Physics_paper_3_tTZ2_HL_

Question Section A

Answer all questions. Answers must be written within the answer boxes provided.
A student investigates the relationship between the pressure in a ball and the maximum force that the ball produces when it rebounds.
A pressure gauge measures a difference ∆p between the atmospheric pressure and the pressure in the ball. A force sensor measures the maximum force Fmax exerted on it by the ball during the rebound.

(a) State one variable that needs to be controlled during the investigation. [1]

The student collects the following data.

The student initially hypothesizes that $F_{\max }$ is proportional to $\Delta p$.

(b) Deduce, using two suitable data points from the table, that the student’s initial hypothesis is not supported.[3]

The student now proposes that $F_{\max }^3=k \Delta p$.
The student plots a graph of the variation of $F_{\max }^3$ with $\Delta p$.

(ii) Plot on the graph the position of the missing point for the $\Delta p$ value of $40 \mathrm{kPa}$.[1]

The percentage uncertainty in $F_{\max }$ is $\pm 5 \%$. The error bars for $F_{\max }^3$ at $\Delta p=10 \mathrm{kPa}$ and $\Delta p=80 \mathrm{kPa}$ are shown.

(d) (i) Calculate the absolute uncertainty in $F_{\max }^3$ for $\Delta p=30 \mathrm{kPa}$. State an appropriate number of significant figures for your answer.[3]

The percentage uncertainty in $F_{\max }$ is $\pm 5 \%$. The error bars for $F_{\max }^3$ at $\Delta p=10 \mathrm{kPa}$ and $\Delta p=80 \mathrm{kPa}$ are shown.

(d) (i) Calculate the absolute uncertainty in $F_{\max }^3$ for $\Delta p=30 \mathrm{kPa}$. State an appropriate number of significant figures for your answer.[3]

(ii) Plot the absolute uncertainty determined in part (d)(i) as an error bar on the graph. [1]

(iii) Explain why the new hypothesis is supported. [1]

▶️Answer/Explanation

Ans:

a height «of drop» $O R$ velocity «of ball» $O R$ kinetic energy uof ball» $O R$
temperature/mass/radius/surface area/volume of ball

b.refers to 2 non-adjacent points
suitable calculation to analyze the proportionality identifies variation/difference in calculated values, «thus hypothesis not supported» $\checkmark$

c(i) $\mathrm{N}^2 \mathrm{~m}^2$ OR $\mathrm{kg}^2 \mathrm{~m}^4 \mathrm{~s}^{-4}$ OR $\mathrm{N}^3 \mathrm{~Pa}^{-1} \checkmark$

(ii)point plotted at $\left(40 \mathrm{kPa}, 49 \times 10^5 \mathrm{~N}^3\right)$

d(i) $15 \%$ seen anywhere $\checkmark$
$
\begin{aligned}
& \star \Delta\left(F^3\right)=» 39.4 \times 10^5 \times 0.15=5.9 \times 10^5 \\
& \pm 6 \times 10^5
\end{aligned}
$

ii)error bar drawn at $30 \mathrm{kPa}$ from $34 \times 10^5$ to $46 \times 10^5 \mathrm{~N}^3 \checkmark$

(iii)a “straight” line can be drawn that passes through origin$\checkmark$

Question

A student conducts an experiment to determine the specific heat capacity of a metal cube. The cube is heated in a beaker of boiling water to a temperature of $100^{\circ} \mathrm{C}$ and then quickly transferred into an insulated vessel of negligible thermal capacity. The vessel contains water at $20^{\circ} \mathrm{C}$ and of known specific heat capacity.

(a) State one other measurement that the student will need to make.[1]

(b) Suggest one modification that the student can make to reduce the fractional uncertainty for the change in temperature of the metal cube.

(c) Some water from the beaker is accidentally transferred with the cube.
Discuss how this will affect the value of the calculated specific heat capacity of the cube. [2]

▶️Answer/Explanation

Ans:

a final temperature of equilibrium/water/cube OR mass of water/cube

b smaller mass of cube OR hotter cube OR more mass of water OR

colder water OR more precise thermometer

c.cube specific heat will be too large/increased value/overestimate 

additional «thermal» energy transferred
OR
temperature rise of water will be larger
OR
temperature drop of cube will be smaller

QuestionSection B

Answer all of the questions from one of the options. Answers must be written within the answer boxes provided.
Option A – Relativity
A wire carries an electric current. An external electron e moves with the drift velocity $v$ of the electrons in the wire. Observer $\mathrm{O}$ is at rest relative to the wire.

(a) State what is meant by a frame of reference. [1]

(b) State and explain the nature of the electromagnetic force acting on electron e in the frame of reference of

(i) observer O. [2]

(ii) electron e. [2]

▶️Answer/Explanation

Ans:

a set of coordinates/axes to record position and time «of an event»

OR
a coordinate system which is at rest/not moving relative to the observer

b i magnetic

«observer O sees» moving charge in a magnetic field

ii electric / electrostatic

positive lattice ions are length contracted relative to e
OR
electrons sees positive wire

Question

Star A and star B are separated by a fixed distance of 4.8 light years as measured in the reference frame in which they are stationary. An observer P at rest in a space station moves to the right with speed 0.78c relative to the stars. A shuttle S travels from star A to star B at a speed of 0.30c relative to the stars.

(a) State the value of the maximum distance between the stars that can be measured in any reference frame.

(b) Write down the speed of shuttle S relative to observer P using Galilean relativity. [1]

(d) Show that the speed of shuttle S relative to observer P is approximately 0.6c. [2]

(e) Calculate the time, according to observer P, that the shuttle S takes to travel from star A to star B.

(f) Identify and explain the reference frame in which the proper time for shuttle S to journey from star A to star B can be measured. [2]

▶️Answer/Explanation

Ans:

a.4.8 «light yearsw $\checkmark$

b.«-» 0.48c $\checkmark$

c.$\begin{aligned} & \gamma=1.6 \\ & D=\frac{4.8}{1.6}=3 \text { «ly« } ~\end{aligned}$

d.$\begin{aligned} & =\frac{0.3 c-0.78 c}{1-\frac{0.78 c \times 0.3 c}{c^2}} \\ & \alpha-\infty 0.63 c\end{aligned}$

e.$
\Delta t_{\mathrm{p}}=\frac{3 \mathrm{ly}}{(0.78-0.627) c}
$
OR
$
\begin{aligned}
& \Delta t_{\mathrm{p}}=1.6\left(\frac{4.8 \mathrm{ly}}{0.3 c}-\frac{0.78 \mathrm{c} \times 4.8 \mathrm{ly}}{c^2}\right) \\
& =19 \text { OR } 20 \text { «years» }
\end{aligned}
$

f.shuttle measures proper time$\checkmark$
as the events occur at the same place for the shuttle / shuttle is at both events $\checkmark$

Question

The spacetime diagram shows the Earth frame with the worldline of a spaceship S moving away from Earth. ct ′ = 0 when ct = 0.

(a) Determine the speed of the spaceship relative to Earth. [1]

A flash of light sent by an Earth observer at ct = 2.0km is directed towards the spaceship.
(b) Estimate, using the spacetime diagram, the time in seconds when the flash of light reaches the spaceship according to the Earth observer. [2]

(c) Determine the time coordinate ct ′ when the flash of light reaches the spaceship, according to an observer at rest in the spaceship.

▶️Answer/Explanation

Ans:

0.6 c
OR

$=1.8 \times 10^8$«$ms^{-1}$»$\checkmark$

b.Line drawn at $45^{\circ}$ from $\mathrm{ct}=2 \mathrm{~km}$ to hit spaceship world line at ct $=5 \mathrm{~km}$ OR
$
\begin{aligned}
& \mathrm{ct}=1.2 /(\mathrm{c}-0.6 \mathrm{c})+2=5 \ll \mathrm{km} » \cdot \checkmark \\
& t=\frac{5000}{c}=1.7 \times 10^{-5} \ll \mathrm{s} »
\end{aligned}
$

c.$
\left(c t^{\prime}\right)^2-0=5^2-3^2
$
OR
$
\begin{aligned}
& \gamma=1.25 \checkmark \\
& c t^{\prime}=4 \text { «km» }
\end{aligned}
$
OR
$
t^{\prime}=13 \ll \mu \mathrm{s} » \checkmark
$

Question

6In the laboratory frame of reference, a kaon decays spontaneously into a positive pion and a negative pion that then move in opposite directions.
$
\mathrm{K}^0 \rightarrow \pi^{+}+\pi^{-}
$
The rest mass of both the pions is $140 \mathrm{MeVc}^{-2}$. The $\pi^{+}$has a momentum of magnitude $340 \mathrm{MeVc}^{-1}$ and the $\pi^{-}$has momentum of magnitude $113 \mathrm{MeVc}^{-1}$.

(a) State the magnitude of the momentum of the $\mathrm{K}^0$ the instant before it decays.[1]

(b) Show that the energy of the $\pi^{+}$is approximately $370 \mathrm{MeV}$.[1]

▶️Answer/Explanation

Ans:

227 «MeVc-1

b.$
E_{\pi^{\prime}}=\sqrt{140^2+340^2}
$
OR
$
=368 \propto \mathrm{MeV} s
$

c.$\begin{aligned} & \text { E Pion minus }=\text { root of }\left(140^2+113^2\right)=\star 180 \mathrm{MeV} » \\ & \text { Rest Energy of } \mathrm{K}^0=\sqrt{(368+180)^2-(340-113)^2} \\ & =499 \approx \mathrm{MeV} \backsim\end{aligned}$

Question

A rocket ship is at rest on the surface of a non-rotating planet $X$. The rocket ship contains a chamber of height $20 \mathrm{~m}$. Photons are emitted with frequency $3.2 \times 10^{10} \mathrm{~Hz}$ and travel from the floor of the chamber to the ceiling of the chamber. A receiver on the ceiling detects the frequency of the photons.

(a) Explain why the frequency of the photons detected at the ceiling is less than the frequency of those emitted from the floor.

(b) The change in frequency detected at the ceiling as compared to the floor was measured to be $1.2 \times 10^{-4} \mathrm{~Hz}$. Deduce the gravitational field strength of planet $\mathrm{X}$. [2]

(c) The rocket ship is launched and accelerates vertically. Explain, with reference to the equivalence principle, why the magnitude of the frequency change observed in photons emitted from floor to ceiling of the rocket ship will increase as it launches. [2]

▶️Answer/Explanation

Ans:

a. some photon energy is used to do work against the gravitational field

b.$\begin{aligned} & \frac{1.2 \times 10^{-4}}{3.2 \times 10^{10}}=\frac{g \times 20}{c^2} \\ & =16.9 \mathrm{~km} \mathrm{~s}^2 \mathrm{v}\end{aligned}$
mention of gravitational redshift $\checkmark$

gravitational effects cannot be distinguished from inertial effects $\checkmark$

«apparent» acceleration within rocket is greater than g
OR
there is a stronger gravitational field
OR
the receiver is moving further away $\checkmark$

QuestionOption B — Engineering physics

6. A student models a rotating dancer using a system that consists of a vertical cylinder, a horizontal rod and two spheres.
The cylinder rotates from rest about the central vertical axis. A rod passes through the cylinder with a sphere on each side of the cylinder. Each sphere can move along the rod. Initially the spheres are close to the cylinder.

A horizontal force of 50N is applied perpendicular to the rod at a distance of 0.50m from the central axis. Another horizontal force of 40N is applied in the opposite direction at a distance of 0.20m from the central axis. Air resistance is negligible.

(a) Show that the net torque on the system about the central axis is approximately 30Nm. [1]

(b) The system rotates from rest and reaches a maximum angular speed of $20 \mathrm{rad} \mathrm{s}^{-1}$ in a time of $5.0 \mathrm{~s}$. Calculate the angular acceleration of the system. [1]

(c) Determine the moment of inertia of the system about the central axis. [2](c) Determine the moment of inertia of the system about the central axis. [2]

(d) When the system has reached its maximum angular speed, the two forces are removed. The spheres now move outward, away from the central axis.

(i) Outline why the angular speed ω decreases when the spheres move outward. [2]

(ii) Show that the rotational kinetic energy is $\frac{1}{2} L \omega$ where $L$ is the angular momentum of the system.[1]

(iii) When the spheres move outward, the angular speed decreases from $20 \mathrm{rad} \mathrm{s}^{-1}$ to $12 \mathrm{rads}^{-1}$. Calculate the percentage change in rotational kinetic energy that occurs when the spheres move outward. [2]

(e) Outline one reason why this model of a dancer is unrealistic. [1]

▶️Answer/Explanation

Ans:

a.$
\Sigma \Gamma=50 \times 0.5+40 \times 0.2
$
OR
$33 » N m » \checkmark$

b.« $\alpha=\frac{20}{5}=» 4 » \operatorname{rad~s} s^{-2} » \checkmark$

c.$
I=\frac{\Gamma}{\alpha}
$
OR
$
\begin{aligned}
& 33=I \times 4 \\
& I=8.25 \ll \mathrm{kg} \mathrm{m}^2 »
\end{aligned}
$

d(i)moment of inertia increases
Angular momentum is conserved $\checkmark$

(ii)$E_k \|=\frac{1}{2} I \omega^2=» \frac{1}{2}(I \omega) \omega=\frac{1}{2} L \omega \checkmark$

(iii)$
» E_k=» 1 / 2 L \omega_1=1 / 2 L \omega_2
$
OR
$
\frac{E_{k 1}}{E_{k 2}}=\frac{\omega_1}{\omega_2}
$
OR
” $L$ is constant so” $E_{\mathrm{k}}$ is proportional to $\omega \checkmark$ $40 \%$ «energy loss» $\checkmark$

e)one example specified eg friction, air resistance, mass distribution not modelled $\checkmark$

Question

A frictionless piston traps a fixed mass of an ideal gas. The gas undergoes three thermodynamic processes in a cycle.

The initial conditions of the gas at $A$ are:
$
\begin{aligned}
\text { volume } & =0.330 \mathrm{~m}^3 \\
\text { pressure } & =129 \mathrm{kPa} \\
\text { temperature } & =27.0^{\circ} \mathrm{C}
\end{aligned}
$

Process AB is an isothermal change, as shown on the pressure volume (pV) diagram, in which the gas expands to three times its initial volume.
(a) Calculate the pressure of the gas at B.[2]

The gas now undergoes adiabatic compression BC until it returns to the initial volume.
To complete the cycle, the gas returns to A via the isovolumetric process CA.

(b) Sketch, on the pV diagram, the remaining two processes BC and CA that the gas undergoes. [2]

(d) Explain why the change of entropy for the gas during the process BC is equal to zero. [1]

(e) Explain why the work done by the gas during the isothermal expansion AB is less than the work done on the gas during the adiabatic compression BC.

(f) The quantity of trapped gas is 53.2mol. Calculate the thermal energy removed from the gas during process CA. [2]

▶️Answer/Explanation

Ans:

use of $p V=$ constant $v$ $P_{\mathrm{B}}=43 » \mathrm{kPa} » \checkmark$

b.concave curved line from $B$ to locate $\mathrm{C}$ with a higher pressure than $\mathrm{A} \checkmark$ vertical line joining $C$ to $A \checkmark$

c.ALTERNATIVE 1
use of $T V^{\frac{2}{3}}=$ constant «so $300\left(3 V_A\right)^{\frac{2}{3}}=T_C\left(V_A\right)^{\frac{2}{3}} » \checkmark$
$
T_{\mathrm{C}}=624 » \mathrm{K} » O R T_{\mathrm{C}}=351 \|^{\circ} \mathrm{C} » \checkmark
$
ALTERNATIVE 2
use of $p V^{\frac{5}{3}}$ to get either $p_{\mathrm{c}}=43(3)^{\frac{5}{3}}$ OR $p_{\mathrm{c}}=268 « \mathrm{kPa} » \checkmark$ « $T_{\mathrm{c}}=268 \times 300 / 129=$ so »
$
T_{\mathrm{C}}=624 » \mathrm{K} » O R T_{\mathrm{C}}=351 \|^{\circ} \mathrm{C} » \checkmark
$

d.ALTERNATIVE 1
“the process is adiabatic so” $\Delta Q=0 \checkmark$
ALTERNATIVE 2
The compression is reversible $“$ so $\Delta S=0 » \checkmark$

e.area under curve $A B$ is less than area under curve $B C \checkmark$

f.$\begin{aligned} & \text { «W }=0 \text { so» } \mathrm{Q}=\Delta \mathrm{U} \\ & \text { «U }=\frac{3}{2} \times 53.2 \times \mathrm{R} \times(351-27) \text { so } » \Delta \mathrm{U}=2.15 \times 10^5 \text { «J» }\end{aligned}$

Question

A large tank is used to store oil of density $850 \mathrm{kgm}^{-3}$ and is filled to a height $h_1$ above the bottom. A valve in the tank wall allows oil to flow out. The centre of the valve is at a height $h_2$ from the bottom of the tank. A circular drainage outlet is at the bottom of the tank.

The drainage outlet has a diameter of $100 \mathrm{~mm}$ and a metal stopper of mass $2.5 \mathrm{~kg}$ is used to plug the outlet.

(a) Determine the minimum force required to lift the stopper when $h_1=4.0 \mathrm{~m}$.[3]

With the metal stopper in place, the valve on the side of the tank is opened to let oil flow out.
Using Bernoulli’s equation, it can be shown that the speed $v$ of oil flowing through the valve can be estimated as $v=\sqrt{2 g\left(h_1-h_2\right)}$.

(b) State two assumptions that were used in obtaining the expression for the speed $v$.[2]

(c) Estimate the maximum radius of the valve so that turbulent flow does not occur. The following data are given: [2]
$
\begin{aligned}
\text { Viscosity of oil } & =0.25 \text { Pas } \\
h_1 & =4.0 \mathrm{~m} \\
h_2 & =0.5 \mathrm{~m}
\end{aligned}
$

▶️Answer/Explanation

Ans:

calculates $P_{\text {bothom }}$ \& $=850 \times 9.8 \times 4=P_{\text {banton }} \|=850 \times 9.8 \times 4 n=33320 \mathrm{~Pa}$ “
OR
calculates weight of oil« Weight $_{\mathrm{od}}=850 \times\left(\frac{0.1^2 \pi}{4}\right) \times 4 \times 9.8 n$
Add the weight of the stopper $\approx F_{\min }=33320 \times\left(\frac{0.1^2 \times \pi}{4}\right)+2.5 \times 9.8 n$
$
F_{\min }=286 \mathrm{~N} \checkmark
$

b.the pressure “at the valve opening and at the top of the oil in the tank» is constant
the velocity «of oil surface» at the top “of the tank» is zero / negligible no energy/head losses “as the oil flows through the valven.
no turbulence/ laminar flow occurs «as the oil flows through the valvew $\checkmark$ fluid not compressible
ideal fluid
no viscosity

c.$\begin{aligned} & \text { Use of } \operatorname{Re}=1000 \text { OR states } \operatorname{Re}=\frac{\sqrt{2 g(4-0.5)} \times r \times 850}{0.25} \\ & \text { « } 1000=\frac{\sqrt{2 g(4-0.5)} \times r \times 850}{0.25} \text { so } \# r=3.6 \ll \mathrm{cm} »\end{aligned}$

Question

A mass vibrating on a vertical spring is driven by a sinusoidal force. The graph shows the variation of the amplitude of vibration with driving frequency for the mass. The dampinginitially applied to the vibrating system has a Q factor of 50.

(a) The damping is changed so that the Q factor is decreased. State and explain one change to the graph.

(b) The driving force is removed and the spring now oscillates freely with a Q factor of 30. Calculate the fraction of the total energy that has dissipated after one cycle is completed. [1]

▶️Answer/Explanation

Ans:

Q decreases so damping increases $\checkmark$ amplitude is lower «everywhere»
OR
peak shifts left / to lower frequency $\checkmark$

b.$\frac{2 \pi}{30} \kappa=\frac{\text { energy dissipated per cycle }}{\text { energy stored }} \cdots=0.21$ or $21 \% \checkmark$

Question

An object is placed in front of a concave mirror with the focal point f as shown.

(a) Construct a ray diagram to locate the position of the image produced. [2]

(b) Describe the features of the image produced.

State the problem illustrated by the diagram and how it is corrected in reflecting telescopes.

▶️Answer/Explanation

Ans:

one ray drawn correctly reflected $\checkmark$
bottom and top of image located within the accepted region $\checkmark$

b.real
OR
inverted
OR
smaller image or magnification<1$\checkmark$

spherical aberration OR rays do not meet at common focus OR blurred image $\checkmark$
parabolic «shape» $\checkmark$

Question

The eye of an observer has a near point of 25cm. A pencil is placed at the near point.

A convex lens of focal length 8cm is then placed between the pencil and the observer as shown. The pencil is positioned at the focal point of the lens.

(b) A student increases the magnification of the pencil by using two 8cm focal length convex lenses placed 25cm apart. The pencil is placed 14cm from one of the lenses.

(i) Show that the magnitude of the magnification of the pencil produced by the lens closest to the pencil is approximately 1.3.

(ii) Calculate the total magnification observed by the student using the two lenses as shown.

(c) The two 8cm focal length convex lenses are now used to construct a telescope in normal adjustment. The diameter of the lenses is much greater than the diameter of the pupil of the eye. State, compared with the naked eye,
(i) one advantage of using this telescope for astronomical observations. [1]

(ii) one disadvantage of using this telescope for astronomical observations. [1]

(d) Describe how international collaboration can improve the quality of the image of radio array telescopes.

▶️Answer/Explanation

Ans:

$M=« \frac{25}{8}=» 3.1$ $\checkmark$

b(i)$\begin{aligned} & \frac{1}{8}=\frac{1}{14}+\frac{1}{v} \text { OR } v=18.7 \\ & M=\frac{18.7}{14} \text { OR } M=1.33 \text { OR } M=1.34\end{aligned}$ $\checkmark$

(ii)$
\frac{1}{8}=1 /(25-18.66)+1 / v
$
OR
$
\begin{aligned}
& v=-30.4 \\
& m=« m_1 \times m_2=1.33 \times \frac{30.4}{(25-18.66)}=1.33 \times 4.8=» 6.4
\end{aligned}
$

c i) better resolution «than the eye» $\checkmark$
c ii) no magnification $O R$ m=1 $\checkmark$

d.) international means base line can be across countries/continents$\checkmark$
greater distances increase the effective diameter of the dish $\checkmark$
great diameter improves resolution$\checkmark$

Question

Signals in an optic fibre require amplification when intensity levels in the fibre have fallen to $1.5 \%$ of the original signal. A light signal of initial intensity $I_0$ is sent down the optic fibre.
(a) The fibre has an attenuation per unit length of $0.30 \mathrm{~dB} \mathrm{~km}^{-1}$. Deduce that the length of the fibre is approximately $60 \mathrm{~km}$ before the signal requires amplification.[2]

A signal is sent down a 27km step-index fibre and received according to the intensity–time graph below.

(b) Calculate the refractive index of the fibre. [2]

▶️Answer/Explanation

Ans:

a.attenuation $=10 \log _{10}(0.015) O R=-18.2 \checkmark$

$
\text { length }=\frac{-18.2}{-0.3}
$
OR
$
60.8 « \mathrm{~km} » \checkmark
$

b.$\begin{aligned} & \Delta t=14 \times 10^{-5} \mathrm{~s} \\ & « \frac{27 \times 10^3}{14 \times 10^{-5}}=\frac{3 \times 10^8}{n} \rrbracket: n=1.56\end{aligned}$

c.refractive index of fibre is less at the edges $\checkmark$ reduces time taken for the longer path signals $\checkmark$

Question

The X-ray attenuation coefficient values for bone and muscle at an energy of $100 \mathrm{keV}$ are shown.
Bone attenuation coefficient $=0.348 \mathrm{~cm}^{-1}$
Muscle attenuation coefficient $=0.173 \mathrm{~cm}^{-1}$
(a) Show that the half-value thickness of bone when using $\mathrm{X}$-ray energies of $100 \mathrm{keV}$ is approximately $2 \mathrm{~cm}$.[1]

A monochromatic X-ray beam of energy $100 \mathrm{keV}$ and intensity $I_0$ is incident on muscle of thickness $4.0 \mathrm{~cm}$.

(b) Calculate, in terms of $I_0$, the final beam intensity that emerges from the muscle.[2]

An X-ray beam of energy $100 \mathrm{keV}$ and intensity $I_0$ is directed at a section of the upper leg that can be modelled using $4.0 \mathrm{~cm}$ of muscle, $2.0 \mathrm{~cm}$ of bone and then $4.0 \mathrm{~cm}$ of muscle as shown.

Additional attenuation coefficient values for bone and muscle are shown for X-ray energies of $1 \mathrm{keV}$ and $10 \mathrm{keV}$.

(d) Compare, with reference to contrast and sharpness, the final images formed when $\mathrm{X}$-rays of $1 \mathrm{keV}$ or $10 \mathrm{keV}$ are incident on the same muscle-bone-muscle section of the upper leg.[2]

(e) Other medical imaging techniques include ultrasound and nuclear magnetic resonance(NMR) imaging.

(i) State one difference between an A scan and a B scan in ultrasoundmedical imaging.

(ii) Explain how position information is obtained in nuclear magnetic resonance (NMR) imaging. [2]

▶️Answer/Explanation

Ans:

$
\frac{\ln 2}{0.348}
$
OR
$
1.99 x \mathrm{~cm} w
$

b.$\begin{aligned} & \frac{I}{I_0}=\mathrm{e}^{-0.173 \times 4} \\ & =0.50\end{aligned}$

c.$\begin{aligned} & I_{\text {muscle }} \times I_{\text {bone }} \times I_{\text {muscle }}=\left(\frac{1}{2}\right)^3 \\ & I=0.125 I_0 \checkmark\end{aligned}$

d.more contrast with $1 \mathrm{keV}$ OR less sharp with $1 \mathrm{keV}$ correct explanation for one of them, i.e. «more contrast with $1 \mathrm{keV}$ » as coefficients have greater ratio $O R$ «less sharp with $1 \mathrm{keV}$ m as the $\mathrm{X}$-rays will be scattered more

e(i) A scan is one dimensional $\checkmark$
$B$ scan is two dimensional $\checkmark$
B scan computed from multiple images $\checkmark$

(ii)gradient field is added to initial strong magnetic field
varies linearly across/along the patient
the flip/Larmor frequency varies linearly across the patient ahence position of proton frequency known»

Question

(a) The Ghost of Jupiter is a nebula.

(i) Outline what is meant by a nebula.

(ii) Astrophysicists have deduced the nature of this nebula from Earth. Outline how they can make these deductions. [1]

(b) Star X and star Y are in our own galaxy. They appear to move with respect to very distant stars when viewed from Earth during a six-month period. The following data are provided.

(i) Deduce which star will appear to move more. [2]

(ii) Calculate, in m, the distance to star X. [1]

(iii) Determine the ratio $\frac{\text { luminosity of star } X}{\text { luminosity of } \operatorname{star} Y}$.

▶️Answer/Explanation

Ans:

a i cloud/body of dust and gas $\checkmark$

a ii observation of light from/passing through nebula $\checkmark$

part of the model of stellar evolution $\checkmark$

b i Star Y$\checkmark$

because parallax angle is greater OR star Y is closer «and that
means movement relative to distant stars is greater» $\checkmark$

b(ii)$\begin{aligned} & \text { « distance }=\left(\frac{1}{0.019}\right) \times 3.26 \times 9.46 \times 10^{15} \\ & 1.6 \times 10^{18} \approx \mathrm{m} »\end{aligned}$

(iii)

$\frac{\text { Luminosity of Star X }}{\text { Luminosity of Star } Y}=\frac{b_x d_x^2}{b_y d_y^2}$
$=10.8 \approx 11$ 、

Question

Three stars A, B and C are labelled on the Hertzsprung–Russell (HR) diagram. L is the Luminosity of the sun.

(a) State the main element that is undergoing nuclear fusion in star C. [1]

(b) Explain why star B has a greater surface area than star A. [2]

(c) White dwarfs with similar volumes to each other are shown on the HR diagram. Construct a line, on the HR diagram, to show the possible positions of other white dwarf stars with similar volumes to those marked on the HR diagram. [2]
(d) Some stars on the HR diagram are likely to evolve into neutron stars. Outline why the radius of a neutron star reaches a stable value. [2]

▶️Answer/Explanation

Ans:

a. Hydrogen

b.stars have same/similar $L$ AND star B has lower $T \checkmark$
correct reference to luminosity formula $\left(L \alpha A T^4\right)$

c.Any evidence of correct identification that three dots bottom left represent white dwarfs$\checkmark$
line passing through all 3 white dwarfs $O R$ line continuing from 3 white dwarfs with approximately same gradient, in either direction$\checkmark$

d.«inward» gravitational force/pressure$\checkmark$
balanced by «neutron» degeneracy pressure/force $\checkmark$

Question

(a) State the Jeans criterion for the collapse of interstellar clouds.[1]

(b) For a main sequence star, the energy it releases during the total time $T$ it spends on the main sequence is proportional to its mass $M$.

(i) Show that $T \propto \frac{1}{M^{2.5}}$.[2]

(ii) For the Sun, $T=10^{10}$ years. Calculate $T$ for a star 20 times more massive than the Sun.[1]

▶️Answer/Explanation

Ans:

$
M>M_{\mathrm{i}}
$
OR minimum mass for interstellar matter to collapse into a star OR «interstellar» potential energy > kinetic energy

bi.ALTERNATIVE 1
$T \propto \frac{M}{L} L \propto \frac{M}{T}$
«and $L$ proportional to $M^{3.5}$ then»
$T \propto \frac{M}{M^{3.5}} \vee$
ALTERNATIVE 2
$L$ proportional $\frac{E}{T}$
AND
$L$ proportional $M \checkmark$
«and $L$ proportional to $M^{35}$ then»
$T$ proportional $\frac{M}{M^{3.5}} \checkmark$

b(ii)$\begin{aligned} & { }_{\text {a }} T_{20} M_{20}^{2.5}=T_{\text {sun }} M_{\text {sun }}^{2.5} \text { OR } T_{20}=10^{10} \times\left(\frac{1}{20}\right)^{2.5} \\ & « T_{20}=\cdots 1.8 \times 10^9 \ll \mathrm{s} » \text { OR } 5.6 \times 10^6 \text { «y» } \\ & \end{aligned}$

Question

(a) Determine the critical density of the universe using a Hubble constant value of $73 \mathrm{kms}^{-1} \mathrm{Mpc}^{-1}$.[2]

(b) Sketch, on the axes shown, the variation of $R / R_0$ (the cosmic scale factor $R$ divided by its present value $R_0$ ) with time for a universe where the density is greater than the critical density.[2]

▶️Answer/Explanation

Ans:

$\begin{aligned} & \rho_c=\kappa \frac{3 \times\left(2.37 \times 10^{-18}\right)^2}{8 \pi \times 6.67 \times 10^{-11}}=\$ 1.0 \times 10^{-26} \ll \mathrm{kg} \mathrm{m}^{-3} » \\ & \end{aligned}$

b.any irregular semicircular shape that returns to zero must pass through present and $\frac{R}{R_0}=1 v$

c.presence of dark energy accelerates the expansion of the universe this increases the rate of cooling

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