted even if earlier parts are incomplete.
Marks: A maximum of 50 marks can be awarded for Section 1. There is a total of ≈ 84
marks allocated to the problems of Question 1 which makes up the whole of Section 1.
Each question in Section 2 is out of 25, with a maximum of 50 marks from two questions
only. Students are recommended to spend about 40 minutes on each question.
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.
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 in the Answer booklet. A formula sheet
should also be made available. Students should ensure that their Name, School and Exam code
written clearly on the Answer booklet. Number each question answered.
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−1 mol−1
Mass of Sun
MS
1.99 × 1030 kg
Radius of Earth
RE
6.37 × 106 m
Specific heat capacity of water
cw
4180 J kg−1 C−1
◦
T(K) = T(◦ C) + 273
Volume of a sphere = 34 πr3
ex
≈ 1 + x + ...
x≪1
(1 + x)n
≈ 1 + nx
x≪1
1
(1 + x)n
≈ 1 − nx
x≪1
tan θ
≈ sin θ ≈ θ
cos θ
≈1−
θ2
2
for θ ≪ 1
for θ ≪ 1
Formula Sheet
Mechanics
Equations of motion
Nov 2022
Waves
s = ut + 12 at2
Refraction
v 2 = u2 + 2as
Double slit fringes
s = 1/2(u + v)t
Doppler effect
Impulse
F ∆t = ∆(mv)
de Broglie wavelength
Work
W = F s cos θ
Photon energy
Centripetal acceleration
a=
v2
Gas law
Work done by a gas
Current
∆Q
I=
∆t
Power
P =VI
Resistance
V = IR
R=
ρ`
A
pV = 13 N mhc2 i
Energy of a molecule
1
2
2 mcRMS
Gravitational potential
R = R1 + R2 + . . .
Resistors in parallel
1
1
1
=
+
+ ...
R
R1 R2
Gravitational field
Electric potential
V = V0 sin ωt
Electric field
SHM
Capacitance
Acceleration
a = −ω 2 x
Displacement
x = A sin(ωt + φ)
r
m
T = 2π
k
Capacitors in series
Period of a spring
Radioactivity
Radioactive decay
Thermodynamics
∆V
∆x
GM
Vg = −
r
GM
Eg = 2
r
Q
V =
4πε0 r
Q
E=
4πε0 r2
Q
C=
V
1
1
1
=
+
+ ...
C
C1 C2
E=−
C = C1 + C2 + . . .
Energy of a capacitor
E = 12 QV
Magnetic force
F = I`B and F = qvB
N = N0 exp(−λt)
λt 1 = ln 2 = 0.693
2
Thermal
Heat transfer
= 32 kT
Capacitors in parallel
EM induction
Decay constant
∆W = p∆V
Pressure of an ideal gas
Field and potential
Resistors in series
AC voltage
pV = nRT
Fields
I = nAvq
Resistivity
E = hf
Gases
Electricity
Electric current
λd
s
fs c
fo =
c ± vs
h
λ=
p
w=
= ω2r
r
p = ρgh
Hydrostatic pressure
n1 sin θ1 = n2 sin θ2
Q = mc∆T
and
Q = mL
∆Q = ∆U + ∆W
ε = −N
dφ
dt
Section 1 — 50 marks maximum
Question 1
a) A steel ball is thrown down with a speed of 3.0 m s−1 on to a hard surface from a height
of 2.0 m. It retains 70% of its energy on each bounce. Calculate
(i) the speed at which it hits the ground for the first time, and
(ii) the maximum height it reaches after the 4th bounce.
[2]
b) A long-distance cyclist uses a cycle computer that is dual-powered: it has an internal
battery and a solar panel. While the cyclist is riding in direct sunlight, the solar panel
on the computer provides energy at a rate equal to 13 of the power consumption. If a
fully-charged cycle computer lasts 10 hours when riding at night, calculate how long a
fully charged computer will run for in direct sunlight.
[2]
c) Riding round a corner at 10 km h−1 a cyclist leans over at an angle of 12◦ to the vertical.
At what angle would they lean over if they went round the corner at 15 km h−1 ?
[3]
d) Io and Europa are both moons of Jupiter. Europa takes twice as long as Io to complete
aIo
an orbit. What is the ratio of the centripetal acceleration of Io and Europa,
?
aEuropa
You may use the result that for this gravitational system, ω 2 r3 = constant, where ω is
the angular velocity and r is the radius of the orbit.
[2]
e) A particle of mass m1 and initial speed u makes an elastic collision with a stationary
particle of mass m2 . The particles move off at speeds v1 and v2 respectively, at equal
angles θ either side of the initial incident direction of m1 .
m1
for which this equal angle condition can occur?
m2
(ii) If m1 = m2 , what is the largest angle of deflection, θ, of particle m1 for this equal
angle condition?
[5]
(i) What is the largest ratio of
f) A body is projected with velocity v up a plane inclined at an angle α to the horizontal.
When it returns through its starting point it is moving with half the speed with which is
was projected.
Determine the coefficient of friction µ, in terms of the angle of the plane.
Hint: The coefficient of friction is given by Ffriction = µN where N is the normal contact
force and Ffriction is the frictional force on the body.
[4]
3
g) A cylindrical container is filled with equal volumes of n different liquids which do not
mix, so that they form horizontal layers each of height h. The densities of the liquids
are ρ, 2ρ, 3ρ, · · · , with the lowest liquid of density nρ. The curved surface area of the
cylinder enclosing each liquid is A.
(i) Give an expression for the force F1 on the area A surrounding the top liquid in terms
of ρ, g, h, A.
(ii) What is the force F2 on the surface A surrounding the second liquid down from the
top, in terms of F1 ?
(iii) Give an expression for the force Fn on the area A surrounding the nth liquid at the
bottom of the cylinder in term of F1 ?
Hint :
1 + 2 + 3 + ··· + n =
n
X
i=
i=1
n
(n + 1)
2
[4]
h) A rocket of mass mr = 5000 kg contains a further mass m0 = 5000 kg of fuel. Once
the fuel is ignited, 50 kg per second of hot gas is expelled downwards at a speed of
2000 m s−1 .
(i) Calculate the thrust, T , applied to the rocket,
(ii) Find an expression for the acceleration of the rocket, a, in terms of its total mass
m, T and g,
(iii) Find an expression for the acceleration of the rocket as a function of time, t, in
terms of T, g, mr , m0 and t0 , the total time for which the thrust acts.
(iv) Calculate the time after launch at which the weight of an astronaut on board will
have appeared to double.
[4]
i) A open topped steel drum is completely filled with oil on a day when the temperature is
5.0 ◦ C. On a warm day the temperature rises, and 2.4% of the oil spills out.
What is the temperature reached on that day?
The volume coefficient of expansion of oil is 7.0 × 10−4 ◦ C−1
The linear coefficient of expansion of steel is 1.2 × 10−5 ◦ C−1
[3]
j) Observed from an air traffic control tower, an aeroplane has a bearing of 068◦ and a range
of 43 km. Five minutes later the bearing of the aircraft is 040◦ with a range of 52 km.
Determine
(i) The speed of the aircraft in m s−1 .
(ii) Its bearing and range 10 minutes after the second sighting.
[4]
4
k) A gas is found to obey the equation relating p, V, n, R, T
−a
p(V − b) = nRT exp
nRT V
where p is the gas pressure
V is its volume
R is the molar gas constant
n is the number of moles
a and b are constants.
(i) Determine the SI base units (m, kg, s) in which a and b are expressed.
(ii) If b ≪ V and a ≪ nRT V , show that this expression approximates to the ideal gas
equation relating p, V, n, R, T at a particular temperature Tc . Determine Tc in terms
of a, b, n and R.
Hint: For x ≪ 1
exp(x) = ex ≈ 1 + x
[4]
l) A thin rod is balanced in a smooth hemispherical bowl fixed to a table, touching both the
interior and the rim as shown in Fig. 1. The rim of the bowl remains horizontal.
Expressed in the simplest form, determine the radius of the bowl r in terms of the length
of the rod ℓ, and the angle θ to the horizontal.
Figure 1: A rod in a smooth bowl.
[6]
m) In the circuit shown in Fig. 2, the three cells
each supply an emf of 5.0 V and have an internal
resistance of 5.0 Ω. The external resistors also each
have a resistance of 5.0 Ω.
What arrangement of switches gives
(i) the maximum current,
(ii) the minimum non-zero current.
(iii) Determine the current in each case.
[3]
Figure 2: Three cells and switches in parallel.
5
n) The circuit of Fig. 3 consists of four resistors and a switch S. When the S is open, the
current flowing through the 5 kΩ resistor is Io . When S is closed, the current flowing
Ic
through the same resistor is Ic . What is the ratio , giving your answer as the ratio of
Io
two integers.
Figure 3: A switched circuit.
[4]
o) The Stefan-Boltzmann law says that the emitted power of a spherical “black body”, P ,
is related to the radius, R, and absolute surface temperature, T , as P ∝ R2 T 4 . Wien’s
“displacement law” says that the wavelength λmax corresponding to the peak value of
the emitted power of this spectrum is inversely proportional to the absolute surface
temperature. The Sun currently has its peak wavelength as 500 nm.
What will be the new peak wavelength when it becomes a red giant, given its radius will
be 200 times larger and its power output 4000 times larger?
[3]
p) One spectral line in hydrogen is caused by photons with an energy of 2.55 eV. The same
line is redshifted in the spectrum of a distant galaxy by 5.4 nm. Calculate
(i) the wavelength of the photon,
(ii) the speed of recession of the galaxy,
(iii) the distance to the galaxy.
How far away is the galaxy? Give your answer in megaparsecs (Mpc).
The Hubble constant, H0 = 70 km s−1 Mpc−1 .
[3]
q) A car drives along a road that has small depressions regularly spaced about 8.0 m apart.
When four 80 kg passengers enter the 800 kg car, it sinks down by 1.8 cm.
At approximately what speed might travelling in the vehicle become very uncomfortable?
[3]
6
r) A pendulum clock is controlled by the swing of a simple pendulum (a mass on the end
of a light rod) and is intended to have a period of 1.00 seconds. However, the clock runs
slow by ten minutes each day. What percentage change should be made in the length of
the pendulum?
[3]
s) A binary star system is 2140 light years away and consists of two stars like the Sun. The
average separation between the stars is 0.00593 light years.
Determine the diameter of the telescope needed to resolve them if using a visible
wavelength of 550 nm.
[2]
t) A glass prism of refracting angle 75.0◦ is shown in Fig. 4 has a refractive index of n =
1.40.
(i) For what range of incident angles will light from air that is incident on face AB
emerge from face AC?
(ii) Show your result on a diagram.
[3]
Figure 4: Glass prism with an apex angle of 75◦ .
u) A particle A, of mass m carrying a charge of Q is suspended by an insulating thread
of length ℓ. Another particle B, of negligible mass but of positive charge +q is brought
towards A, which is repelled. When B arrives at the point previously occupied by A, the
system is in (neutral) equilibrium.
1
Calculate the work done in terms of m, g, ℓ, q, Q, and k, where k =
.
4πϵ0
[6]
v) A capacitor of value 1.0 F discharges through a device whose resistance R varies linearly
with applied potential difference, V , so that R = AV + B, where A and B are constants.
The resistance of the device has a value of 10.0 Ω when V = 6.0 V, and 4.0 Ω when
V = 0.06 V.
The capacitor is initially charged to a potential of 6.0 V.
Determine how long it takes for the capacitor to discharge to 1% of this initial value.
[6]
7
w) The 238
92 U decays according to
238
92 U
4
→ 234
90 Th + 2 He
Determine the kinetic energy of the emitted α−particle in MeV.
c = 2.998 × 108 m s−1
−25
Mass of the 238
kg
92 U nucleus is 3.85395 × 10
−25
234
Mass of the 90 Th nucleus is 3.78737 × 10 kg
Mass of the α−particle is 6.64807 × 10−27 kg
[5]
END OF SECTION 1
8
Section 2 — attempt two questions only
Question 2
This question is about Centre of Mass and Centre of Gravity.
Centre of mass has many applications in systems, and can enable otherwise complex
calculations to be simplified, often in collisions and rotational dynamics.
a) Two weights with masses m1 and m2 are attached to the ends of a light rod at distances
x1 and x2 respectively, measured from a point that is along the line of the rod but beyond
the end. A normal force, F , applied to some point on the rod will hold it in equilibrium.
(i) What is meant by the the word equilibrium?
(ii) By taking moments about the origin point of x1 and x2 , show that a single force, F ,
applied normal to the rod will hold it in equilibrium, if it acts at the point given by
XCM =
m1 x1 + m2 x2
m1 + m2
(iii) If an excess force ∆F is applied to the centre of mass of the rod, what is the ratio
of the kinetic energy gained of m1 to that of m2 ?
(3)
b)
(i) A 75 cm long uniform plank of mass 0.24 kg hangs over the edge of a table as in
Fig. 5 and supports a 0.35 kg mug of tea with a diameter of 8.0 cm which sits on
the end of the plank with one side on the edge. A pile of textbooks, each of mass
0.48 kg is used weigh the plank down on the table, with 15 cm of plank resting
under the books that are also 15 cm wide. How many books are required to support
the system?
(2)
Figure 5: Cup of tea on a plank supported by books.
(ii) Two uniform, similar rods of the same material AB of lengths LAB and BC of length
LBC respectively, are joined rigidly at B. They are suspended from A on a string
and the rods hang at angles α and β to the vertical.
LAB
Obtain an expression for
in terms of α and β.
LBC
(3)
9
(iii) A wooden stick of length L and density ρw has a rectangular cross section of side b.
It floats in a liquid of density ρl . If it floats vertically, obtain an expression for the
vertical difference in heights of the centres of mass of the liquid displaced and the
stick, in terms of L, ρw and ρl
(1)
(iv) A light rod of length L = 30 cm has masses of 0.40 kg and 0.90 kg attached to
the ends of the rod. It is thrown vertically upwards with a flick of the wrist, from
a height h above the ground, rotating at eight times each second, and returning to
height h. It is given 110 J of energy initially.
How many rotations does it make before it lands?
(3)
c) A small ball of mass 31 m travelling at speed 2v collides linearly and inelastically with a
ball of mass 23 m moving at speed v. A set of equal time interval photographs are taken
of the collision, and are set out in Fig. 6.
(i) Sketch this and add to it the position of the centre of mass of the particles.
(ii) From measurements of the tracks on this paper, estimate the fraction of the KE lost
in the collision.
(2)
Figure 6: Collision of two particles.
10
d)
(i) A thin uniform string of mass m and length 2a hangs over a light, smooth pulley
of negligible radius. When displaced from equilibrium, it slides off and leaves the
pulley at speed v. Obtain an expression for v in terms of a and g.
(4)
(ii) In a pendulum clock, the length of the pendulum must be kept constant,
independent of temperature changes, if the clock is to run at the correct rate. In the
diagram of Fig. 7, a triangular support for the pendulum mass is made of an
isosceles triangle of rods, the horizontal rod of length L1 with coefficient of
thermal expansion α in ◦ C−1 and two sloping rods of length L2 of a different metal
with coefficient of thermal expansion β. The pendulum mass is attached at its
centre and the pendulum is supported so that it can swing in and out of the page.
If the pendulum is temperature compensated for small temperature changes ∆T ,
L1
what is the required ratio of the lengths
in terms of α and β?
L2
(3)
Figure 7: A temperature compensated pendulum constructed from a triangular support of rods, so that
it can swing in and out of the page.
(iii) A light rod of length ℓ has two similar masses m attached to the left and right ends.
It is suspended horizontally by a string in a non-uniform vertical gravitational field
which increases linearly in strength from g at the left end to 1.1g at the right end.
i. When the rod is balanced, as a fraction of ℓ, what is the difference between the
centre of mass of the system and the point of suspension (the centre of gravity)?
ii. If the string is now attached to the centre of the rod, a third mass m must also
be attached to the rod in order to balance it. In terms of ℓ, how far from the left
end of the rod must this mass be attached?
(4)
[25 marks]
11
Question 3
This question is about investigating materials with X rays and electrons.
a) An accelerated charge radiates electromagnetic radiation. A synchrotron electron storage
ring uses this idea to produce a continuous spectrum of electromagnetic radiation as the
electrons are accelerated in a circular trajectory.
(i) Particular wavelengths of light can be selected using a monochromator, some of
which make use of diffraction and interference from crystals. The diagram shows
two rays from the continuous spectrum striking two adjacent crystal planes
separated by a distance a. The crystal planes act as partially reflecting mirrors. For
certain angles θ, measured to the reflecting plane, the emerging rays are
monochromatic.
Figure 8: Reflection from two atomic surfaces in a crystal.
(ii) What is the path difference between the two rays shown, in terms of a?
(iii) Determine the wavelength in terms of θ and a.
(iv) If a = 0.31 nm, calculate the first order wavelength of light strongly scattered at an
angle of θ = 15◦ , and
(v) the direction and magnitude of the change in momentum of the photons of this
wavelength. You may use a diagram.
(5)
b) In practice two crystals are used such that the light enters and leaves the monochromator
in the same direction. The height, h between the incoming and outgoing beams is fixed.
Using a suitable approximation to first order, obtain a formula for x in terms of the
wavelength λ, the crystal lattice spacing a, and the height h.
(3)
Figure 9
12
c) Monochromatic light of part (b) is incident on a sample of copper so that electrons are
emitted via the photoelectric effect. The energy of the emitted electrons are investigated
using a hemispherical electron energy analyser, shown in Fig. 10.
E0 , referred to as the pass energy, is the energy of the electron travelling from the
analyser entrance to the exit slit of the electron detector along the equipotential path
defined by, R0 = Rout2+Rin , in which Rin and Rout are the inner and outer hemisphere
radii, respectively.
These charged spheres produce a radial electric field between them. Only electrons of
the particular energy E0 , describe a perfectly semi circular path and are detected. The
inner sphere is at a potential Vin and the outer one is at a potential Vout so as to make the
electric field radial.
Figure 10: Hemispherical electron energy analyser.
Figure published in Progress in Materials Science 2020 X-ray photoelectron spectroscopy: Towards reliable binding energy referencing G.
Greczynski, L. Hultman
(i) Draw a sketch of the electric field lines between the hemispheres.
(ii) Write down the potential Vin of the inner hemisphere in terms of the charge Q on
the hemisphere and its radius Rin .
(iii) In the simplest form possible, determine an expression for the pass energy, E0 in
terms of the radii of the hemispheres Rin and Rout , the difference in the potentials
(i.e. ∆V = Vin − Vout ), and the electronic charge e.
(iv) If the mean radius of the hemispheres is 50 mm with a separation of 10 mm, what
potential difference between the plates gives a pass energy of 5.0 eV?
(7)
d) In a solid such as copper, the electrons exist in very closely spaced energy levels, which
can be investigated. A 5.0 eV free electron is produced, for example, when an electron is
in an energy level 4.0 eV below the top level in the solid and is released without collision
in photoelectric emission. The workfunction of copper is 4.6 eV.
(i) What was the photon energy used?
13
(ii) what wavelength is this and which part of the electromagnetic spectrum does it
correspond to?
(3)
(iii) When an electron travels from the solid into the vacuum, it changes speed, similar
to the behaviour of light crossing a boundary. Whilst inside the material the electron
was travelling in the direction shown in Fig. 11, at θ = 50◦ to the surface normal.
i. Calculate a value for the (relative) refractive index µ > 1, corresponding to the
electron going from the vacuum into the copper.
ii. For the the electron leaving the copper at θ = 50◦ , at what angle does it emerge
into the vacuum?
(2)
Figure 11: An electron crossing a surface from the vacuum.
e) Light of wavelength λ = 640 nm passes through an evacuated tube of length ℓ = 8.0 cm
with thin glass ends. The beam interferes with a second coherent beam of light on a
screen, causing destructive interference, as shown in Fig. 12. Air has a refractive index
given by µair = 1 + 0.00029 PP0 where P0 = 1.0 × 105 Pa. What minimum pressure of
air in the tube would cause the interference pattern to become constructive interference
at the screen?
(2)
Figure 12: Interference between two beams from a coherent source of light. One beam passes through
a partially evacuated tube.
f) The refractive index of a material for electrons depends upon the wavelength of the
electron. So if we instead consider X rays reflecting off adjacent layers of a crystal, we
14
Figure 13: X ray interference between layers of atoms.
can take into account the effect of refraction within the material.
An example is illustrated in Fig. 13 in which adjacent layers of atoms are laid down on a
substrate. X rays are incident at angle θ1 and refracted in the second layer at angle θr . If
there are multiple layers and reflections, then these will dominate intensity of the
reflection at the first surface from the vacuum. So we we can look at the path difference
between the emerging rays G and H. In this very simplified model, the layers have been
given the same thickness a.
(i) Determine the path length BCD.
(ii) What is the optical path difference between rays G and H?
(3)
[25 marks]
15
Question 4
This question is about the physics of the DART mission to deflect an asteroid.
The Double Asteroid Redirection Test (DART) was a mission by NASA to measure the
effectiveness of the deflection of an asteroid through a high-speed impact. Such a strategy
would be a cheap way of achieving planetary defence against such asteroids should one of
comparable size threaten the Earth. The DART probe successfully impacted into Dimorphos,
an asteroid moon of the asteroid Didymos, on 26th September 2022 (see Fig. 14). The target
was chosen as it would only affect the orbit of the moon and so there was no danger of it being
pushed into an Earth-crossing orbit, and Dimorphos is comparable to the 140 m minimum size
to be considered a potentially hazardous asteroid, so is like the real threats we might face.
(a)
(b)
Figure 14: Observations before and after impact.
Fig. (a): The view of the asteroid Didymos (bottom left) and its moon Dimorphos (top right) taken by
DART as it was heading towards the moon for impact. Credit: NASA/Johns Hopkins APL
Fig. (b): The view of the pair taken with the probe LICIACube just after its closest approach, from a
vantage point a little behind the DART impactor. There is clearly a considerable amount of material
ejected from the surface of Dimorphos, as expected from a successful impact. This amplifies the effect
of the impact on the orbit of Dimorphos, giving a larger change in orbital velocity than by just the
probe alone. Credit: ASI/NASA
a) DART was fitted with the most up-to-date ion drive available (NEXT-C) to ensure it could
accelerate up to the required impact velocity with only a fraction of the mass of fuel that
would be needed for conventional rockets. It does this by applying a very small thrust
for a very long time. The fuel used was singly ionised xenon-131 ions.
The accelerating potential was 1800 V.
(i) What was the ejection speed of the xenon? Assume the xenon ions started from
rest.
(ii) The current of ions was 3.52 A. What is the thrust generated?
(iii) The probe hit Dimorphos 306 days after launch. If the ion drive had been run
continuously for that time (and there was enough xenon to do so), what would
16
have been the speed at impact?
Assume linear acceleration of the probe in deep space starting from rest, and take
the average mass of the probe to be 600 kg during the acceleration phase.
(6)
Dimorphos was discovered through dips in the brightness of Didymos as it passes in front and
behind the asteroid (known as transits).
b) Radar observations of Didymos measured its diameter to be 780 m (assuming a spherical
shape) and the centre-to-centre distance to Dimorphos to be 1.20 km (assuming a circular
orbit). The relative depth of the dips in brightness meant that the diameter of Dimorphos
was derived to be 164 m (again assuming a spherical shape), and the period of the orbital
motion was 11 h 55 min.
(i) By considering the forces acting on each of the objects in their circular orbits,
calculate the total mass of the binary system.
(ii) Assuming both asteroid and moon have the same density, calculate the mass of
Didymos and Dimorphos.
(3)
c) The DART probe had a mass of 570 kg at impact and was travelling at a speed of
6.14 km s−1 (relative to the rest frame of Dimorphos).
(i) By conserving momentum and assuming a head-on inelastic collision, determine
the change in Dimorphos’ orbital speed, ∆v. Give your answer in mm s−1 .
(ii) The impact will have made the orbit elliptical, although by considering the total
energy of the orbit you can calculate the radius of an equivalent circular one.
i. obtain an expression for the total energy of a satellite, ms , in orbit about a
central mass, mc , in terms of G, mc , ms and the radius of orbit r.
Using this approach, calculate
ii. the expected decrease in orbital radius, ∆r, and hence
iii. the expected decrease in orbital period, ∆T , giving your answer in minutes.
(10)
d) Less than a month after the impact, it was announced that the observed orbital period
shortening was 32 minutes. This is larger than the one in part (c) due to the momentum
transfer enhancement factor, β, which quantifies how much the ejecta from the surface
and deformation of Dimorphos has amplified the effect. To calculate β accurately is
quite complicated and will need months of observations, but a very rough estimate can
be made using the approximation β ∝ ∆v.
Treating the value in part (c) as the β = 1 case, find the observed value of ∆v and hence
a ballpark estimate for the value of β.
(6)
[25 marks]
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Question 5
This question is about the physics involved in star formation.
The James Webb Space Telescope (JWST) has only been publicly releasing images since July
2022 and has already shown it is a more than worthy successor to the Hubble Space Telescope
(HST). In Fig. 15 is a recent side by side comparison is of the ‘Pillars of Creation’ in the Eagle
Nebula – a famous area undergoing considerable star formation.
Figure 15: The ‘Pillars of Creation’ – a famous star-forming region in the Eagle Nebula – as viewed
with the Hubble Space Telescope (left) and the James Webb Space Telescope (right). Since the JWST
operates at infrared wavelengths it can see through the dust and reveal so many of the new stars that
have been born. Credit: NASA, ESA, CSA, STScI; Joseph DePasquale (STScI), Anton M. Koekemoer
(STScI), Alyssa Pagan (STScI).
Some of the new stars being born will be large and hot, so will have strong stellar winds and
considerable ionising radiation that will carve out holes around them, meaning that over time
the dust columns as we see them will disappear. The ionised bubble is known as a Strömgren
Sphere and is created by ionising photons (those with energies greater than 13.6 eV) ionising
any hydrogen atoms in the vicinity. At the edge there is a balance between new ionisations and
electrons recombining with hydrogen nuclei.
The formula for the radius of the sphere is:
1
3N 3 − 23
rS =
nH
4πα
where N is the number of ionising photons per second, α quantifies the probability that a
proton and electron recombine into a hydrogen atom, and nH is the number density of
hydrogen nuclei in the sphere (assumed to be the same as the number density of electrons).
Some Solar data:
Mass of the Sun, MSun = 1.99 × 1030 kg
Radius of the Sun, RSun = 6.96 × 108 m
Luminosity of the Sun (power output), LSun = 3.85 × 1026 W
Effective surface temperature of the Sun, TSun = 5780 K
Peak wavelength in solar spectrum, λSun = 500 nm
Distance from Sun to Earth = 1.50 × 1011 m
a) Infrared measurements by JWST suggest that in this region some O and B class stars
will be formed. An example of such a star has a mass of 23MSun , radius of 10.0RSun ,
luminosity of 1.47 × 105 LSun , and effective surface temperature of 35 800 K.
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(i) Given that the peak wavelength in a blackbody spectrum (i.e. the wavelength λmax
at which the power emitted is a maximum) is inversely proportional to the effective
surface temperature, T , so T λmax = constant, verify by calculation that photons
emitted at the star’s peak wavelength will be ionising.
(ii) In the Pillars of Creation, nH ≈ 109 m−3 and ionisation will lead to an equilibrium
temperature inside the sphere of 8000 K for which α = 3.1 × 10−19 m3 s−1 .
Assuming that the star emits monochromatically at just the peak wavelength,
estimate N and hence estimate rS . Give your answer in light years (ly).
(iii) The height of the pillar on the left is 5 ly. How many such large stars will be needed
to photoevaporate the pillar (given long enough), such that it is no longer visible?
Assume they are evenly spread out throughout the pillar.
(5)
As well as the ionising effects on its surroundings, a new, bright star will exert radiation
pressure, and this can trigger further collapses of regions of the nebula into other stars.
Assuming the region outside the Strömgren Sphere is purely neutral hydrogen, the mass that
will be collapsed by an external pressure p0 is given as:
Mcollapse =
1.18kB2 Tneb2
1
3
p02 G 2 m2H
This is known at the Bonner-Ebert mass, where kB is the Boltzmann constant, Tneb is the
temperature of the neutral hydrogen in the nebula, and mH is the mass of a hydrogen atom. The
radiation pressure is given as:
L
prad =
4πr2 c
where L is the luminosity of the star (the power output), r is the distance from the star, and c is
the speed of light.
b) Taking the temperature of the nebula to be Tneb = 50 K, and taking p0 = prad , calculate
the cloud mass that will begin to collapse just outside the Strömgren Sphere and form a
new star (cluster). Give your answer in solar masses.
(2)
Radiation pressure can also be important in some stars in keeping them in equilibrium against
the gravitational forces trying to compress them. The gravitational pressure at the centre of a
star is given by:
Z Rstar
GM (r)ρ(r)
dr
pGrav =
r2
0
where M (r) is the mass enclosed within radius r, and ρ(r) is the density. Since in real stars
both M and ρ are complicated functions of r, this integral is not straightforward to do and
generally can only be done numerically. In equilibrium, this will be balanced by contributions
from the gas pressure and blackbody radiation pressure:
ρc kB Tc 4σ 4
+
T
m
3c c
where ρc and Tc are the central density and central temperature, respectively, σ is the StefanBoltzmann constant, and m is the average mass of the particles. Generally, in the core of the
star is a mix of fully ionised hydrogen and helium, so
ptot = pgas + prad =
m
2
=
mH
1 + 3X + 0.5Y
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where X and Y are the mass fractions of hydrogen and helium, respectively.
c) At the centre of the Sun, lots of its initial hydrogen has already been turned into helium,
so X = 0.35, Y = 0.65, ρc = 1.53 × 105 kg m−3 and Tc = 1.57 × 107 K.
(i) Estimate the pressure at the centre of the Sun, ptot .
(ii) Calculate prad as a percentage of ptot . Comment on your answer.
(iii) Consider a solar mass, Earth-sized white dwarf and assume it has a constant density
equal to its average density. Carry out the integration and estimate pGrav . How many
times larger is it than the solar ptot ?
REarth = 6370 km
(10)
The photons generated by the nuclear fusion in the core of the Sun are gamma ray photons,
however by the time they reach the surface of the Sun they are mostly visible light. This is
because the photons spend a very long time being absorbed and then re-emitted by particles in
the plasma, and initially this is via Compton scattering, which lowers the energy of the photon
with each emission, and then later via Thomson scattering, which does not affect the photon
energy but does mean it continues to bounce around in the star.
Throughout all the scattering, the photons follow a “random walk”, meaning that after N steps
of average step size ℓ (known as the mean free path), the final displacement, d, is
√
d=ℓ N
4σ 4
E
T and the mean temperature of the
d) The energy density of blackbody photons is =
V
c
6
Sun is 4.73 × 10 K.
(i) By considering the total energy stored as EM radiation in the Sun and the current
luminosity, estimate the time it takes for a photon to diffuse from the core to the
surface. Give your answer in years.
(ii) Hence find the mean free path, ℓ, and the number of interactions, N .
(4)
Unlike photons, neutrinos produced in the fusion reactions hardly interact at all with the
dense, opaque plasma in the core. Consequently, they can tell us about the energy production
happening now, which can be compared to the observed luminosity and alert us to any
changes over the photon diffusion timescale. In 2018, the Borexino experiment detected, for
the first time, neutrinos from the proton-proton I branch from the Sun. This branch is
responsible for 92% of the Sun’s energy output, and each neutrino corresponds to 13.1 MeV of
energy provided to the star by fusion reactions. The measured neutrino flux at the position of
the Earth was (6.1 ± 0.5) × 1010 neutrinos cm−2 s−1 .
e) Use the photon output luminosity to predict the neutrino flux at Earth and compare it to
the measured value. Comment on the implications of your answer.
(4)
[25 marks]
END OF SECTION 2
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