S354 Assignment Booklet I - archive.org

The Open Supplementary Material

University S354 Understanding Space and Time

S354 AB I

S354

Assignment Booklet I

Contents

2 CM A S354 41

(covering Units 1, 2 and 3)

5 TMA S354 01

(covering Units 1, 2 and 3)

Cut-off date

16 April 1993

23 April 1993

Continuous assessment: Please remember that Block 3 and Part C of

Block 6 are not included in the programme of study, nor therefore in the assessment.

The course timetable has been adjusted accordingly, to allow one revision period

between Blocks 2 and 4 and a second between Blocks 4 and 5. The numbering of

CMAs and TMAs has been kept in accordance with the block structure. That is

why there is no TMA 03, nor any CMA43. The continuous assessment of Parts A

and B of Block 6 consists solely of CMA46; there is no corresponding TMA.

Substitution: The marks for one of the 4 TMAs and one of the 5 CMAs are

substitutable. Remember that both your overall mark for continuous assessment

and your preparation for the examination will be improved by the submission of a

complete set of assignments. Substitution may upgrade the marks for your worst

CMA and/or your worst TMA; but the degree of upgrading depends on the results

of all assignments and on your examination performance.

Copyright © 1993 The Open University

13.1 DT/T SUP 24408 1



Computer Marked Assignment

Course and assignment number:

S354 41

Make sure you know how to use the CM A form: detailed instructions are

given in your student handbook (or supplement).

You are strongly advised to attempt every question in this assignment.

If you do not wish to answer a question, pencil across the ‘don’t know’ cell

on If you think that a question is unsound in any way, pencil across the

‘unsound’ cell (‘U’) in addition to pencilling across either an answer cell or

the ‘don’t know’ cell.

Note For each question you must pencil across either the required

number of answer cells or the ‘don’t know’ cell.

Covering: Units 1-3

Cut-off date:

Friday 16 April 1993

Ql to Q9 mainly concern Unit 1.

Q1 Points A and B have position vectors rA =

(5,7,2) and rB = (3,4,—4), where all lengths are measured in metres. What is the distance between these points? Select one option from the key for Ql. Pencil across one cell in row 1.

Q4 and Q5 refer to the motion of a certain particle. As viewed from an inertial frame S the position vector of the particle is

r(<) = (cos 4t, cos t, \/2sin t),

where the time t is measured in seconds and all lengths are measured in metres.

KEY for Ql

A 1 in

B 3 m

C 5 m

D) 7m

E 8 m

F 10 m

G 15 m

II 17m

Q2 What is the angle between the two displacement vectors

u = (l,-l,\/6) and v = (-1,1, V6)

where all lengths are measured in metres? Select one

option from the key for Q2. Pencil across one cell in row 2.

KEY for Q2

Q4 Select the one option from the key that gives

the distance in metres of the particle from the origin

of S at time t = it/4 s. Pencil across one cell in row 4.

KEY for Q4

A \/5 m

B \/3 m

C 5 m

D 3m

E 1 m

F 2 m

G y/2m

H \/3/2m

Q5 Select the one option from the key that gives

the speed of the particle in ms-1 in S at time t = 7t/4s. Pencil across one cell in row 5.

A 0° E 75° KEY for 5

B 30° F 90° A y/5 m s 1 E 1ms 1

1 45° G 120° B \/3 m s-1 F 2 ms-1

0

O

1

II 135° C 5ms-1 G v^ms"1

Q3 A triangle ABC is defined by the three D 3 ms'1 H \/3/2 m s -l

displacement vectors

AB = (1,1,3), BC = (2,1,1), AC - (3,2,4),

where all the lengths are measured in metres. What is the area of the triangle? Select one option from the key for Q3. Pencil across one cell in row 3.

KEY for Q3

A V6m2 E (\/l3/2) m2

B vTT m2 F 3%/lOm2

C \/l5m2 G 11m2

D \/30m2 H 6 m2

Q6 to Q9 In a given reference frame, a particle has position vector x = (a;1, x2, x3)., velocity vector v =

(v1, v2, v3) and acceleration vector a = (a1, a2,a3).

For each of Q6 to Q9, you are asked to select from the

key one combination of components of these vectors that is invariant under the transformation indicated. Each option may be used more than once.

Q6 A quantity that is invariant under any translation in space. Pencil across one cell in row 6.

Q7 A quantity that is invariant under any rotation in space. Pencil across one cell in row 7.

2

Q8 A quantity that is invariant under any rotation

about the 1-axis, but is not invariant under a rotation about the 2-axis. Pencil across one cell in row 8.

Q9 A quantity that is invariant under any reflection of the 1-axis, but is not invariant under a

reflection of the 2-axis. Pencil across one cell in rowr 9.

KEY for Q6 to Q9

A a1#3 + a2x2

B a1#1 + a3x2

C a2x3-a3x2

D v3a1 + v1a3

E x1a1 + x2a2 + x3a3

F x1v1 — x2v2 + x3v3

G — x^v1 + x2v2 + x3v3

Q10 to Qll refer mainly to Unit 2

Q10 and Qll refer to two planets that are spherically

symmetric and attract each other by Newton’s law of gravitation. In a certain inertial frame S, one planet

has mass m and position vector (a;1, x2, x3), while the other planet has mass M and position vector

(A1, A2, A3). G denotes the gravitational constant, and

R= y/ix1 - A1)2 + (x2 - A2)2 + (x3 - X3)2.

Q10 Select from the key the correct expression for

the 2-component of the acceleration imparted to the planet of mass m by the gravitational attraction of

the planet of mass M? Pencil across one cell in row 10.

Qll Select from the key the correct expression for the 2-component, of the acceleration imparted to the

planet of mass M by the gravitational attraction of

the planet of mass m. Pencil across one cell in row

11.

KEY for Q10 and Qll

A Gm(x2 - X2)/R2

B —Gm(x2 — X2)/R2

C Gm(x2 — X2)/R3

D -Gm{x2 - X2)/R3

E GM(x2 — X2)/R2

F -GM(x2 - X2)/R2

G GM(x2 — X2)/R3

H -GM(x2 -X2)/R3

Q12 and Q13 share the same key and concern a star, S, with two planets, A and B, each of which describes

a circular orbit around S and has a mass that is negligible in comparison with that of S. The distance from A to S is two astronomical units, i.e. twice the

mean distance from the Earth to the Sun. The period

of A’s orbit is one year. The distance from B to S is 8 astronomical units.

Q12 Select from the key the mass of S in units of the Sun’s mass. Pencil across one cell in row 12. & Q13 Select from the key the number of years that it takes for B to orbit S. Pencil across one cell in row 13.

(r

KEY for Q12 and Q13

D 1

E 2

F 4

G 8

II 16

Q14 and Q15 The entries in the key for Q14 and Q15 refer to some famous empirical laws.

Q14 Which two of the laws listed in the key provide the best support for the idea that the ratio of

gravitational mass to inertial mass is the same for all bodies? Select two options from the key. Pencil across two cells in row 14.

Q15 Which one of Kepler’s laws provides the best support for the idea that the gravitational force between two particles acts along the straight line joining them? Select one option from C, D and E in the key. Pencil across one cell in row 15.

KEY for Q14 and Q15

A Galileo’s second law

B Galileo’s first law

C Kepler’s third law

D Kepler’s second law

E Kepler’s first law

Q16 and Q17 The entries in the key for Q16

and Q17 refer to some Newtonian assumptions about Space and Time.

KEY for Q16 and Q17

A Space is isotropic.

B Time is homogeneous.

C Space is homogeneous.

D Reflections are discontinuous linear transform¬ ations.

E Rotations are continuous linear transform¬ ations.

F The Galilean transformation describes uniform linear boosts.

G None of A-F is relevant.

Q16 Which, if any, of the assumptions listed in the

key is closely asociated with the fact that equation (12) of Unit 2 involves only the displacement vector Xi — x2, rather than Xx and x2 separately? Select

one option from the key. Pencil across one cell in row 16.

Q17 Which, if any, of the assumptions listed in the key singles out the fact that Newton’s universal law of gravitation is an inverse square law, rather than

an inverse cube law? Select one option from the key. Pencil across one cell in row 17.

3

Q18 to Q22 may refer to material discussed in any of Units 1, 2 and 3.

Q18 Which two of the statements given in the key

are correct? Pencil across two cells in row 18.

KEY for Q18

., A In Newtonian mechanics, a reference frame that

is stationary in the laboratory is inertial.

B In Newtonian mechanics, all inertial reference

frames are at rest in absolute space.

C In Newtonian mechanics, it is impossible to

extend the axes of an inertial frame over all

space.

, D From the Principle of Relativity, it follows that

two equally calibrated rotating reference frames

cannot be distinguished from one another.

E From the Principle of Relativity, it follows that

y two equally calibrated inertial reference frames

that are related by a rotation in space are

physically indistinguishable.

F An inertial frame is the only type of frame

in which the homogeneity of space, the

homogeneity of time and the isotropy of space

all apply.

Q19 Which, if any, of the statements A to F in the

key is correct? Pencil across one cell in row 19.

Pencil across two cells in row 20.

KEY for Q20

A |aA | = fc|x2 + 3xx.x2|

B |ax| = fc|(xx - x2)2 + xx.x2|

C |ai| = fc|xx — 2x2|2

D |ai| = fc|xf + X2 — 2xi.x2|

E |ai| = fc|xf+x|+2xi.x2|

F |ax| = fc|(vx - v2).(Xl +x2|

G |ax| = fc|vx.xx+v2.x2|

H |a-L| = fc|xx.vx + x2.v2 - xi.v2 - x2.vx|

Q21 and Q22 refer to a head-on collision between

two particles. The first particle, having mass mi kg

moves along the x-axis in an inertial frame from

the left with velocity (u,0,0) towards the second

particle, having mass m2 kg, which initially sits at

rest at the origin. The force of interaction between

the two during the collision is characterized by a

time independent potential energy function. After

the collision, the first particle moves with velocity

(u',0,0) and the second with velocity (?;,0,0).

Throughout the entire process, you should assume

that the two particles are isolated, so that they

experience no forces other than those between them.

Solve Q21 and Q22 within the context of the

Newtonian World View of Units 1, 2 and 3.

KEY for Q19

A

B

C

V

D

The physical content of the Principle of

Relativity in Newtonian mechanics was first

pointed out by Albert Einstein.

Newton’s second law of motion and the law

of universal gravitation are consistent with the

Principle of Relativity in Newtonian mechanics.

It is a consequence of the Principle of Relativity

in Newtonian mechanics that, if two inertial

observers look at a given event, they must

describe it in the same way.

The Principle of Relativity only applies to free

particles.

If the Principle of Relativity were exactly true,

the period of a given pendulum would be the

same on the Moon as it is on Earth.

F It is a consequence of the Principle of Relativity

in Newtonian mechanics that a person who

jumps off a tower and falls freely observes the

same laws of physics as someone who stands on

the ground.

G None of A-F is true.

Q21 What is the velocity of the second particle

after the collision when they have separated enough

for the force between them to be negligible? Pencil

across one cell in row 21.

KEY for Q21

A (u,0,0)

B (-«,0,0)

° fe“'°'°) D (-2—14,0,0^

\ m2 )

E

F

G

II

2 m2 ix, 0,0

mi + m2

2 m2

m i + m2

2m i

u, 0,0

mi -f m2

2m i

mi + m2

u,0,0

u, 0,0

Q22 The kinetic energy of the second particle after

the collision is |m2u2. Suppose the same experiment

were run many times with different masses m2 but

all other things being held the same. For what mass

m2 would the kinetic energy lost by the first particle

be a maximum? Pencil across one cell in row 22.

Q20 The key lists proposals for the magnitude of

the acceleration of one particle (particle ‘one’) as

it interacts with another particle (particle ‘two’) in

all possible circumstances. Which two proposals are

consistent with the Principle of Relativity? In the

key, fc is a positive constant, and X!,vx and x2,v2

are the positions and velocities of the two particles.

KEY for Q22

A mijy/2 E mx/2

>B mx F Zero

C \/2mi G Infinite

D 2mx H None of the above

4

The Open University

Supplementary Material

S354 Understanding Space and Time


Tutor Marked Assignment S354 01

Make sure you know how to complete and send in your TMA and PT3 form:

detailed instructions are given in your student handbook (or supplement). Covering: Units 1-3

Cut-off date:

Friday 23 April 1993

This assignment relates to Units 1-3 and consists of a single question, divided into nine parts, (a)-(i).

Question 1

An isolated physical system, consisting of three

particles, A, B and C, is viewed from an inertial frame S. Particles A and B are identical and have

mass m. Particle C has mass m/4. The initial positions (in metres) and initial velocities (in metres per second) are

xa(0) = (2,0,0) vA(0) = (0,1,0)

xB(0) = (-2,0,0) vB(0) = (0, —1,0)

xc(0) = (0,0,4.5) vc(0) = (0,0,0)

At a subsequent time t0. it is found that

xb(*o) = (0, —1. 0.5) vB(to) = (u, v,w)

xc(*o) = (0,0,0.51 vc(t0) = (0,0,12)

where u,v,w are to be calculated in parts (f), (h) and (e) respectively.

(a) Write down the inrial positions and velocities

in a frame S' obtained fn im S by a rotation of 180 degrees about the 3-axis.

(6 marks)

(b) In a few sentences, give a concise, but thorough,

statement of the Principle >f Relativity in Newtonian mechanics.

(14 marks)

(c) From the symmetry implied by part (a) and your answer to part i b . explain why particle C cannot leave the 3-axis :: r later times.

{15 marks)

(d) In a manner similar to the argument given in part (c), write down me coordinates of the position of A and its velocity vec: rs in S at time t0.

{6 marks)

(e) The net linear momentum of an isolated system of particles is the vector sum of the linear momenta of

the particles, and is conserved. Use the conservation of linear momentum to find w. Show your working.

(12 marks)

(f) The net angular momentum of an isolated system of particles is the vector sum of the angular

momenta of the particles, and is conserved. Use the conservation of angular momentum to find u. Show your working.

{12 marks)

(g) You are now given that m = 2 kg, that the forces acting on the particles are (in newtons)

Fa = —2(xa - xc)

Fb = —2(xb - xc)

Fc = —Fa - Fb

and that the potential energy function is (in joules)

u = |xA - xc|2 + |xB - xc|2.

The energy of an isolated system of particles is

the sum of the particles’ kinetic energies and the potential energy function. Show that the total

energy of the system is conserved. Show all your steps, being sure to take into account the fact that

Fa, Fb, Fc, xa, xB and xc are vectors.

{20 marks)

(h) Use the conservation of energy to find v. Show your working.

{10 marks)

(i) The following are solutions to Newton’s second law:

xa(<) = (2cost,sint, |(1 - cos3t))

xb(0 = (-2cost, - sint, |(1 - cos31))

xc(t) = (0,0, |(1 + 8cos3t)).

Show that they satisfy the initial conditions, and verify that your answers to (f), (h) and (e) correctly give the velocity of B at time tQ =tt/2.

{5 marks)



S354 AB II

S354

Assignment Booklet II

Contents Cut-off date

2 CMA S354 42

(covering Units 4-7) 11 June 1993

5 TMA S354 02

(covering Units 4-7) 18 June 1993

6 CMA S354 44

(covering Units 10 12) 6 August 1993

9 TMA S354 04

(covering Units 10-12) 13 August 1993






why there is no TMA 03, nor any CMA 43. The continuous assessment of Parts A

and B of Block 6 consists solely of CMA 46; there is no corresponding TMA.





CMA and/or your worst TMA; but the degree of upgrading depends on the results

of all assignments and on your examination performance.

Copyright (c) 1993 The Open University

13.1 DT/T SUP 24409 2

The Open University



Computer Marked Assignment


S354 42

Make sure you know how to use the CMA form: detailed instructions are



If you do not wish to answer a question, pencil across the ‘don’t know’ cell 0?’).

If you think that a question is unsound in any way, pencil across the

‘unsound cell (‘U’) in addition to pencilling across either an answer cell or the ‘don’t, know’ cell.


number of answer cells or the ‘don’t, know’ cell.

Covering: Units 4-7

Cut-off date:

Friday 11 June 1993

Q1 to Q5 are concerned with electromagnetic forces and waves.

with centre at the origin, carrying a steady flow of electrons. Pencil across one cell in row 3.

In each of questions Ql to Q4, a source of electric or magnetic field is specified. In each case, you are

to select from the key one item that gives a pair of

vectors that could describe the motion of a positive

test charge at the point (1,0,0) moving with velocity

v and having an acceleration a produced by the source. (All positions and motions are referred to the same inertial frame of reference.)

KEY for Ql to Q4

A v = (v1,0,0) with v1 > 0

a = (0,0, a3) with a3 > 0

B v = (v1,0,0) with?;1 > 0

a = (0,0, a3) with a3 < 0

C v = (u1,0,0) withv1 > 0

a = (a1,0,0) with a1 > 0

D v = (v1,0,0) with v1 > 0

a = (a1,0,0) with a1 < 0

E v = (0,v2,0) withv2 > 0

a = (0, a2,0) with a2 0

F v = (u1,0,0) with v1 > 0

a = (0, a2,0) with a2 ^ 0

G v = (0, v2,0) with?;2 > 0

a = (0,0, a3) with a3 ^ 0

Q4 The source is a very long, neutral, straight

wire, lying stationary along the 3-axis, carrying a steady flow of electrons, each of which has a velocity u = (0,0, u3) with u3 > 0. Pencil across one cell in row 4.

Q5 Which of the quantities in the key have exactly

the same value for all inertial observers, according to

Einstein’s theory of special relativity? Choose three

items from the key. Pencil across three cells in row 5.

KEY for Q5

A The electric charge of a chosen particle

X B The acceleration of an isolated free particle

C The magnitude of the electric field at a chosen event

D The magnitude of the magnetic field at a chosen event

E The product of the quantities e0 and /i0

appearing in equations (1) and (7) of Unit 4

F

G

The spatial distance between two chosen events

The current carried by a given wire.

Q6 to Q10 are concerned with relations between events in space-time.

Ql The source is a negative charge, stationary at the origin. Pencil across one cell in row 1.

Q2 The source is a bar magnet, stationary at the origin, aligned along the 3-axis. Pencil across one cell in row 2.

Q3 The source is a neutral, circular loop of wire, stationary in the plane of the 2-axis and the 3-axis,

Each of the questions Q6 to Q10 refers to the events

labelled £a~£g in Figure 1, which is a space-time

diagram for an observer O who is at the origin at

time t — 0. Each event occurs on the .x1-axis and produces a flash of light, along with other possible effects. The light cone of O at time t — 0 is indicated by dotted lines.

2

at

€c

8a

■ • £D

•'' • 8f £e

Scr.

FIGURE 1

Q6 Which of the events could O photograph at time t = 0? Pencil across one cell in row 6. £

Q7 Which is the last event that O could see after time t, = 0, if he remains stationary? Pencil across one cell in row 7. y

Q8 Which of the events could be caused by actions

taken by O at time t, = 0? Pencil across two cells in row 8. C-, [

Q9 Which of the events could have caused any of

the actions taken by O before time t, = 0? Pencil across no more than two cells in row 9. _

Q10 Which two events could not have been caused

by event £q? Pencil across two cells in row 10.

KEY for Q6 to Q10

and O' are collinear and their origins are located so that they coincide at t = 0 = t'.

Qll For what value of V, if any, would events 82 and £3 be simultaneous according to O'? Pencil across one cell in row 11.

Q12 For what value of V, if any, would events 82 and £3 occur at the same point in space according to O'? Pencil across one cell in row 12. j ;

Q13 For what value of V, if any, would events 8\ and £4 be simultaneous according to O'? Pencil across one cell in row 13.

Q14 For what value of V, if any, would events £, and £4 occur at the same point in space according to O'? Pencil across one cell in row 14.

KEY for Qll to Q14

A 0

B \e

C -JC

D JC

L

E

F

G

H

_ 1 2'

c

—c

Not physically possible

Q15 Now suppose that a clock is taken, at constant speed u = 0 (as observed by O), from the space-

time origin of O (the event (0,0)) to event £3, and then at constant speed c/2 from £3 to £4. Assume

that the sudden acceleration at £3 does not affect the

clock’s mechanism. What (to two decimal places)

is c(At) clock> where c(At,)c\ock is the time recorded on the dock in going from (0,0) to £3 to £4? Pencil across one cell in row 15. f

Q16 An identical clock is taken directly from the

origin of O, at space-time point (0,0), directly to

event £4 at constant speed c/3. What (to two

decimal places) is c(At)r\ock for this trajectory? Pencil across one cell in row 16.

Cr KEY for Q15 and Q16

A Ea E Ee A 0.00 E 2.00

B Ed F Ef B 0.73 F 2.73

C Ec G Eg G 1.41 G 2.83

D Ed D 1.73 H 3.46

Qll to Qll refer to certain events in space-time.

An observer O attached to an inertial coordinate

system with space-time axes x and ct records events £1 to £4 as follows:

£1 with space-time coordinates (ctx,xQ = (1,2)

£2 with space-time coordinates (ct2,x2) = (0,4)

£3 with space-time coordinates (ct3, •As) = (1,0)

£4 with space-time coordinates (ct4,x4) = (3,1)

A second observer, O', with axes (ct',x') moves at some velocity V with respect to O, where V may be

positive, negative or zero. The x and x' axes of O

Q17 The key lists pairs of events. You are to select the single correct response as follows: if for the pair

£<-; and Si, it would in principle be possible to take a clock from En to £5, then choose the corresponding

entry in the key, but if none of the pairs is possible choose entry H in the key. Pencil across one cell in row 17.

KEY for Q17

YA £2 to E1 E £3 to

x B £x to E2 NF £2 to v C £4 to £1 G £2 to

D £1 to £4 H None

3

Q18 to Q24 are concerned with the relativistic energies and momenta of two particles, which collide.

A proton of mass 1.67 x 10~2'kg is travelling at a speed v = 1.8 x 108ms-1 along the 1-axis of an

inertial frame S in a head-on collision course towards a second proton which lies at rest in S on the 1- axis. Assume that before and after the collision the

two protons are far enough apart for the Coulomb repulsion between them to be negligible.

Q22 What is the relativistic momentum in S of the

system of two particles, in kg ms-1, to one decimal place, after the collision? Pencil across one cell in row 22.

KEY for Q20 to Q22

6- A 0.0 E 3.0 x 10-19

B 1.1 x 10~19 F 3.6 x 10-19

C 1.5 x 10~19 G 3.8 x 10~19

D 1.9 x 10"19 H None of A-G

Q18 What is the relativistic energy in S of the incident proton, in joules, to one decimal place? Pencil across one cell in row 18.

Q19 What is the relativistic energy in S of the

system of two particles, in joules, to one decimal place? Pencil across one cell in row 19.

KEY for Q18 and Q19

A 0.3 x 10~10 E 1.9 x 10-10

B 0.4 x 10"10 F 3.0 x Hr10

G 0.5 x 10"10 G 3.4 x 10“10

D 1.5 x 10-10 H None of A-G


incident proton, in kg ms-1, to one decimal place? Pencil across one cell in row 20.


system of two particles, in kgms-1, to one decimal

place? Pencil across one cell in row 21.

The two protons collide head-on and then separate, both moving on the 1-axis of S.

Now suppose the collision is observed from a second inertial frame S' in which the total relativistic

momentum of the two proton system is zero.

Q23 What is the speed in ms-1 of either proton in S' before the collision in ms-1, to one decimal place?

Pencil across one cell in row 23.

D

D

KEY for Q23

A 0.0 E 1.0 x 108

B 0.6 x 108 F 1.2 x 108

G 0.7 x 108 G 2.4 x 108

D 0.9 x 108 H None A-G

Q24 What is the total system of two protons in

place? Pencil across one

I relativistic energy of the S' in joules to one decimal

cell in row 24.

KEY for Q24

A 0.7 x 10"10 E 3.2 x nr10

B 1.9 x lO"10 F 3.4 x 10“10

C 2.9 x 10"10 G 3.8 x 10"10

D 3.0 x 10"10 H None of A-G

4





Make sure you know how to complete and send in your TMA and PT3 form: Covering: Units 4-7

detailed instructions are given in your student handbook (or supplement).

Cut-off date:

Friday 18 June 1993

This assignment consists of three questions.

Question 1

This question carries 40% of the marks for this assignment and compares some of the maternal of Units 4-7 with that of Units 1-3.

In this question, you are asked to relate certain

aspects of the world-view of Special Relativity, as described in Units 4-7 to the older view of Galileo and Newton, as described in Units 1-3. You are

asked to answer concisely and clearly the questions listed using very little, if any, mathem.ati.es. In doing

so, imagine that you are writing notes to yourself or, better perhaps, to a fellow student. Try to

compose your answers so that if you read them in

six months’ time they would still make sense to you,

provided the Course material was to hand. Marks

will be given for any relevant reference to the Unit

text or Glossary supporting your explanation. In

the interests of conciseness and precision, restrict yourself to a total of about 600 words when answering

all these questions below. Write in sentences.

These are the questions:

1(a) What is an observer? In particular, what is an ‘event,’ and how might the positions and times of

events be measured in the two world-views?

(b) Are the postulates Al to A8 of Unit 1 altered by the Special Theory? If so, how are they changed?

(c) What is the Principle of Relativity in the Galilean

view, and what does it say about physical laws? How is it altered by the Special Theory?

2 With the assertion of the ‘constancy’ of the speed of light (page 10 of Unit 5), the Special Theory

introduces a new feature. Together with the Lorentz

transform (and its implications), what does this have to say about postulates A9 and A10 of Unit 1? If you can, give specific examples.

3(a) According to Newtonian mechanics, the net (linear) momentum of an isolated system of two

or more particles is conserved in an inertial frame. How does Einstein’s Special Theory adapt this

statement? Namely, what is the linear momentum of a free particle in an inertial frame, in Special

Relativity, and how does it relate to the momentum in Newtonian mechanics?

(b) What is the ‘time component’ of momentum of

a particle in Einstein’s Theory, and how is it related by different inertial observers?

(c) With reference to Section 6 of Unit 7, how does the principle of relativity lead to the possibility of the creation, or annihilation, of particles?

Question 2

This question carries 50% of the m.arks for this assi.gnm.ent and relates mainly to Unit 6.

Two observers, O and O' are located at the origins of their respective inertial frames. Suppose they arrange things so that their axes, x and x'

respectively, are collinear and their origins coincide at t = 0 = t!. Suppose also that O' moves along the

positive x direction at speed 2c/3 with respect to O.

d hen the following sequence of events occurs:

Event S\: The origins of O and O' coincide.

Event £2: at time T, O sends a pulse of light, towards O'.

Event £:i: O' receives the pulse and instantly sends a pulse back towards O.

Event £4: O receives this latter pulse.

(a) Draw a clearly labelled, two-dimensional, space-

time diagram with axes x and ct showing the world¬

line of O', the world-lines of the two light pulses, and events £\ to £4 according to O.

(b) Draw a similar diagram from the point of view of O', with axes marked x' and ct.'.

(c) Find the coordinates, (ct,x), in terms of c and T, for all four events according to O.

(d) Find the coordinates (ct',x') of all four events according to O'.

(e) What is the time elapsed between £\ and £4 as recorded by O' and as recorded by O? Why m.ust the former exceed the latter?

Question 3

This question carries 10% of the marks for this assignment.

If a body emitting radiation is moving at an angle to the line of sight, the effects of time dilation and change of travel time to the observer both occur

and can cancel one another, so that the frequency received is the same as that transmitted. Find the

angle at which this would occur for a body moving at 5 x 107ms-1 in the frame of a given observer.

5



Computer Marked Assignment Course and assignment number:

S354 44

Make sure you know how to use the CM A form: detailed instructions are



If you do not wish to answer a question, pencil across the ‘don’t know’ cell

('?’)•


‘unsound’ cell (‘U’) in addition to pencilling across either an answer cell or tile ‘don’t, know’ cell.

Note For each question you must pencil across either the required number of answer cells or the ‘don’t know’ cell.

Covering: Units 10-12

Cut-off date:

Friday 6 August 1993

Q1 to Q8 are largely qualitative in character and are

important to the understanding of general relativity.

Q1 Which three of the statements in the key are true? Pencil across three cells in row 1.

KEY for Q1

A The principle of equivalence only applies to

test-bodies that do not experience any non- gravitational forces.

B The principle of equivalence asserts that in

any region of space-time it is impossible to

distinguish the effects of uniform acceleration from those of gravitation by any kind of experiment.

C The principle of equivalence asserts that in any

local region of space-time it is impossible to

distinguish the effects of uniform acceleration from those of gravitation by any kind of experiment.

D The principle of equivalence would be contradicted by the following observation:

two identical particles, initially separated by a distance of 100 metres, are dropped

simultaneously from the same altitude above the Earth; when they strike the Earth, they are separated by less than 100 metres.

E The mathematical proof of the principle of

equivalence is based on Newton’s laws of motion and Newton’s law of gravitation.

F The principle of equivalence implies that within

the framework of Newtonian mechanics, the gravitational mass of any body is proportional to the inertial mass of that body.

G The principle of equivalence implies that, if the co-ordinates of a freely falling frame are used to write down the metric of space-time, then

there will be a local region in which the metric is approximately equal to the metric of the space- time of special relativity.

Q2 Which two statements in the key are true? Pencil across two cells in row 2.

KEY for Q2

A The universality of free fall in its unrestricted form explains why space-time is curved.

B The universality of free fall in its unrestricted

form follows from the principle of equivalence.

C A failure of the principle of equivalence would

necessarily imply the failure of the universality of free fall in any local region of space-time.

D The universality of free fall in its unrestricted

form suggests that freely falling test particles move on geodesics in space-time.

E Within the framework of Newtonian mechanics,

the universality of free fall in its unrestricted form implies that the inertial mass of any body

is proportional to the gravitational mass of that body.

F A failure of the universality of free fall in any

local space-time region would not imply the failure of the principle of equivalence.

G The universality of free fall in its unrestricted form is believed to be true only under the

conditions in which the laws of Newtonian

mechanics provide an adequate description of the motion of a test-body.

Q3 Which two of the statements in the key about the claim that ‘any freely falling frame of reference is an inertial frame of reference’ are true? Pencil across two cells in row 3.

KEY for Q3

A When applied to a local region of space-time,

the claim is incompatible with Newton’s first, law of motion, even when all speeds are small.

B When applied to a finite region of space-time, the truth of the claim actually depends on the

precision with which observations are made in order to test the claim.

C When applied to a local space-time region,

the claim would not, imply a failure of the universality of free fall in that same region of space-time.

6

D When applied to a local space-time region, the claim can never be true when the space-time

region concerned is in the vicinity of any large aggregate of matter.

E The claim is only justified when made by an observer using a globally inertial frame of

reference and applied to a local space-time region within that globally inertial frame.

F The claim implies the universality of free fall in unrestricted form.

Each of questions Q4 to Q6 consists of two

statements, (a) and (b). You are to decide first whether each statement is true or false. Second, if you decide that both statements are true, you are to

decide whether or not the truth of statement (b) is a significant part of the reason (possibly the whole

reason) for the truth of statement (a). Each answer will thus consist of two or three items from the key.

KEY for Q4 to Q6

A Statement (a) is true.

B Statement (a) is false.

C Statement (b) is true.

D Statement (b) is false.

E The truth of statement (b) is a significant part of the reason for the truth of statement (a).

!• Statements (a) and (b) are independently true.

Q4 Statement (a) Within a small region of space-

time, the motion of a test-body is independent of its composition, provided the body is not subjected

to any non-gravitational forces. (The test-body is assumed to be small enough that it does not disturb its surroundings.)

Statement (b) In a small enough region of space- time, the effects of uniform acceleration cannot be

distinguished from those of gravity (where the precise

meaning of ‘small enough’ depends on the precision

of the measurements being considered.)

Pencil across two or three cells in row 4.

Q5 Statement (a) The geometry of a smooth two- dimensional surface, such as a sphere or a cylinder, is

completely specified by giving the form of the metric coefficients.

Statement (b) Only one set of metric coefficients can describe any one surface.


Q6 Statement (a) The distribution of energy density, momentum density and momentum flux

uniquely determines the metric coefficients of space time.

Statement (b) If the distribution of energy density, momentum density and momentum flux is zero at

a given event, then all components of the Riemann curvature must vanish at that event.


Q7 The key contains statements about the curvature of four-dimensional space-time. Select the item that is a true statement. Pencil across one cell in row 7.

KEY for Q7

A If a region of space-time contains no matter, the Riemann curvature within that region must be zero.

B The source terms in the field equations uniquely determine the metric coefficients of space-time.

0 The Riemann curvature at a given event can be zero while the Ricci curvature at the same event is non-zero.

D The Ricci curvature at a given event in space-

time depends only on the source terms at that event.

E None of statements A-D is true.

Q8 Three data takers, A, B and C, are respectively located in the nose, mid-point, and tail of a rocket that is undergoing uniform acceleration with respect to an inertial frame of reference. Each data taker is

equipped with a clock that is identical with the clocks being used by the other two data takers. Which three

of the statements in the key are true? Pencil across three cells in row 8.

KEY for Q8

A A’s clock is running slower than C’s.

B C’s clock is running slower than A’s.

C Clocks A and C are running at the same rate.

D According to B, the clocks being used by A and C are both running at the same rate.

E According to B, the clocks being used by A and C are running at different rates.

I- According to an inertial observer, instantan¬

eously moving with the same velocity as the rocket, the clocks being used by A and C are both running at the same rate.

C According to an inertial observer, instantan¬

eously moving with the same velocity as the rocket, the clocks being used by A and C are running at different rates.

Q9 to Q13 are concerned with the geometric properties of two-dimensional surfaces.

Q9 to Qll These questions refer to a two-

dimensional surface S that is described by the metric

a; = [(Ag1)2 + cos2(A,1)(A92)2]1/2

where A is a positive constant.

Q9 What is the curvature of the surface S at the

point q1 = 0,q2= 0? Pencil across one cell in row 9.

KEY for Q9

AO E A2/2

B -A/2 F —A2

C A/2 Q y2

D —A2/2

7

Q10 A small but finite circle centred on the point

q1 = 0, q2 = 0 is drawn on the surface S with the aid of a measuring tape. Which of the statements in the

key would best describe the relationship between C, the circumference of the circle, and r, the radius of the circle? Pencil across one cell in row 10.

KEY for Q10

A C is greater than 27rr.

B C is equal to 2nr.

C C is less than 2i\r.

D The metric does not give sufficient information to choose between options A, B and C.

Qll A small but finite triangle (whose sides are geodesics) is drawn on the surface S with one vertex

at q1 = 0, q2 = 0. Which item from the key describes the sum of the interior angles of the triangle? Pencil across one cell in row 11.

KEY for Qll

A The sum exceeds 7r.

B The sum is less than 7r.

C The sum equals 7r.

D None of the above options must apply.

Q12 On a different surface S' (not the surface of Q9 to Qll), circles are constructed. It is found that

the ratio of circumference to radius of these circles is greater than 27t, regardless of where on the surface

the centres lie. Which if any, of the surfaces A F in

the key might produce this result? Pencil across one cell in row 12.

KEY for Q12

A A right circular cylinder

B A sphere viewed from the outside

C A sphere viewed from the inside

D An ellipsoid of revolution (Fig. 50a Units 11/12)

E A paraboloid of revolution (Fig. 50b Units 11/12)

F A torus (Fig. 51, Units 11/12)

G None of A-F

Q13 The metric of a surface, in the co-ordinate system (x},x2) is given by

Al = [(Ax1)2 + (x1)2(Ax2)2}1/2

Two curves are drawn on this surface. Curve C, has

the parametrization x1 = a. and x2 = bs 4- c. Curve C2 has the parametrization x1 = as + b and x2 = c.

For both curves a, b and c are arbitrary constants,

which may be non-zero, and s is the parameter. Pick from the key, which is given as a table, the one option

that correctly describes whether the surface is flat or

curved, and whether curves Ci and C2 are geodesics or not. Pencil across one cell in row 13.

KEY for Q13

option flat curved Ci geodesic C2 geodesic

A yes no yes yes B no yes yes yes

C yes no yes no D no yes yes no E yes no no yes F no yes no yes G yes no no no H no yes no no

Q14 to Q18 consider the effects of gravity on the frequency of electromagnetic radiation.

An astronaut is floating freely deep in space. Inside his spacecraft, he measures the frequency of

electromagnetic radiation emitted by his portable laser to be 1014 Hz. For each question, select from the key the answer closest to your own.

Q14 What frequency would the astronaut measure if he remained in the spacecraft, while the laser

transmitted radially outwards from the surface of a spherically symmetrical planet of mass 1025 kg and

radius HP m? Assume the spacecraft is stationary and very distant from the planet, and use G =

6.67 x 10“11 kg"1 m3 s-2 for Newton’s gravitational constant. Pencil across one cell in row 14.

Q15 What frequency would the astronaut measure

if he was holding the laser while falling freely towards the planet? Pencil across one cell in row 15.

Q16 What frequency would the astronaut measure if he was on the surface of the planet and the laser

transmitted radially inwards from the stationary (and still distant) spacecraft? Pencil across one cell in row 16.

Q17 What frequency would the astronaut measure if he was beside the laser on the surface of the planet? Pencil across one cell in row 17.

Q18 What frequency would the astronaut measure if he climbed a vertical tower of height 100 metres on the surface of the planet, and the laser transmitted

radially outwards from the planet’s surface? Pencil across one cell in row 18.

KEY for Q14 to Q18

A 1014 + (6.2 x 107)Hz

B 1014 - (7.4 x 106)Hz

C 1014 + (7.4 x 106) Hz

D 1014 — 7.4 x 103 Hz

E 1014 + 7.4 x 103 Hz

F 1014 Hz

G 0 Hz

H None of the above

The Open University



Tutor Marked Assignment Course and assignment number:

S354 04

Make sure you know how to complete and send in your TMA and PT3 form: Covering: Units 10-12


Cut-off date:

Friday 13 August 1993

This assignment consists of two equally weighted questions.

Question 1

This question carries 50% of the marks for this

assignment and relates to the whole course to this point, especially Units 10 and 11/12.

Write an essay comparing and contrasting Einstein’s

General Theory with the world-views of Newtonian mechanics, as described in Block 1, and of the Special

Theory, as described in Block 2. You should conduct

your essay along lines suggested by the following

considerations, which may be blended in any order into your presentation:

(i) The definition of ‘inertial frame’ in the general

theory and its relation to the meaning of this term in the two earlier theories;

(ii) The geometric aspect of the general theory, and in particular its description of the motion of test

particles and light rays under the influence of matter and energy, and the importance of the proper time;

(iii) Ways in which the general theory can be said to

contain, or reduce to, the two earlier theories.

You should try to quote very little, if any,

mathematics, but, rather, to describe the physical

content. You may want to mention experimental evidence that is relevant to the points you are making, but this should not dominate your essay.

Try to make the essay a self-contained summary for yourself or other similarly knowledgeable person. Favour encapsulated descriptions of content over

abstract references to material in the Units.

Note that this question requires you to combine

relevant material from several Blocks of the course, in a coherent manner, as is frequently the case with

questions set in Part III of the final examination. The following advice, given for essay questions in the exam., also holds here.

• Full marks be may obtained for a well-organized and clearly expressed essay of about 600-1000 words.

• As guidance for the content of the essay, topics of principal importance are listed.

• Approximately 75% of the marks for this question are reserved for the coverage of these principal points.

• The remaining 25% of the marks for this

question are for your explanation of supporting points, and for the overall clarity and style of the essay.

Question 2

This question carries 50% of the marks for this assignment and relates m.ainly to Units 10 and 11/12.

Two identical clocks, A and B, tick at the same rate

when they are both in a locally inertial frame of

reference. In this question, however, the clocks are

at the equator of a hypothetical rotating planet with

no atmosphere to complicate observations. Clock

A is at ground level while clock B is on top of a

tower. An observer O, who is distant from the planet and in its equatorial plane, observes the clocks to

tick at the same rate. He has read Unit 10 and

Block 2 and reasons that this is because the effects of

the ‘gravitational redshift’ and ‘time dilation’ cancel out. Under the assumption that the planet is a

spherically symmetric mass distribution, and that

space-time around the planet can be described by the Schwarzschild metric, you can, below, analyse

the problem in an alternative way. In the following sequence of steps, show your working.

(i) First consider a clock that rotates with the planet at the equator and at some fixed distance r from the

centre. Suppose the time between ticks, as observed by O, is At. If the planet rotates at angular speed uj, then the azimuthal angle changes by A(j> = w At

during this time. Show that the time between ticks for such a clock, (At),., is

(Ar)r = (A(){l A rVn1/2

C2 J where k = 2mG/c:2, m is the planet’s mass, G is the gravitational constant and c is the speed of light.

(ii) Suppose the planet’s radius is R. Assuming that k/R and Rw/c are small compared to unity, show that

(At)a “ At + R2w2\

c2 )

[Hint: for x small in magnitude vT + x = 1 + x/2.]

(iii) Suppose the tower’s height is h. And suppose that h is small compared to R. Find an expression

9

for (At)b and then show that

(Ar)BS(A,)A + (A

[Hint: A further useful expansion is

when x is small in magnitude.]

(iv) Show, with these approximations for (At)a and (At)b, that the clocks will be observed by O to

tick at the same rate provided Rw2 = <7, where g —

mG/R2 is the ‘acceleration of gravity’ at the planet’s

surface.

(v) Suppose the radius R and mass m of the planet were the same as those for Earth, namely R. = 6.4 x 106 m and m = 6 x 1024 kg. Then find (to one

decimal place) the value of cu and period of rotation T = 2-k/uj, for which the clocks are observed by O

to run at the same rate. In this analysis, both. k/R and Rw/c were assumed small. Is that assumption justified for the values of R. and m given here?

(vi) Which term in the expression for (Ar)A corresponds to the Lorentz dilation effect?

Printed in the United Kingdom by

The Open University



S354 AB III

S354

Assignment Booklet III

Contents Cut-off date

2 CM A S354 45

(covering Units 13 and 14)

5 TMA S354 05

(covering Units 13 and 14)

6 CM A S354 46

(covering Block 6 Parts A and B)

17 September 1993

24 September 1993

8 October 1993






why there is no TMA 03. nor any CMA43. The continuous assessment of Parts A

and B of Block 6 consists solely of CM A 46: there is no corresponding TMA.





CMA and/or your worst TMA: but the degree of upgrading depends on the results

of all assignments and on your examinat ion performance.

Copyright © 1993 The Open University

13.1 DT/T SUP 24410 6




S354 45

Make sure you know how to use the CMA form: detailed instructions are Covering: Units 13 and 14 given in your student handbook (or supplement).

You are strongly advised to attempt every question in this assignment. Cut-off date:

If you do not wish to answer a question, pencil across the ‘don’t know’ cell Friday 17 September 1993 (*?’).


‘unsound' cell (‘If) in addition to pencilling across either an answer cell or t lie ‘don’t know’ cell.


number of answer cells or the ‘don’t, know' cell.

Q1 Which two of the following statements are true? Pencil across two cells in row 1.

KEY for Q1

A Hersehel confirmed earlier indications that the

Universe is basically isotropic.

B The formation of galaxies seems to have been completed in the first billion years. (Ignore

the point of whether a collision constitutes ‘formation’.)

C Invisible planets an' t hought to constitute about 10% of the matter in our Galaxy.

1) The darkness of the night sky precludes an infinite universe.

E Discounting supernovae, all stars visible individually to the unaided eye belong to our Galaxy.

Q2 1 he so-called “standard model’ is first broached in Section 4.1 of Unit 13. Which one of the state¬

ments in the key is assumed to apply to a cluster of galaxies in applying the standard model? Pencil across one cell in row 2.

KEY for Q2

A It is not in free fall, because the distance between it and any other cluster of galaxies is increasing.

B It is not in free fall, because it is described by constant co-moving co-ordinates.

C It is not in free fall, because it is subject to the

gravitational influence of all the other clusters of galaxies in the Universe.

D It is in free fall.

For Q3 to Q7, imagine a star (in another galaxy) at

a distance of 3 x 10(> light years from Earth, with a mass 3 x l() inkg. a radius 9 x 10s m and a period

of rotation 3 x 10(> s. Suppose the star’s atmosphere

is turbulent with a typical speed of 7000 ms-1. Take the Hubble parameter to be 2 x 10-l8s-1. An Earth-bound astronomer estimates from these

data the relative shift, Avj1/. in a certain hydrogen

line, where v is the frequency of the hydrogen line in the laboratory. In Q3 to Q7, you should

consider each separate contribution to Av/u positive (negative) according to whether it tends to raise

(lower) the received frequency compared to that of

the laboratory standard. The astronomer proceeds as follows:

Q3 She calculates the relative gravitational redshift, {Av/v)^ x 10fi, of the line due to the star’s

mass. Select the item from the key nearest to what she finds. Pencil across one cell in row 3.

KEY for Q3

A 2.1 E -2.1

B 2.5 F -2.5

C 4.6 G -4.6

D 7.3 H -7.3

Q4 She then estimates the range of values of (Av/u)r x 10**, where (Av/v)r is the relative Doppler shift due to the star’s rotation. Choose the item from the key nearest to what she finds. Pencil across one cell in row 4.

Q5 Then she estimates the range of values in (Av/v){ x 10(>, where (Av/v){, is the Doppler shift due to the turbulence in the star’s atmosphere.

Choose the item from the key nearest to what she finds. Pencil across one cell in row 5.

KEY for Q4 and Q5

A —2.5 to 2.5

B —6.3 to 6.3

C -7.3 to 7.3

D -15 to 15

E ' -23.3 to 23.3

F -30.3 to 30.3

G -43 to 43

H -51.8 to 51.8

Q6 Suitably interpreting equation (18) of Unit 13, the astronomer then estimates the cosmological redshift, (Au/i/)c, for this star. She is interested

in the measured shift (Au/u)m. Including also the effects in Q3 to Q5, she then estimates that

(Avfu)m x 10'* = A ± B, where B represents the range. W liich item in.the key is nearest to her results

2

for A ± B? Pencil across one cell in row 6.

KEY for Q6

A -187 ±30 E -—183 ± 21

B -187 ±17 F -196 ±21

G -192 ±30 G -192 ± 27

D -192 ±17 H -192 ±34

Q7 Select from the key t he two items t hat represent possible further contributions to tlie* measured shift of the hydrogen line that the astronomer has not

considered in Q3 to Q6. Pencil across two cells in row 7.

KEY for Q7

A An effect due to the time-dependence in /?(/), the scale factor in the Robertson Walker metric.

R An effect due to the overall expansion in the Universe.

C A Doppler shift associated with the motion of the star in its galaxy relative to Earth in its Galaxy.

D The gravitational frequency shifts associated with the star’s galaxy and the Earth’s Galaxy.

E The absorption of the star’s light by dust and Other interstellar matter.

P A general relativistic effect associated with

k, the spatial curvature parameter in the Robertson Walker metric.

G The shift associated with the curvature of

space-time in the star's immediate vicinity caused by its mass.

Q8 Which two of the following statements are

true for the model, homogeneous, isotropic, universe

described by the Robertson Walker metric, equation

(2) of Unit 13? Pencil across two cells in row 8.

KEY for Q8

A Space-time is flat only if k = 0 and R = 1.

B If k = 0, space-time can be curved.

C If k = 0, space-time cannot be curved.

I) If k > 0, the circumference of a circle is less than

2n times its radius for any /?(/).

E The co-moving co-ordinates and a2 of two galaxy clusters change with time.

F Whatever k and R(t) may be, the time taken

for a light signal to pass between two galaxy clusters does not change with time.

G If k < 0, the sum of the angles in a large triangle

will be, at any given time, greater than tt.

Questions Q9 to Q15 concern a matter-dominated universe in which the mass energy density is less than the critical density. In each question, you are to select one item from the key to describe how a particular quantity behaves as R approaches infinity.

KEY for Q9 to Q15

A It tends to zero, becoming .proportional'to 1//?.

B It tends to zero, becoming proportional to 1 /R-.

G It tends to zero, becoming proportional to I /RA.

D It tends to zero, becoming proportional to 1 //?'.

E It becomes infinite.

F It retains a finite negat ive value.

G It approaches a finite positive value.

Q9 How does the mass energy density p behave Y- a-s R approaches infinity? Pencil across one cell in

row 9.

v QlO How does dR./dt behave*as R approaches infinity? Pencil across one cell in row 10.

Qll How does the spatial curvature parameter k ' behave as R approaches infinity? Pencil across one

cell in row 11.

f Q12 How does d-R/dt'2 behave as R approaches ' infinity? Pencil across one cell in row 12.

Q13 How does the Hubble parameter II behave

as R approaches infinity? Pencil across one cell in row 13.

Q14 How does t he deceleration parameter q behave

as R approaches infinity? Pencil across one cell in row 14.

Q15 How does the critical density p( behave as R approaches infinity? Pencil across one cell in row 15.

Q16 Xow consider a mat ter-dominated, universe in

which the mass energy density is equal to the critical

density. Which two of the following statements art1 correct? Pencil across two cells in row 16.

KEY for Q Hi

A The space time is flat, and the space is flat.

B The space time is curved, and the space is flat.

(' The scale factor R is constant.

pH I he scale factor R depends on time.

E The space time is curved, and the spatial curvature is positive.

F The space-time is curved, and the spatial curvature is negative.

G Only one of the above responses is correct .

3

Q17 to Q20 refer to a hypothetical universe, not necessarily the universes of Q9 to QIC, in which the

scale factor R(t) = At1^, where A is a constant,

for each question, select one item from the key that gives the range in which the power of t. to which the

quantity referred to is proportional, falls.

KEY for Q17 to Q20

A —2.1 to —1.7 E 0.1 to 1.1

15 -1.7 to -1.1 F 1.1 to 1.7

C -1.1 to-0.1 G 1.7 to 2.1

D -0.1 to 0.1

Q17 How does f lu' Iluhhle parameter H vary with time? Pencil across one cell in row 17.

Q18 How does the deceleration parameter q vary with time? Pencil across one cell in row 18.

Q19 How does the spatial curvature parameter k vary with time? Pencil across one cell in row 19.

Q20 How does the pressure p vary with time? Pencil across one cell in row 20.

Q21 Concerning our Universe, and not the

universe of Q17 to Q20, which two of the statements

in the key are correct? Pencil across two cells in row 21.

KEY for Q21

A The Universe was matter dominated at the time of cosmological nucleosynthesis.

B Matter and radiation were''decoupled at the time of cosmological nucleosynthesis.

(' A significant fraction of the nuclei formed during

t he first 15 minutes after the Big Bang are still in existence today.

D If there were a large number of unobserved black

holes in intergalactic space, the present density of the Universe could be greater than the critical density pc.

E If the neutrino were found to have a non-zero mass, it would not affect, the estimated mass-

energy density of the Universe because neutrinos hardly ever react with matter.

Q22 Suppose it was discovered that all previously estimated cosmological distances needed to be increased by a factor of 2. By considering the effect this would have on the density p and Hubble

parameter H, how would this affect the estimate of p/p( where pc is the critical density? Assume

Euclidean geometry and the cosmological principle. Pencil across one cell in row 22.

KEY for Q22

A It would lower it by a factor of about 8.

B It would lower it by a factor of about 4.

C It would lower it by a factor of about 2.

D It would be unchanged.

E It would raise it by a factor of about 2.

E It, would raise it by a factor of about 4.

G It would raise it by a factor of about 8.

Q23 Suppose that the present density of the

Universe was known to be in the range from 10~27

t° 8 x 10 kg m-1. Also suppose that it was possible to measure the freeze-out concentration

(mass fraction) of each of the elements listed in

the key to an accuracy of 1%. Which one of the

measurements would provide the best estimate of the

present density of the Universe? Pencil across one cell in row 23.

KEY for Q23

A 4 He C 2H

B 3 He D 7 Li

4

4

The Open University





Make sure you know how to complete and send in your TMA and PT3 form: Covering: Units 13 and 14


Cut-off date:

• Friday 24 September 1993

This assignment consists of two questions, based on Block 5.

Question 1

This question carries 50% of the marks for this assignment, and relates mainly to Units 13 and li,.

In this question, you are asked to discuss some underlying physical aspects of the ‘Standard Model’

of cosmology centred around the Robertson-Walker metric, together with the Einstein field equations

given by equations (1) and (2) of Unit 14. You should

concentrate on conveying the basic ideas without

recourse to much mathematics. Limit the length of your answer to, say, 500 to 600 words overall. Do,

however, write in sentences.

(i) The cosmological principle asserts that the

large-scale distribution of mass-energy in the Universe is, and always has been, homogeneous and

isotropic. Write down short definitions of the terms

1 ‘homogeneous’ and ‘isotropic’, and briefly explain

what the term ‘large-scale’ means in the context of the cosmological principle. What is the minimum

distance that can reasonably be regarded as ‘large- scale’ in the context of cosmology?

(ii) List six of the main (known) contributions t.o

the mass-energy of the Universe, and write a short account of the extent to which experimental support can be found for each of the following claims about

the large-scale distribution of mass and energy:

(a) The distribution is now isotropic;

(b) The distribution is now homogeneous;

(c) The distribution has always been isotropic and homogeneous.

State any assumptions that underlie the interpretation of the evidence you quote.

(iii) How can you possibly have any idea of what

the Universe was like when it was only 20 minutes old?

(iv) What is outside a closed universe?

Question 2

This question carries 50% of the marks for this

assignment and relates mainly to Units 13 and 14.

In this question, you are asked to consider a

few possibilities allowed by the so-called ‘Standard Model’ of cosmology, as discussed in Units 13 and

14. At the core of this model are the following:

(a) The Robertson Walker metric, and the assump¬ tions leading to it, given mainly in Section 4 of Unit 13.

(b) The Einstein field equations, equations (1) and (2) of Unit 14, and the discussion of them given

in that unit. Also note especially the three

assumptions made at the bottom of page 7.

In what follows, you should take the value

of the Hubble parameter H to be 2 x 10-l8s

and the gravitational constant G to be 6.67 x 10“11 kg-1 m3 s-2.

(i) Find an expression for the density pc(t.) at any time t. for which space would be flat. Explain its

significance. Estimate that, value at, the present time,

P\iow Is space-time flat when p(t) = pc(t)7 Explain briefly.

(ii) Show that, whether the universe is open or

closed, (\2R(t)/dt2 can never be positive. Is this true in a radiation-dominated universe? Why?

(iii) Show that, the deceleration parameter, defined

by <7(0 = —RR/(R)2, must, be non-negative. Is this true iu a radiation-dominated universe? Why?

(iv) For this part, assume the Universe is closed. For times starting shortly after the Big Bang and

extending as far into the future as is possible within the compass of the assumptions of the ‘Standard

Model’, make sketches showing how the following quantities depend on time:

R, R, H, R, pc, and k.

Be sure to show the main features, and indicate £now, the present time, on your sketches.

(v) For a matter-dominated flat universe, derive expressions for i?., //, pc and q as functions of time.

Sketch the time dependence of these four quantities on the same basis as you drew for part (iv).




S354 46

Make sure you know how to use the CM A form: detailed instructions are given in your student handbook (or supplement).

Y°u are strongly advised to attempt every question in this assignment.

Tf you do not wish to answer a question, pencil across the ‘don’t, know’ cell

If you tlnnk that a question is unsound in any way, pencil across the

‘unsound’ cell (‘U’) in addition to pencilling across either an answer cell or the ‘don’t know’ cell.

Note For each question you must pencil across either the required number of answer cells or the ‘don’t know’ cell.

Covering: Block 6, Parts

and

Cut-off date:

Friday 8 October 1993

Q1 Suppose that, at some stage in the Universe’s history, a spaceship moves in such a way that it

records a maximum temperature for the cosmic background radiation of 10 K and a minimum temperature of 2.5 K. What is the speed of

the spaceship relative to the cosmic background? ( Warning: Is v/c small?) Select the item from the

key 1-hat is closest to your answer. Pencil across one cell in row 1.

KEY for Ql

A >0.7c E 0.3c

B 0.6c F 0.2c

C 0.5c G 0.1c

D 0.4c

cosmic background radiation to change in the next

1 000 years? Select the item from the key that is closest to your answer. Pencil across one cell in row 4.

KEY for Q4

A °K D —6 x 10~8 K

B —6 x 10-18 K E —2 x 10“7 K

C —6 x 10-15 K

Q5 I'he key contains statements about the 3K

cosmic background radiation. Select options that are correct according to the model of our Universe

described in Unit 13 to Block 6B. Pencil across one cell in row 5.

Q2 Suppose that the cosmic background radiation in Ql is detected by an observer who is at rest with

respect to it. What temperature does the observer

assign to this background radiation? Select the item fiom the key that is closest to your answer. Pencil across one cell in row 2.

KEY for Q2

A 7.5 K

B 6.25 K

C 5.0 K

D 4.5 K

Q3 The key for this question lists some particles that were present in the early Universe. Which of

these particles had the greatest direct influence on

the cosmic background radiation? Pencil across one cell in row 3.

KEY for Q3

A Helium nuclei D Electrons

B Hydrogen atoms E Neutrons

C Deuterium nuclei F Protons

Q4 If the temperature of the cosmic background radiation were now 3.2 K, and if the current value of the Hubble parameter were 2 x 10-18s-1, by

h°w much would you expect the temperature of the

E 4.0 K

F 3.75 K

G 3.0 K

KEY for Q5

A y B

C

The cosmic background radiation began to

acquire its thermal character before the era of decoupling.

The number of photons in the cosmic background radiation has varied widely since the era of decoupling.

The temperature of the cosmic background radiation was equal to that of the matter in the Universe until the appearance of galaxies.

In a closed universe, the cosmic background radiation would eventually appear as visible light.

E The total energy of the cosmic background radiation is currently much greater than that of matter.

F The microwave spectrum collected on Earth is dominated by signals of cosmic origin.

/G The temperature of the cosmic background radiation first began to decrease immediately after the era of decoupling.

Q6 The. discovery of the microwave background is considered as a major advance in cosmology.

The key lists a number of aspects of the standard cosmological model described in Unit 13 to Block

6B. For which two of. these do the discovery and

6

W >

experimental study of the microwave background provide important evidence? Pencil across two cells in row 6.

KEY for Q6

A Our Local Group does not have large

(> 1000 km s'1) proper motion.

B Our Local Group does have large

(> 1 OOOkms-1) proper motion.

G The Hubble parameter s;2x 10~18s-1.

D Decoupling temperature « 3 000K.

E The decoupling event now has a redshift. of 2 m 1 000.

F Galaxies formed at about 5 x 108 years.

G The temperature at cosmological nucleosynthesis was about 10!)K.

Q7 A steady-state cosmological model, in which the mean density of the Universe is unchanging, has been proposed from time to time. Choose from the following key two items with which such a theory agrees. Pencil across two cells in row 7.

KEY for Q7

A The decoupling of radiation from matter at a given stage in the evolution of the Universe

B The conservation of baryon number

G The cosmological principle

D The upper limit on the age of the Universe given by the reciprocal of the Hubble parameter

E The redshift of distant galaxies

Q8 The key for this question contains statements about the existence of black holes. Select two options

that ^.re believed to be correct in the standard theory. Pencil across two cells in row 8.

KEY for Q8

Any star with a mass greater than 3 solar masses is a black hole.

A black hole would be formed if 104:{ kg of matter were compressed to an average density of 1 kgm-3.

No black hole can exist with a mass less than that of our Sun.

D The minimum mass of a rapidly rotating stellar

black hole is greater than the minimum mass of a static stellar black hole.

E A black hole would be formed if 10,okg

of matter were compressed to the size of a hydrogen atom.

Q9 The key for this question contains plan views of the spatial x1, a;2-plane with a static black hole

at. the origin surrounded by an event-horizon. Each option illustrates a world-line that passes beyond the

event-horizon. Which of these options are possible? Pencil across two cells in row 9.

KEY for Q9

A

x'

7

Q10 The key for this question contains statements

about the behaviour of matter and radiation in the vicinity of a static black hole with an event-horizon at radial co-ordinate k. Select two options that are

true. Pencil across two cells in row 10.

KEY for Q10

A The matter inside the event-horizon must have a much higher density than that, normally

encountered on Earth.

B The radiation produced by the Hawking process has a minimum wavelength of about k.

C Visible light of wavelength 10"'m can be produced by the Hawking process if the black

hole has any mass greater than 10“1 kg.

D Radio messages cannot be sent outwards across

the photon-sphere surrounding the black hole.

E An observer at rest at r = Ak/Z observes that spectral lines of very distant atoms have their

frequency doubled when measured with respect, to his proper time.

F An observer who maintained a constant radial co-ordinate near the black hole would

obtain a null result for the Michelson Morley experiment.

Qll A light signal is emitted outside the event- horizon of a. static black hole and propagates radially

outwards. The signal is emitted from a stationary

source, A, at radial co-ordinate 4A-/3 and is detected

by a stationary receiver, B, at radial co-ordinate 2k.

The event-horizon has radial co-ordinate k. If the

frequency of the signal at A was 6 x 1011 Hz, what,

to one decimal place, is its frequency at B? Pencil across one cell in row 11.

for Qll

A} 8.5 x 1014 Hz E 4.2 x 1014 Hz

B 7.1 x 1014 Hz F 3.0 x 1014 Hz

C 6.0 x 1014 Hz G 7.1 x 1013 Hz

D 5.4x1014 Hz

Q12 A black hole is just capable of producing

electrons (of rest mass 9.1 x 10-31 kg) by the Hawking process. What is the temperature of the

radiation that it produces by the Hawking process? Select, the option from the key that is closest to

your answer (use h. = 6.63 x 10-34 Js for Planck’s constant). Pencil across one cell in row 12.

KEY for Q12

A 4.2 x 10fi K

B 1.1x107K

C 7.5 x 107 K

D 1.6 x 108 K

E 3.0 x 108 K

F 7.2 x 109 K

G 1.6 x 1010 K

H 8.0 x 1010 K

Q13 Assuming that the Universe is open, what, according to the Hawking model, is the maximum lifetime of the black hole referred to in Q12? Pencil across one cell in row 13.

KEY for Q13

A 9.3 x 10(> years E 2.0 x 1010 years

B 2.0 x 108 years F 2.3 x 1012 years

G 9.4 x 108 years G 1.0 x 1014 years

D 1.0 x 109 years H ; 1.0 x 1022 years

The remaining questions in CMA Jt6 have elements

related to the entire Course, and have a more general, nature than Ql to QlS.

Q14 to Q16 concern the assumptions of Newtonian physics as described in Block 1. The keys for these questions list some of those assumptions.

Q14 Which, if any, of the assumptions listed under A-E in the key for Q14 and Q15 were modified or discarded by the Special Theory of Relativity and

its ramifications? Pencil across up to three cells in row 14. Y

Q15 Which, if any, of the assumptions listed under A E in the key for Q14 and Q15 were discarded or

put into doubt by the General Theory of Relativity

and the related developments of cosmology [black holes, Big Bang]? Pencil across up to three cells in row 15. 1 C A CI

KEY for Q14 and Q15

A A1 (page 10, Unit 1)

B A2 (page 12, Unit 1)

C A3 (page 12, Unit 1)

D A4 (page 13, Unit 1)

E A5 (page 14, Unit 1)

F None of items A-E

G All of items A-E

Q16 Which two of the assumptions listed in the

key were not discarded by the Special Theory of Relativity and its ramifications? Pencil across two cells in row 16.

KEY for Q16

A A7 (page 16, Unit 1)

B A8 (page 17, Unit. 1)

C A9 (page 34, Unit 1)

D A10 (page 34, Unit 1)

E All (page 35, Unit 1)

F A13 (page 4, Unit 3)

Printed in the United Kingdom by

The Open University

The Open University


CMA 41 Answers (1993)

Q1 D is the correct answer.

rA - rs = (5, 7,2) -<3, 4,-4) = (2, 3, 6)

so lrA - i*bi = V2^ + 3^ + 6^ = 7

D is the correct answer,

lul = Vl + 1 + 6 = V~8

Ivl = V1 + 1 + 6 = V"8

uv = -1 -1 + 6 = 4 u-v 4 1 lullvl ~ 8 “ 2 cos 0 =

Q3 This question was zero-weighted. The correct analysis is given as follows:

Let a = (1,1, 3) and b = (2,1,1). Then the area of the triangle is

| laxbl = iV(l-3)2 + (6-l)2 + (l-2)2 =

= i V 4 + 25 + 1 =

Q4 B is the correct answer,

r (rc/4) = (-1,1,1)

lr (7t/4)l = Vl + 1 + 1 = V3

Q5 G is the correct answer.

^ r (t) = (-4 sin 4t, - V2 sin t, V2 cos t)

|r(7t/4) = (0, -1,1)

l|r(7c/4)l = V0+ 1 + 1 = -nT2


The components of x are affected by translation, but those of v and a are not

Q7 E is the correct answer.

It is the scalar product of two vectors and is, therefore invariant under all rotations.


Since x-v is unaffected by any rotation and -x1 v1 is unaffected by any rotation about the 1-

axis but is affected by rotations about the 2-axis.

Q9 B & C are the correct answers.

For B since a1 x1, a3 and x2 are unaffected by a reflection of the 1-axis, but x2 is affected by

a reflection of the 2-axis. C does not contain a 1-component but is not invariant to a

reflection of the 2-axis.

Q10 H is the correct answer.

If x is the position vector of m and X the position vector of M then the force on m owing to GmM

M is FmM = —(x - X). Hence the 2-component of the acceleration caused to m by

M is -GM (x2 -X2)/R3.

Qll C is the correct answer.

The force on M owing to the gravitational attraction of m is FMm = -FmM- Hence the 2-

component of the acceleration of M is + Gm (x2 - X2)/R3.


If Ms is the star's mass then the magnitude of A's acceleration is G Ms /R2 (where Ra = 2

astronomical units). But from equation (6) on page 10 of Unit 2 this is also

R* i (27c)2 ^ where Ta is the period of A.

A A

SO (27C)2 Ra

Ms = *-q-y • A similar formula applies for a circular orbit around the sun, which ta

has mass Msun- Hence Ms = 3 |Ten2

Ita, Msun where Rg = 1 unit and Tg = 1 year.

But Ra = 2 and Ta = lyr. Hence Ms = 23 Msun.

Q13 G is the correct answer.-

Use Kepler's Third law. Then

^ =T^/R|andT| =

Q14 B and C are the correct answers.

See pages 10, 18 and 19 of Unit 2.


See page 8 of Unit 2.

Q16 C is the correct answer.



All of assumptions A to F do not rule out an inverse cube law.

Q18 E and F are true. Not A since a lab frame rotates with the earth. Not B since some inertial

frames are related by boosts. Not C since, for Newton, space is infinite. Not D since rotating

frames are not inertial. For E and F see The Glossary ("Inertial Frames of Reference") and section

2.3 of Unit 3.

Q19 B is true. Not A since although a ship is not strictly inertial, Galileo had the idea. For B see

Section 4.1 of Unit 3. Not C since the coordinates of an event may be different for different inertial

observers. Not D; for instance see section 2.1 of Unit 3. Not E since both frames are not strictly

inertial, and anyway the two frames are not related by a Galilean translation of the pendulum and

other masses.

Q20 D and H are invariant under the Galilean transformation. Note that, in D and H, you can

wnte kl(xi - X2)2I and kl(xj - X2)- (vj - V2)l for the right-hand-sides. See Unit 3, SAQ 11, and its

solution.

Q21 G is correct. From the considerations, expecially of Unit 3, Sections 4.3 and 5.6, linear

momentum and energy are conserved. Thus, respectively,

miu = miu" + m2 v and ^mi u2 = Jmi u + \ m2 v2.

Solve these efficiently by writing R = mi/m2. Then they are equivalent to R(u - u0 = v and R (u -

u )(u + u') = v2. Use the 1st of these equations in the second to get v = u + u'. Use the 1st equation

again, with this latter result, to get v = 2 uR/(l+R).

Q22 B is correct. The kinetic energy of the second particle after collision is

4 m2V2 = 2m2 u2 R2/(l + R2) = 2 mi u2 R/(l + R2)

This is the energy lost by the first particle. Since mi is held constant one need only maximize R/(l +

R2). This maximum occurs when R = 1 or m2 = mi.

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The charge is attracted towards the origin along the 1-axis.

Q2 F is the correct answer.

The force, and therefore the acceleration (q v x B) must be perpendicular to v and to B.

From Figure 23 of Unit 4 you can see that B lies along the 3-axis.


As suggested by Figure 24 of Unit 4 the magnetic field lies along the 1-axis, so the

acceleration is perpendicular to this direction and the velocity.

Q4 B is the correct answer.

As suggested by Figure 20 of Unit 4, the magnetic field (at point (1,0,0)) lies along the

negative 2-direction. If v lies along the position 1-direction then a lies along the negative 3-

direction.

Q5 A, E and B are the correct answers.

The charge of a panicle and the constants eo and po are the same for all inertial observers.

Also laws of physics are form invaiiant, so acceleration of a free particle is zero for all inertiai

observers.


O lies on the light cone of ^g*


Consider light cones centred at the various events. The ray from crosses the t-axis at the

Q8 C and D are the correct answers.

%c lies in the future (see Figure 14 of Unit 7) and lies on the light cone and so could be

triggered by a pulse of light. Note that %c could be (for instance) caused by 0 by launching

an object with some speed less than c.

Q9 A is the correct answer.

The negative times of O lie in the region of absolute future only of event %\, and so could be

influenced by it. O's point (0,0) does lie on the light cone of but the problem specified

times before t = 0.

Q10 A and B are the correct answers.

Only events and do not lie in the region of absolute future of the light cone of %q.

Qll C is the correct answer.

t2-‘3=y(l2 - *3 -^(*2 - X3)j=y^- \ ~ ^(4)

This vanishes when V = -c/4.


X2-X3' = Y(*2 - *3 -V O2 - *3)) = Y^4 -V ^

This vanishes only for the impossible value V = - 4c.


, / . /

*1 -*4 = J. . v,,. 2

- t4 -"2V-V1 -*4jj- j j- c

This vanishes only for the impossible value V = -2c.


x\ -xt,' = 7 (xi - X4 -V (fi - t4)) = y 1 - v \jj

This vanishes for V = - c/2.


Since the speed are constant (for each leg) we can use

where the proper time interval At is the time recorded on the clock. See Section 5.2 of Unit 7.

From O to ^3 is c(At)o,3 = V1-0 = 1

From to%4 is c(Ax)3t4 = V22 - l2 = ^3

Hence c(At)o,3,4 = 1 + V3 = 2.73.


c(At)0i4 = V32- l2 = V8 =2.83.


The clock must be taken forward in time at speeds less than c. Only item D meets that

criterion.


E = y m c2

= 1.67 x IQ-27 x 9 x 1016 = 1.9 x Kh10 J.


ETqt = y m c2 + m c2

= |x 1.67 x 10~27 x 9 x 1016 = 3.4 x 1(H°J.


p =ymv=|x 1.67 x 10~27 x 1.8 x 108 = 3.8 x 10-19 kg mr1.

G is the correct answer.

The proton to be struck lies at rest and so has no linear momentum.


Linear momentum is conserved (p26 of Unit 7).


Let the incident particle approach from the left in S. Then S' moves to the right, with respect

to S, at some speed V. In S' the struck particle moves (to the left) with velocity -V and the

incident particle moves (to the right) with velocity V. We can use the velocity addition

formula, equation (29) of Unit 6, to write

where v=1.8xl0^ms 1 = 0.6 c. The above equation can be rearranged to give the

quadratic equation

so that

Vr fc\fV\ ir -2 - - +1=0 C VC

x=£+v 4-1 C V Vv2

v V 5 4 VI With - = 0.6, — = ^ ± y Only the minus sign makes physical sense. Hence — = ~

c j

[This problem can also be done by the optional material in Unit 7, pp 28-30].


Both move at V = ^ c. Hence the "centre-of-mass energy" is

t- _ n _ me2 2 me2 3 2

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Q1 E and B are the correct answers.

A is wrong because there were no previous indications.

C gives a percentage which is far too high.

D is not true because expansion and absorption are enough to make the sky dark, even in an infinite

universe.


In the standard model as set out in this course, the gravitational influence of other galaxies is

assumed to be negligible (It is accepted that there are superclusters, but it is supposed that they are

not bound).


See the solution to SAQ 2 of Unit 13.

(Av\ = _ G_m = 6.67 x 1Q-H x 3 x 1Q30

[ v Jg R c2 9 x 108 x 9 x 1016 -2.5 x 10-6.


As for SAQ 2 of Unit 13,

v _27tR _ 27t x 9 x 108

c " cT “ 3 x 108 x 3 x 106 = 6.3 x 10-6

This rotation speed can approach or recede along the line of sight so

— I = ± 6.3 x 10-6.

QS E is the correct answer.

As for SAQ 2 of Unit 13, since the turbulent motion may approach or recede from the observer.

Av'

v = + X = + 7000

3 x 108 = ± 23.3 x 10-6.


Equation (18) of Unit 13 gives an expression for the relative change in wavelength. This can be re¬

expressed as

_ Hr _ 2 x 10-18 x 3 x 10$ x 9.46 x IQ1*

^ Jc c - 3 x 108 -189.2 x 10-6

You must convert from light years to m. Use the table at the end of Unit 14. And note that the

cosmological redshift lowers the frequency.

(-189.2 - 2.5 ± 6.3 ± 23.3) x 10"6 = -191.7 ± 29.6.


Both A and B are alternative statements of the cosmological redshift [Q6]. The absorption by dust

might dim the light but would not shift its frequency. G is a restatement of the shift calculated in Q3.

Note that the ± signs represent ranges, so the overall range is got by adding the maxima (minima).

Q8 B and D are the correct answers.

For a general discussion see section 4 of unit 13. For option F see Section 5.2 of Unit 13.




From the last equation on page 9, Unit 14, dR/dt tends to the value c, since k is a negative

constant and the density falls as HR}.

Qll F is the correct answer,

k is a constant.

- (See also SAQ 3).

l//?(r)2 and approaches zero as t tends to infinity.

Q13 A is the correct answer. JL dtf(0 R(t) dr

We have already seen that dR(t)/dt becomes a positive finite quantity, so H varies as 1 /R{t) and

approaches zero with time.


From equations (1) and (2) of Unit 14 d 2R

dr2

Because p(r) « 1 //?(r)3, the right-hand side is


From equation 16 of Unit 13 q = - ~j

But from Q13, H ~ 1 /R(t) and from Q12, d2 R{t)ldr2 ~ l//?(r)2

Thus q approaches zero as \/R(t).


By definition, the critical density at time t is

pc(r) = ?>c1H2{t)l%TiG, which varies as l//?2(r), from Q13.

Q16 B and D are the correct answers.

From the analysis on page 10 of Unit 14, k vanishes, so space is flat. From SAQ 9 of Unit 14, R is

proportional to the two-thirds power of time. So by section 4.4 of Unit 13, space is flat and space-

time curved.


H = —. This is proportional to r1.

Q18 D is the coiTect answer.

1 R. , , v q = -jp • This 1S constant.


£ is constant.


Add equations (1) and (2) of Unit 14, rearrange to get - p = H2 - 2q H2 +-^|- •

The first two terms on the right are proportional to r2 and the third goes like r7/4, and this

dominates eventually.



The critical density pc is proportional to H2 and hence to R~2 But p goes like R~3. So p/pc goes

like 1 /R.


From Figure 12 of Unit 14 it is clear that in the given range the proportion of deuterium ^H) is the

fastest changing and hence the most selective.

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From the analysis on page 17 of Unit 6A, and not assuming v/c to be small, gives

f 1 +

\

similarly

r |=T0

1 -

1+VV

y To solve for - divide one equation by the other and solve for v/c to get v/c =

3/5.


Use either equation given for Ql, say the first, to get 10 = To '> - ^

1 - J

= 2 T0, T0 = 5.


The original radiation, before decoupling, interacted mostly with charged particles, and above all

with the lightest charged particles. These were electrons [page 11 of 6A].


From page 12 of 6A, T We are dealing with small changes in R, so the analysis on page 41

Unit 13 gives R(ti) = R(t0) + R(t0) H (to) (ti - to)

_1_

R(t0)(l+H(t0) (ti-to)) So T (ti)

n 1 . T(tl) _ 1 But T (to) - R(t0) SO T(to) - ! + H (to)(ti - to)

Now use \ ■ — 1 - x (for x small) to get 1 l A

=l-H(to)(ti-to)or

T(ti)-T(to) = -T(to)H(to)(ti-to)

use T (to) = 3.2, H (to) = 2 x 10-18 s-1 and (since there are 3.16 x 107 s in one year)

ti - to = 1(P x 3.16 x 107, to get the result.

Q5 A and D are the correct answers. d A

Note that G is not correct. See, for instance, Figure 8 of Unit 14.

Q6 A and E are the correct answers.

Neither D nor G, for these processes were characteristic of the basic physics of nuclei and atoms.

Q7 c and E are the correct answers.

E because there is an expansion; see COSMOLOGICAL REDSHIFT in the Glossary. See also pp

23-4 of Block 6A.

Q8 B and D are the correct answers. 2Gm

For a black hole you must have the event horizon k lie outside of the radius. For B, k = —and cz

would form a black hole. For E you find k = 1.5 x 10-17 m and (for a hydrogen atom) r ~ 10-10 m

so this would not form a black hole.

That option D is correct follows from the discussion on page 33 of Block 6B.

Q9 A and D are the correct answers.

Inside the event horizon all light cones tilt enough towards the singularity so that all motion must

tend inwards.

Q10 B and E are the correct answers.

B follows from page 37 of Block 6B: since E = hcA for photons, equation (13) is equivalent to ^

he3 * ~ 2mG or ^ > k where k = 2mG/c2.

E is true by the analysis on page 17 of Block 6B; taking the inverse of equation (1)

f gives vr =

k\ 2 1 ~ 7 Voo. Taking r = 4k/3 gives vr = 2 Vc*.

QH E is the correct answer.

By inverting equation (1) of Block 6B you get for the frequency at r

( kV^ Vr = — —J It follows that

VB=VA _IA

1

\

. Use va = 6 x 1014 Hz,

V *J 4

ta = 3 k and rg = 2k to get

vB = 4.24 x 1014 Hz.


Let m be the rest mass of an electron and M the black hole's mass. Then from equation (13) on page

37 of Block 6B (with E = me2)

o he3 m°2 ~ 2MG Solve lhis for M.

Then (page 38 of Block 6B)

T = 6.18 x 10-8 x ^ = 6.18 x 10-8 x M he

use m = 9.1 x 10-31 kg? etc. t0 get t « 7.5 x 107 K.

[Note that G and Ms are given at the end of Unit 14.]

Q13 H is the conrect an swer.

Solving for the mass of the black hole (See the solution to Q12) gives he

M = 2mG (where m = 9.1 x 10"31 kg)

The life-time can be estimated from the equation at the bottom of page 38 of Block 6B to get

3

Q16 A and F are the correct answers.

F is true, in fact, also in General Relativity (except, presumably, at the centre of a black hole!),

provided an internal frame is (briefly!) defined as freely falling and non-rotating.

S354 Assignment Booklet I - archive.org

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Transcript of S354 Assignment Booklet I - archive.org