7/23/2019 ADG-sugg.pdf

1/2

PHY 202 Suggested Problems

Q 1 a)Calculate the Fourier transforms of the following functions

(i) f

x

x2 when

1 x

1

0 when x 1

(ii) f

x

x when a x a

0 when

x a

(iii) f

x

x when 0 x 1

2 x when 1 x 2

0 when x 2

, f

x f

x

(iv) f

x 11 x2

(v) f

x e

x

e

2

x]

b)If

f

x

1

0A

cos

xd

then show that

(i) x2f

x 1

0A

cos

x dwhere A

d2Ad2

(ii) f

ax 1a

0 A a

cos

x d a 0

c)Show that the function f

x

i when 0 x

0 when

x 0

can not be represented by a Fourier integral.

Q 2 a)Calculate the Laplace transform of

(i)

a bt 2

(ii) sin

t

(iii) f

t

t when 0 t a

0 when t a

b)Calculate the inverse Laplace transform of

(i) 5s

3

(ii) 1s2 25

(iii) a1s

a2s2

a3s3

c)Solve the initial value problem

y

4y

3y 0

y

0

3 y

0

1

using Laplace transforms.

Q 4 a) Consider the cyclic group of order 4 consisting of the elements a a 2 a3 and a4 e. Identify the

normal subgroups of this group. How many classes does this group have?

Construct the character table of this group. A particular representation of this group has the characters

e 4

a

a2

a3 0

Show that this representation is reducible and reduce it into irreducible pieces.

1

7/23/2019 ADG-sugg.pdf

2/2

b) Consider a mechanical model of the water molecule in which the two O-H bonds are considered to be

two springs of spring constant kand the two H atoms are also assumed to interact via a weaker spring of spring

constant k

. Using group theory based methods calculate the normal frequencies of small oscillations of this

molecule.

Q 5 a)Solve the integral equation

y

x F

x

2

0cos

x

y

d

b)Obtain an approximate solution to the integral equation

y

x x2

1

0sin

x y

d

by replacing sin

x by the first two terms in its power series development.Q 6a) A pendulum vibrating in the vertical plane consists of a massmattached to a fixed point of suspension

by a spring. Set up the Lagrangian. Use the variational principle to derive the equations of motion for this

system.

b)The distance between two infinitesimally close points on the surface of an unit sphere is given by

dl2 d2 sin2d2

Find the differential equation obeyed by geodesics on the unit sphere. Hence show that the equator is a geodesic.

2