UfflVERSm TEKNOIOOI HAUYStt

R E S E A R C H U N I V E R S I T Y

UNIVERSITITEKNOLOGI MALAYSIA FAKULTI SAINS

FINAL EXAMINATION SEMESTER I SESSION 2011/2012

CODE

COURSE

SSCM 1033/SSH 1033

MATHEMATICS METHODS II

LECTURER MR. CHE LOKMAN JAAFAR

PROGRAM 2 SSA

DATE 13 JANUARI2012

DURATION 3 HOURS

INSTRUCTION ANSWER ALL QUESTIONS

(THIS QUESTION PAPER CONSISTS OF 7 PRINTED PAGES INCLUDING THIS PAGE)

SSCM 1033

Answer All Questions

1. (a) Use appropriate test to determine convergence or divergence of the following series.

oo 2

® I2;n= 1

n\

00 i

n=2 n\nn

[4 marks]

[4 marks]

(b) Show that the Maclaurin series for

f (x ) = sin* is

Hence

(i)

x3 x5

X-------- + —3! 5!

obtain the power series for cos x, and deduce that

3'cos2 x2 = 1 -x4 +—X*

(ii) estimate(•0.8 2 2 2

x cos x dxJo

using power series.

[4 marks]

[4 marks]

[4 marks]

2. (a) A quantity T is given as T = 2n — . If the maximum percentage error in measuring

x and y are respectively 1% and 3%, estimate the maximum percentage error in calculating T.

[8 marks]

f 'N JC

y.

(b) Given that f (x,y) = ^cos\

Find fx 5 fy and show that xfx + yfy - f

Hence show that xf^ + yfyx - 0 [8 marks]

(c) Find all critical points for the graph z = 2y4 + x2 -12xy. Classify the points as

minimum, maximum or saddle point. [8 marks]

2

SSCM 1033

3. (a) Evaluate IJ ey dAR

where R is the region bounded by the lines y = 2x,y = 3 and they-axis.

[8 marks]

(b) Use double integral to find the volume in the first octant bounded above by the plane octant x+ y + z = 4.

[8 marks]

(c) Use polar coordinates to evaluate

J2 JJ8 y dxdy [8 marks]

J1 J' fJO Jo Jo

answer.

4. (a) Evaluate f f f xy2z3 dxdydz. Give one complete interpretation of yourJo Jo Jo

[6 marks]

(b) Use cylindrical coordinates to find the volume inside the paraboloid z - 5 - x2-^2 and

bounded below by the plane z = 1. [8 marks]

5. Let G be the solid inside the sphere x 2+ y1 + Z2 = 4 and above the plane z = V2 .

Use triple integral in spherical coordinates to

(a) show that the volume of G is — - 5-/2 W,3 [8 marks]

(b) evaluate iffoG

Hence find the center of gravity of the solid G assuming constant density.

[10 marks]

3

FORMULA SSH 1033

Differentiations Integrations

d— [fc] — 0,k constant. dx

J kdx = kx + C, k constant.

f xn+l / xndx = +C,njt 1. J n+1

J ezdx — ex + C.

s P - w i - i .r dxJ — =ln|x|+C.

— [cost] = — sin i. dx

J sin xdx = — cos x + C.

df l—Ism xj = cos i. dx

J cos x dx = sin x + C.

d , 2 —Itanxl = sec i. dx

/ sec2 xdx — tan x + C.

d . 2 —Icotxl = —cosec x. dx

J cosec2 x dx = — cot x + C.

d ,— [secx] = sec x tan x. dx

J secx tan xdx = sec x + C.

d r— Icosec xl = —cosec i cot x. dx

J cosec x cot xdx = —cosec x+C.

— [cosh x] = sinh x. dx

J sinh xdx = cosh x + C.

—[sinhx] - - coshx. dx

/ cosh x dx = sinh x + C.

— [tanh x] = sech2x. dx

J sech2x dx -- tanh x + C.

—[coth x] = —cosech2x. dx

y cosech2xdx = — cothx + C.

-̂ -(sechx] = — sech x tanh x. y sech x tanh xdx = —sech x + C.

—[cosech x] = —cosech x coth x. dx

J cosech x coth x dx= —cosech x + C.

4

r

SSH 1033

Differentiations of Inverse Functions

£[sin_lx! = 7ra'1x1 <L

[cos 1x] = ...................|x| < 1.dx y/Td r -i , 1— [tan x] = ----------------.

dx 1 +x2

— [cot­ax

-1 1 + x2

-^-(sec ‘ij = | A'37. N > 1- dx_____________ |x| yx2 *̂ 1___________

—[cosec 1x] =------------- . * Ixl > 1.dx__________ 1 Ixlv/^ l̂’

•y-[sinh 1 x] = — ,1-=. dx y/x* + 1

-̂ -(tanh = 1 |x| < 1. dx 1 — x'4

— [coth 1 x] = 1 ■, |x| > 1. dx 1 — xz

— [sech Ix] = — , 0 < x < 1- dx ' xVl -xl _____

— [cosech 1x] = ------------ . , x ̂ 0.dx____________ lx|y/TT^

Integrations Resulting in Inverse Functions

/dx

= sin l(x) + C.

/dx

1 +x:= tan 1 (x) + C.

/dx

\x\Vx2 — 1= sec 1 (x) + C.

/dx

Vx2 + 1= sinh 1 (1) + C.

/dx

\/x2 — 1= cosh 1 (x) + C, x > 0.

/dx

1 — x:= tanh *x + C, |x| < 1.

/dx

:2 -1: coth lx + C, |x| > 1.

/:dx

x>/l — x*= —sech 1 (x) + C, (x| < 1.

/dx

Xy/l+X*= cosech 1 Ixl + C.

Trigonometric Hyperbolic

cos2 x + sin2 x = 1 1 + tan2 x = sec2 x cot2 x + 1 = cosec2x

sin(x ± y) = sin x cos y ± cos x sin y

cos(x ± y) = cosxcosy sin xsiny. , . tan x ± tan y

tan(x ± y) =----------------------- —1 tan x tan y

sin 2x = 2 sin x cos x cos 2x = cos2 x — sin2 x

2 cos2 x — 1 = 1 — 2 sin2 x

2 tan itan 2x =------------- -—

1 — tan2 x2 sin x cosy = sin(x + y) + sin(x — y)

2 sin x sin y = — cos(x + y) + cos(x — y)2cosxcosy = cos(x + y) + cos(x — y)

_ £ 1sinh x = ---------------

2e*+e _ I

coshx =----------------2

cosh2 x — sinh2 x = 1 1 — tanh2 x = sech2x

coth2 x — 1 = cosech2 x sinh(x ± y) = sinh x cosh y ± cosh x sinh y cosh(x ± y) = cosh x cosh y ± sinh x sinh y

tanhx i tanhytanh(x ± y) = —------------- -------- —

1 dh tanh x tanh ysinh2x = 2 sinh x coshx

cosh 2x = cosh2 x + sinh2 x = 2 cosh2 x — 1 = 1 + 2s inh 2 x ’

2 tanh xtanh 2x =--------------- =—

1+ tanh x

Inverse HyperbolicLogarithm

sinh-1 x = ln(x + y/x2 + 1), —oo < x < oo cosh-1 x = ln(x + Vx2 — 1), x > 1

tanh-1 x = - In ( | , —1 < x < i 2 \l-xj'

a* = « * , E O log,,x

loga I - . logt a

5

SSH 1033

Polar Coordinates:

x = r cos 8, y — r sin 6.

Jl f (X ,y )dA=Jj f (r cos 8, r sin 6) rdrdd.

n n

Cylindrical Coordinates:

x = rcos9, y = rsind, z — z.Ill/(I- z)dV = JII f (r cos 9, r sin 9, z) dz r dr dS.

Spherical Coordinates:

x = p cos 6 sin <p, y = p sin 6 sin <f>, z = p cos <jb.

JJJ H*,y, = f (p cos 0 sin 0, p sin 9 sin <j ) , p cos <p) p2 sin <f>dpd<t>d9.

6

SSH 1033

Mass and first moment formulas

THREE-DIMENSIONAL SOLID

IlhMass: M — JJI S dV S = 8 (x,y, z ) is the density at (t, v, z).

d'

First moments about the coordinate planes:

MyX = IllXSdK = IIiy5dV’ Mxy=X IIIZSdVD D D

Center of mass:_ MyZ _ MxZ — ^xy

X = ~M’ y = ~W’ Z^~M

TWO-DIMENSIONAL PLATE

Mass: M — JJS dA S = S (x,v ) is the density at (x, y).

R

First moments: My = JJx 8 dA, Mx = JJy S dA

R R

~ - MxCenter of mass: x = -rj , y = -z -rM M

/

UTMUNIVERSITI TEKNOLOGI MALAYSIA

R E S E A R C H U N I V E R S I T Y

UNIVERSITI TEKNOLOGI MALAYSIA FAKULTI SAINS

FINAL EXAMINATION SEMESTER I SESSION 2011/2012

CODE

COURSE

SSH 1523

LINEAR ALGEBRA

LECTURER DR. FONG WAN HENG

PROGRAM 2 SPM, 4 SPC, 4 SPF

DATE 18 JANUARY 2012

DURATION 3 HOURS

INSTRUCTION ANSWER ALL QUESTIONS

(THIS QUESTION PAPER CONSISTS OF 6 PRINTED PAGES INCLUDING THIS PAGE)

SSH 1523

Answer ALL Questions.

1. (a) Suppose A and B are two matrices of order nxn and A is not singular.

Show that A~ lBT A= B . [3 marks]

"1 2 3 N ' 3 0"

(b) Given B 1 = 0 1 -1 . Find a matrix A such that BA = -2 1

J 0 2, v1 2,[3 marks]

a b c d e f

(c) If d e f = 3, find a b c

g h i -d + g -e + h I +

[3 marks]

(d) Using row operation and properties of determinant, find the determinant of the

matrix

1 2 1 -3n

- 1 - 2 0 1

2 0 2 -1

12 1-5

without using cofactor expansion. [4 marks]

2. (a) Solve the following system of linear equations.

2x + 3y- z = 3

-x - y + 3 z = 2 [4 marks]

x - 2y + z = - 4

(b) Given a system of linear equationsx + 2y + 3z = 2

2x + 5y + 6z = a + 4

x + y + {b2 - l)z = 0

where a,6el. Reduce the system’s augmented matrix obtained from the system to row echelon form. Hence determine a and b so that the system

(i) has a unique solution,(ii) has many solutions,(iii) has no solution. [8 marks]

2

SSH 1523

3. (a) Show that W = { p(t ) e P2: p(t ) = at2 -1, a e is not a subspace of P2.

[3 marks]

(b) Given v, =(l,2,-l), v2 =(-2,-4,2) andv3 =(2,h, -2) are vectors in vector

space R3. Find h so that v3 € Span {v,,v2|. [4 marks]

(c) Given S = {(1,0,2), (2,1,1), (1,-1,3), (2,1,0)j is a subset of the

vector space M3.

(i) Determine whether set S is linearly dependent or not. [2 marks]

(ii) Is S a basis for M3 ?If not, find a subset of S which is a basis for span S. [4 marks]

(d) Let S = {w,, u2} and S' = [u , u \} be ordered bases of vector space M2 where

'2' rnf°l

r-ru, =

J,,«2 =

,0,1 = ,u\ =

(i) Find a transition matrix P which satisfies the equation [v] = P[v] s for

every v e.

(ii) Compute the coordinate vector [v]s, where v =

(iii) Compute [v]s, using part (i) and (ii).

v 3 y

[3 marks]

[2 marks]

[2 marks]

3

SSH 1523

'1 0 2"

4. (a) Given the matrix A = 0 1 7 . Find a basis and dimension for the row

,"2 1 3,space of the matrix A. [4 marks]

(b) Show that for any vector vel", the norm of unit vector -jj—rr is always 1.v

(C) Given u = [a, 0, b) and v =

orthonormal.

/ 1 -2 -1_ N

V6 ’V6 ’V6

[2 marks]

. Find a and b so that u and v is

[4 marks]

4

UNIVERS1TI TEKNOLOGl MALAYSIA

UN1VERSITI TEKNOLOGl MALAYSIA FAKULTI SAINS

FINAL EXAMINATION SEMESTER II SESSION 2010/2011

COURSE CODE

COURSE

LECTURERS

SSP 1163

WAVES, SOUND & OPTICS

DR. ASIAH YAHAYA DR ZUHAIRI IBRAHIM

PROGRAMME

DATE

TIME

1SSZ, 1SSF

03 MAY 2011

2 HRS 30 MINUTES

INSTRUCTION TO STUDENTS:

ANSWER FOUR (4) QUESTIONS ONLY

THE LIST OF FORMULAE IS GIVEN

THIS EXAMINATION PAPER CONSISTS OF (6) PRINTED PAGES

SSP 1163

Answer part A or part B

Part A

Explain with an example, how a wave can be generated?

(5 marks)

(b) Write an equation of a harmonic wave travelling along the x-direction, at a

time t = 0, its amplitude is 2 m and a wavelength 10 m.

(5 marks)

(c) Find the resultant wave of the superposition of two harmonic waves in the

form

E = E0 cos (a-cot)

with amplitudes of 5 cm and 6 cm and phases of 71/8 and n/4 respectively.

Also both waves have a period of 1 s.

(8 marks)

(d) A wave is represented by the equation (in S.I. units),

ip(x, t) — 5 cos 7r(3 x 106x + 9 x 1014t)

Assuming the phase is contant, calculate the speed of this wave.

(7 marks)

Part B

(a) The differential wave equation is given by

d2cp __ 1 d2<pSx 2 v 2 S t 2 '

Show that cp(x,t) =A sin (kx +wt) is a solution of the differential equation.

(5 marks)

(b) Explain what is meant by an electromagnetic wave? Discuss your answer

with appropriate examples.

(5 marks)

(c) Two parallel waves,n

E1 — 5 sin (cot + —)

nE2 = 6 sin (cot + —)

propagates in space. Find the resultant of the superposition.

SSP 1163

(8 marks)

(d) Standing waves are produced by the superposition of the wave,

and its reflection in a medium with negligible absorption with x in cm and t in

seconds. For the resultant wave, calculate;

(i) wavelength,

(ii) length of the loop,

(iii) the net velocity,

(iv) the period.

(ii) Define the terms pitch, loudness and quality with reference to sound.

(b) (i) A metal bar with a length of 1.50 m has a density of 6400 kgm-3

Longitudinal sound waves take 3.90 x 10-4 s to travel from one end of the

bar to the other. What isYoung’s modulus of this metal?

(ii) An 80 m long metal brass rod is struck at one end, A person at the other

end hears two sounds as a result of two longitudinal waves, one travelling in

air and the other in th metal rod. What is the time interval between the two

sounds. (/ = 9.0 x 1010 Pa,p = 8.6 x 103 kgm-3)

(3 marks)

(c) (i) What is the difference between the fundamental frequency of an open and

a closed pipe. Draw suitable diagrams to illustrate your answer.

(4 marks)

(ii) Two open organ pipes, one 2.5 m and one 2.4 m in length are sounded

Together. Calculate the number of beats per second that will be produced

(7 marks)

2. (a) (i) Explain how sound can be produced. (3 marks)

(3 marks)

(3 marks)

between the fundamental tones. Assume the speed of sound is 330 ms'1.

(4 marks)

3

SSP1163

(d) A Doppler effect was observed when a train was moving towards an observer

who was standing on the railway platform. The speed of the train was

10 ms’1. At the same time the whistle from the train was blown with a

frequency of 400 Hz. Is the frequency heard by the observer, > 400 Hz or

< 400 Hz or = 400 Hz? Prove it.

(5 marks)

(a) When unpolarized light is incident on two crossed polarizers, no light is

transmitted. Inserting a third polarizer between the other two, some

transmission will occur. How can adding a third filter increase transmission?

(5 marks)

(b) Unpolarized light of intensity 20.0 Wm-2 is incident on two polarizing filters.

The axis of the first filter is at an angle of 25° counter clockwise from the

vertical (viewed fin the direction of the light travelling) and the axis of the

second filter is at 62.0° counter clockwise from the vertical. What is the

intensity of the light of the light after it has passed through the second filter?

(6 marks)

(c) Coherent light from a sodium vapour lamp is passed through a filter that

blocks everything except light of a single wavelength. It falls on two slits

separated by 0.460 mm. In the resulting interference pattern on the screen

2.20 m away, adjacent bright fringes are separated by 2.82 mm. What is the

wavelength of the light?

( 7 marks)

(d) You wish to coat a flat glass (n = 1.50) with a transparent material (n = 1.25)

so that reflection of light at wavelength 600 nm is eliminated by interference.

What minimum thickness can the coating have to do this?

(7 marks)

4 (a) Describe both the similarities and differences between Fraunhofer and Fi esnel

diffraction

(5 marks)

4

SSP 1163

(b) In a single slit diffraction pattern,

(i) what is the intensity at the point where the total phase different between the

top and bottom of the slit is 66 rad?

(3marks)

(ii) If the point is 7.0° away from the central maxima, how many wavelengths wide

is the slit?

(3 marks)

(c) Monochromatic light of wavelength 441 nm is incident on a narrow slit. On a

screen 2.00 m away, the distance between the second diffraction minimum

and the central maximum is 1.50 cm.

(i) Calculate the angle of diffraction Q of the second minima.

(3 marks)

(ii) What is the width of the slit?

(3 marks)

(d) (i) How many bright fringes appear between the first diffraction-envelope minima

to either side of the central maximum in a double-slit pattern if A = 550 nm, d =

0.150 mm and a = 30.0 urn?

(4 marks)

(ii) Using the same data as in (i), what is the ratio of the intensity of the third

bright fringe to the intensity of the central fringe?

(4 marks)

5

i

SSP 1163

List of Formulae and constants

B = 1.42 x 105 Pa

^sound ~~ 344m/s

Pmax = BkA

I = -eqcE

I =2 pv

1.22 AD

1 = 1,sin(5/2\

P/2 '/'sin (3/2

y(x,t) = 2j4sin(/cx)sin(ojt) = (i4sw sin(kx)) sin(ooO

E„2 = Ex2 + E2

2 + 2E1E2 cos(S2 - 50

tan S =

s0 = 8.85 x 10

sin<5x + E2 sin<52

Ei cos Si + E2 cos S2

(C.s)2-12

kg.w?= i Mogf- '1

O ± vL) ^fL =

O ± Vs)

H0 = An x 10 7 kg.m

(A.sy

/sinb/z\ . ,1 = 4,0 ( itt) cos ( a / 2 )

for P — ~rO. sin 8 and a = ^ d sin 6A A

6

UNIVERSITI TEKNOLOGI MALAYSIA FAKULTI SAINS

PEPERIKSAAN AKHIR SEMESTER I SESI 2010/2011

KOD MATA PELAJARAN SSP1223

NAMA MATA PELAJARAN FIZIK MODEN

NAMA PENSYARAH EN. ABD RASHID ABD RAHMAN

KURSUS 1SPM/3SSF/3SSH

TARIKH 30 NOVEMBER 2010

MASA 3 JAM

ARAHAN KEPADA PELAJAR :

1. Jawab semua soalan di Bahagian A2. Jawab empat (4) soalan sahaja di Bahagian B

(KERTAS SOALAN INI MENGANDUNGI (14) HALAMAN BERCETAK)

SEKSYEN A

Jawab SEMUA soalan, dengan SERBA RINGKAS mungkin. (40 markah)

1. (a) Nyatakan dua perbezaan utama di antara Transformasi Galileo dengan Transformasi Lorentz.

[2 markah]

(b) (i) Nyatakan sumbangan serta kelemahan hukum/teori sinaran jasad-hitam yang diutarakan oleh Wien dan Rayleigh-Jeans.

(ii) Terangkan dengan ringkas bagaimana Max Planck berjaya mengatasi kemelut sinaran jasad-hitam.

[5 markah]

(c) Nyatakan dengan lengkap, dua postulat kerelatifan khas Einstein.[3 markah]

(d) Kira tenaga jisim rehat satu proton dalam unit S.l. (J), berdasarkan

[4 markah]jisimnya yang diberikan sebagai 1.67262 x 10 27 kg.

(e) Lukiskan gambarajah yang mencirikan spektrum sinaran jasad hitam.[4 markah]

(f) Apakah hipotesis Einstein mengenai tabiat cahaya ?[3 markah]

(g) Nyatakan dengan ringkas hipotesis de Broglie[3 markah]

(h) Lakarkan gambarajah aras tenaga (bertanda) bagi atom hidrogen Bohr yang menunjukkan penghasilan pancaran siri Lymann.

[5 markah]

(i) Nyatakan 4 (empat) nombor kuantum

[4 markah]

(j) Apakah yang dimaksudkan dengan masa wajar ?[2 markah]

(k) Terangkan mekanisme penghasilan sinar-X selanjar dan cirian. Gunakan beberapa gambarajah yang sesuai.

[5 markah]

2

SEKSYEN B

Jawab HANYA 4 (EMPAT) soalan sahaja. (60 Markah)

2. (a) Pergerakan suatu jasad P dalam dua dimensi menurut satu pemerhati pegun O diberikan oleh persamaan:

jc = 5 + It3 - 5Z2, dan y = f3 - 2t + 2,

O ’ ialah satu pemerhati yang sedang bergerak pada halaju malar 10 ms'1 dalam arah-x. Menurutnya, O ’, P berada di kedudukan (x y ’).

(i) Dapatkan persamaan untuk x ’ dan y

(ii) Dapatkan persamaan komponen halaju P menurut O dan O

(iii) Dapatkan komponen pecutan P menurut kedua-dua pemerhati.

[5 markah]

(b) Zarah muon (ju) merupakan zarah yang tidak stabil dan akan mereput kepada beberapa zarah yang lebih stabil. Purata tempoh hayat satu muon dalam keadaan pegun ialah 2.20 *1 O'6 s. Jika ia bergerak dengan suatu halaju v yang amat tinggi maka tempoh hayatnya akan bertambah kepada 15.6 xlO-6 s. Kira halaju v itu.

[4 markah]

(c) Dua proton sedang bergerak dengan kelajuan yang sama dalam arah yang bertentangan. Kedua-dua proton itu terus wujud (tidak termusnah) selepas berlanggar secara berdepan untuk menghasilkan satu pion yang neutral (;r°). Pion itu terhasil dari tenaga kinetik asal kedua-dua proton itu. Kesemua zarah berada dalam keadaan pegun sejurus selepas perlanggaran tak kenyal ini. Jisim rehat zarah pion itu ialah mK = 2.40 x 10'28 kg.

(i) Kira tenaga yang diperlukan untuk menghasil pion itu.

(ii) Tentukan halaju (kerelatifan) asal setiap proton itu.

(iii) Kira momentum proton itu.

[6 markah]

3

Nyatakan 4 keputusan penting dari eksperimen fotoelektrik - dengan lakarkan graf yang releven.

[4 markah]

Fungsi keija bagi Na yang mempunyai permukaan yang bersih ialah2.3 eV. Dua sinar cahaya, A.i = 4500 A dan hi = 5500 A, menyinari permukaan logam tersebut.

(i) Kira tenaga foton setiap cahaya untuk menentukan samada fotoelektron boleh terhasil.

(ii) Kira tenaga kinetik maksimum fotoelektron yang terpancar.

(iii) Kira frekuensi ambang bagi Na.

[5 markah]

Satu foton sinar-X dengan panjang gelombang 5.5 A melanggar satu elektron pegun lalu mengalami serakan Compton pada sudut 125° merujuk kepada arah asal foton tuju itu.

(i) Kira tenaga asal foton tuju itu.

(ii) Kira tenaga foton yang terserak.

(iii) Tentukan halaju elektron yang tersentak.

[6 markah]

Terangkan dengan ringkas bagaimana hipotesis de Broglie dapat memberikan asas kepada andaian Bohr mengenai pengkuantuman momentum sudut, Ln = nh.

[4 markah]

Terangkan dengan ringkas eksperimen belauan elektron yang telah membuktikan kesahihan hipotesis de Broglie.

[5 markah]

Kira panjang gelombang jirim bagi zarah berikut - abaikan kesan kerelatifan:

(i) Satu elektron berhalaju 106 ms'1.

(ii) Satu elektron yang dipecut dengan voltan 200 V.

[6 markah]

5. (a) Terangkan dengan ringkas kebaikan serta kelemahan model atom Dalton.

[2 markah]

(b) (i) Terangkan dengan ringkas model atom "plum pudding" olehThomson.

(ii) Apakah kelemahan model atom Thomson ?

[4 markah]

(c) Model "plum pudding" tidak dapat menerangkan beberapa penemuan penting serta pemerhatian yang bercanggahan. Rutherford telah cuba merungkaikan masalah ini melalui beberapa siri eksperimennya ke atas zarah-a.

(i) Bincangkan dengan ringkas idea di sebalik eksperimen zarah-a Rutherford.

(ii) Bincangkan dengan ringkas hasil eksperimen zarah-« di atas.

(iii) Apakah penerangan Rutherford mengenai hasil tersebut ?

(iv) Bincangkan dengan ringkas kelemahan model atom Rutherford.

[9 markah]

6. (a) Model Rutherford menghadapi masalah kestabilan. Masalah ini telahdirungkai oleh postulate dan andaian Bohr.

(i) Nyatakan postulat (andaian) Bohr mengenai atom yang stabil.

(ii) Nyatakan beberapa kej ayaan model atom Bohr.

(iii) Apakah kelemahan model tersebut ?

[7 markah]

5

(b) Aras tenaga bagi orbit ke-w dalam atom hidrogen Bohr boleh ditulis sebagai:

(i)

(ii)

r k2e*mE„ = — —

2nlh2fc2 (1)Nyatakan maksud setiap simbol yang digunakan di dalam persamaan di atas.

Dari persamaan di atas tunjukkan bahawa persamaan bagi transisi elektron dari aras tenaga n, ke n/. adalah:

hv = me8 slh1

_1___l_nl n]

\ j ‘ J(2)

(iii) Kira panjang gelombang foton yang terpancar akibat perpindahan elektron dari aras n = 3 ke aras tenaga asas.

[8 markah]

7. (a) Bincangkan dengan ringkas kesan ke atas spektrum sinar-X selanjar dan cirian akibat perubahan berikut (gunakan beberapa rajah yang sesuai) :

(i) Perubahan beza keupayaan (voltan operasi),

(ii) Perubahan anod sasaran.

[8 markah]

(c) Satu tiub sinar-X beroperasi dengan beza keupayaan 18 kV. Sinar-Xditujukan ke atas satu hablur LiF (lithium floride), dengan pemalar kekisi d- 2.82 A, untuk analisis belauan Bragg.

(i) Kira panjang gelombang minimum (Kmin) sinar-X selanjar yang dihasilkan oleh tiub ini.

(ii) Kira sudut Bragg yang terkecil bagi nilai Xwa ini.

[7 markah]

6

Persamaan Schrodinger tak bersandarkan masa (TISE) dalam satu dimensi boleh ditulis sebagai:

Fungsi gelombang bagi suatu zarah yang diterbitkan dari persamaan di atas ialah y/{x).

(i) Nyatakan makna setiap simbol dalam persamaan Schrodinger di

(ii) Apakah yang dimaksudkan dengan y/(x) mesti temormal ?

Fungsi gelombang bagi satu elektron yang berada di dalam suatu perigi keupayaan segi empat tak terhingga selebar L diberikan oleh persamaan berikut:

Tenaga elektron itu diberikan sebagai:

(i) Kira amplitud gelombang dalam sebutan L, dengan menggunakan syarat pemormalan ke atas fungsi gelombang.

(ii) Kira tenaga elektron jika L = 2 A bagi n— 1,2 dan 3.

(iii) Lukiskan bentuk gelombang de Broglie elektron itu bagi n = 1,2

(3)

atas.

[5 markah]

(4)

dan 3.

[10 markah]

ENGLISH VERSION

SECTION A Answer ALL questions, as BRIEFLY as POSSIBLE. (40 marks)

1. (a) State the two main differences between Galilean Transformation and Lorentz Transformation.

[2 marks]

(b) (i) State the contributions and the weakness of the black-bodyradiation laws/theory as proposed by Wien and Rayleigh-Jeans.

(ii) Explain briefly how Max Planck succeeded in overcoming the black-body radiation dilemma.

[5 markah]

(c) State in full the two Einstein’s postulates of Special Relativity.[3 marks]

(d) Calculate the rest mass energy of a proton in S.I. unit (J), based on its

[4 marks]given mass of 1.67262 * 10'27 kg.

(e) Draw a diagram to characterise the spectrum of a black body radiation.[4 marks]

(f) What is Einstein's hypothesis on the nature of light ?[3 marks]

(g) State briefly de Broglie's hypothesis.[3 marks]

(h) Sketch a labelled energy level diagram for Bohr’s hydrogen atom indicating the production of the Lymann series.

[5 marks]

(i) State the 4 (four) quantum numbers.[4 marks]

(j) What is meant by proper time ?[2 markah]

(k) Explain briefly the production mechanism of the continuous and characteristic X-rays. Use appropriate diagrams.

[5 marks]

8

SECTION B

Answer ONLY 4 (FOUR) questions. (60 Marks)

2. (a) The motion of a body P in two dimension according to a stationary observer O is given by the equations :

x - 5 + 2J3 - 5Z2, dan y = ? - It + 2,

O’ is an observer that is moving with a uniform velocity of 10 ms'1 in the x-direction. According to O P is at the position (x yr).

(i) Derive the equations for x' and y ’.

(ii) Derive the expression for the velocity components of P according to O and O

(iii) Derive the acceleration compponents of P according to both observers.

[5 markah]

(b) Muon (jx) is an unstable particle that will decay into more stable particles. The average half-life of a stationary muon is 2.20 x 1 O'6 s. If it moves with a very high speed v its half-life would increased to 15.6 *1 O'6 s. Calculate the speed v.

[4 marks]

(c) Two protons are moving with equal speeds in the opposite directions.They continue to exist (not annihilated) after a head-on collision to produce a neutral pion (/r°). The pion was produced from the initial kinetic energies of the two protons. The particles were at rest immediately after the inelastic collision. The rest mass of the pion is mK= 2.40 x 10"28 kg-

(i) Calculate the energy needed to create the pion.

(ii) Determine the initial (relativistic) speed of each proton.

(iii) Calculate the momentum of the proton.

[6 marks]

9

State 4 important results of the photoelectric experiment - with the relevent graphical sketches.

[4 marks]

Work function of a Na with clean surface is 2.3 eV. Two photons with wavelengths, A.i=4500 A dan = 5500 A, strike its surface.

(i) Calculate the energy of each photon to determine if photoelectron could be produced.

(ii) Calculate the maximum kinetic energy of the ejected photoelectron.

(iii) Calculate the cut-off frequency of Na.

[5 marks]

An X-ray photon with a wavelength 5.5 A being scattered off (Compton scattering) by a stationary electron with an angle 125° relative to the incident photon.

(i) Calculate the energy of the incident photon.

(ii) Calculate the energy of the scattered photon.

(iii) Determine the velocity of the scattered electron.

[6 marks]

Explain briefly how does de Broglie’s hypothesis managed to provide the basis for Bohr’s assumptions on the quantisation of angular momentum,L„ - nh.

[4 marks]

Explain briefly an electron diffraction experiment that verified de Broglie’s hypothesis.

[5 marks]

Calculate the de Broglie wavelength for the following bodies, disregarding any relativistic effects:

(i) An electron with a velocity of 106 ms'1.

(ii) An electron accelerated through a potential of 200 V.

[6 marks]

Explain briefly the success and the weakness of Dalton's atomic model.

[2 markah]

(i) Explain briefly the "plum pudding" atomic model by Thomson.

(ii) What are the weaknesses of Thomson's atomic model ?

[4 markah]

The "plum pudding" model could not explained some new important discoveries and with contradicting observations. Rutherford had tried to resolve the situation through his series of experiment on a-particle.

(i) Discus briefly the idea behind the Rutherford's or-particle experiment.

(ii) Discus briefly the results of the experiments above.

(iii) What are Rutherford's explanation on these results ?

(iv) Discus briefly the weaknesses of Rutherford's atomic model.

[9 markah]

Rutherford's model was faced with the problem of instability. This was resolved by Bohr's postulates and assumptions.

(i) State Bohr's postulates (assumptions) regarding a stable atom.

(ii) State some of the successess of Bohr's atomic model.

(iii) What are the weaknesses of the model ?

[7 markah]

Energy level for an n-orbit in a Bohr’s hydrogen atom can be written as:

E=- 2 * 22 n%(1)

(i) Specify the meaning of each symbol used in the above equation.

(ii) Using the above equation, show the equation for transition of electron from energy levelw, to n/. is :

hv = me

8 e]h2

1 1

\”2f n. (2)

(iii) Calculate the wavelength of the photon emitted due to the transition of electron from energy level n = 3 to ground state.

[8 marks]

Discuss briefly the effects on the continuous and characteristic X-ray spectrum due to changes on the following parameters (use appropriate diagrams as illustrations):

(i) change in potential difference (operation voltage).

(ii) change in material of target anode.

[8 marks]

An X-ray tube is operating at a potential of 18 kV. The X-rays is incident onto a NaCl crystal, with a lattice parameter of </=2.82 A, for Bragg diffraction analysis.

(i) Calculate the minimum wavelength (Xnin) of the continuous X-ray produced by the tube.

(ii) Calculate the smallest Bragg angle for this value of /Lm.

[7 marks]

8. (a) The one dimensional time-independent Schrodinger equation (TISE) may be written as:

(3)

The wave function for a particle that is a general solution of the above equation is i//(x).

(i) State the meaning of each symbol used in the Schrodinger equation above.

(ii) What does it means that y/(x) must be normalised ?

(b) The wave function for an electron inside an infinite square potemtial well of a width L is given by the following equation :

The energy of the electron is given by:

(i) Calculate the wave amplitude in terms of L, using normalisation condition on the wave function.

(ii) Calculate the energy of the electron if L = 2 A for n = 1,2 and 3.

(iii) Draw the shape of the de Broglie matter wave of the electron for n= 1,2 and 3.

[5 marks]

(4)

[10 marks]

13

Pemalar Fizik dan pertukaran unit

Cas electron e = 1.60218 x 10'19C

Jisim rehat electron me = 9.10938 x 10'31 kg= 0.511 MeV/c2

- 5.48580 x 10"* u.j.a

Jisim rehat proton mp = 1.67262 x 10'27 kg

938.3 MeV/c2

1.00728 u.j.a

Jisim rehat neutron mn = 1.67493 x 10'27 kg

- 939.6 MeV/c2

= 1.00866 u.j.a

Halaju cahaya c = 2.99792 x 108 m/s

Pemalar Wien b = 2.898 x 10 3 m.K

Pemalar Planck h = 6.62607 x 10‘34 J.s

4.13567 x 10‘15 eV.s

Pemalar Coulomb k = 8.98755 x 109Nm2C‘:

Pemalar Rydberg = 1.09737 x 107 m*1

Ketelusan elektrik £0 ~ 8.85419 x 10'12 C2N"

1 eV = 1.60218 X 1019 J

1 u.j.a = 1.66054 x 10'27 kg

1 A = 10'10 m

Rumus

L = Lq.Ii — —r- = — (l - C°M)°v c2 m c

E = E 0 +K W = K m +*o

E 0 = m 0c 2 £2 = /?2c2 + (m0c2 )2

«A = 2t/sin<9 *

14