# Answer to Question #26258 in Mechanics | Relativity for zameer

Question #26258

a) Write the equation of motion of a simple harmonic oscilla

tor which has an amplitude of

5 cm and it executes 150 oscillations in 5 minutes with an init

ial phase of 45

°

. Also

obtain the value of its maximum velocity.

(3+2)

b) If the displacement of a particle executing SHM be 10 cm

and 12 cm when the

corresponding velocities are 16 cms

−

1

and 14 cms

−

1

respectively, calculate the amplitude

of motion.

(5)

c) Derive expressions for potential energy and kinetic energy

of an oscillating spring-mass

system.

(5+5)

d) Discuss the principle of superposition. Two collinear SH

Ms, with amplitudes 5 cm and

12 cm are superposed. Calculate the resultant amplitude when

the SHMs differ in phase

by (i) 60

°

, (ii) 90

°

and (iii) 120

°

.

(4+6)

e) Establish the differential equation for a damped oscillat

or. Show that, for weak damping,

the solution of the differential equation for the damped oscillat

or is given by

)

(

cos

)

(

exp

)

(

0

φ

+

ω

−

=

t

bt

a

t

x

d

(4+6)

f) What do you understand by weakly damped forced oscillator and it

s transient and steady

states? Show that the average power absorbed by a forced oscill

ator is given by

]

4

)

[(

2

2

2

2

2

0

2

2

0

ω

+

ω

−

ω

ω

=

>

<

b

m

bF

P

(5+5)

g) What do you understand by the normal modes of coupled oscillators

? If a coupled

system has many normal modes, do all normal modes have the sa

me frequency?

Calculate the velocity of elastic longitudinal wave along a

stretched steel wire, given

density of steel = 8000 kgm

−

3

, Young’s modulus of elasticity = 2

×

10

11

Nm

−

2

. (3+2+5)

2. a) Two points

x

1

and

x

2

at

x

= 0 and

x

= 1 m are observed. The transverse motion of the two

points are found to be as follows:

t

t

x

y

π

=

3

sin

2.0

)

,

(

1

and

π

+

π

=

8

3

sin

2.0

)

,

(

2

t

t

x

y

Calculate the frequency, wavelength and speed of the wave. (5)

4

b) A sinusoidal wave is described by

cm

)

95.5

20.4(

sin

0.4

)

,

(

t

x

t

x

y

−

=

where

x

is the position along the wave propagation. Determine th

e amplitude, wave

number, wavelength, frequency and velocity of the wave. (

2

×

5 = 10)

c) Two waves, travelling along the same direction, are given

by

)

(

sin

)

,

(

1

1

1

x

k

t

a

t

x

y

−

ω

=

and

)

(

sin

)

,

(

2

2

2

x

k

t

a

t

x

y

−

ω

=

Suppose that

ω

1

and

k

1

are respectively slightly greater than

ω

2

and

k

2

. (i) Derive an

expression for the resultant wave obtained by their super

position. (ii) Explain the

formation of wave packet.

(5+5)

d) Standing waves are produced by the superposition of

two waves given by

m

)

2

(

sin

2.0

)

,

(

1

x

t

t

x

y

−

π

=

and

m

)

2

(

sin

2.0

)

,

(

2

x

t

t

x

y

+

π

=

(i) Obtain the resultant displacement of the particle at

x

at time

t

. (ii) At what value of

x

is

displacement zero at all times. (iii) What is the distance

between two nearest values of

x

at which the displacement is zero. Is it related to wavelength

? (3+3+4)

e) A 125 cm long string has a mass 2.0 g and it is str

etched with a tension of 7.0 N between

fixed supports. What is the speed of the transverse wave

on the string?

tor which has an amplitude of

5 cm and it executes 150 oscillations in 5 minutes with an init

ial phase of 45

°

. Also

obtain the value of its maximum velocity.

(3+2)

b) If the displacement of a particle executing SHM be 10 cm

and 12 cm when the

corresponding velocities are 16 cms

−

1

and 14 cms

−

1

respectively, calculate the amplitude

of motion.

(5)

c) Derive expressions for potential energy and kinetic energy

of an oscillating spring-mass

system.

(5+5)

d) Discuss the principle of superposition. Two collinear SH

Ms, with amplitudes 5 cm and

12 cm are superposed. Calculate the resultant amplitude when

the SHMs differ in phase

by (i) 60

°

, (ii) 90

°

and (iii) 120

°

.

(4+6)

e) Establish the differential equation for a damped oscillat

or. Show that, for weak damping,

the solution of the differential equation for the damped oscillat

or is given by

)

(

cos

)

(

exp

)

(

0

φ

+

ω

−

=

t

bt

a

t

x

d

(4+6)

f) What do you understand by weakly damped forced oscillator and it

s transient and steady

states? Show that the average power absorbed by a forced oscill

ator is given by

]

4

)

[(

2

2

2

2

2

0

2

2

0

ω

+

ω

−

ω

ω

=

>

<

b

m

bF

P

(5+5)

g) What do you understand by the normal modes of coupled oscillators

? If a coupled

system has many normal modes, do all normal modes have the sa

me frequency?

Calculate the velocity of elastic longitudinal wave along a

stretched steel wire, given

density of steel = 8000 kgm

−

3

, Young’s modulus of elasticity = 2

×

10

11

Nm

−

2

. (3+2+5)

2. a) Two points

x

1

and

x

2

at

x

= 0 and

x

= 1 m are observed. The transverse motion of the two

points are found to be as follows:

t

t

x

y

π

=

3

sin

2.0

)

,

(

1

and

π

+

π

=

8

3

sin

2.0

)

,

(

2

t

t

x

y

Calculate the frequency, wavelength and speed of the wave. (5)

4

b) A sinusoidal wave is described by

cm

)

95.5

20.4(

sin

0.4

)

,

(

t

x

t

x

y

−

=

where

x

is the position along the wave propagation. Determine th

e amplitude, wave

number, wavelength, frequency and velocity of the wave. (

2

×

5 = 10)

c) Two waves, travelling along the same direction, are given

by

)

(

sin

)

,

(

1

1

1

x

k

t

a

t

x

y

−

ω

=

and

)

(

sin

)

,

(

2

2

2

x

k

t

a

t

x

y

−

ω

=

Suppose that

ω

1

and

k

1

are respectively slightly greater than

ω

2

and

k

2

. (i) Derive an

expression for the resultant wave obtained by their super

position. (ii) Explain the

formation of wave packet.

(5+5)

d) Standing waves are produced by the superposition of

two waves given by

m

)

2

(

sin

2.0

)

,

(

1

x

t

t

x

y

−

π

=

and

m

)

2

(

sin

2.0

)

,

(

2

x

t

t

x

y

+

π

=

(i) Obtain the resultant displacement of the particle at

x

at time

t

. (ii) At what value of

x

is

displacement zero at all times. (iii) What is the distance

between two nearest values of

x

at which the displacement is zero. Is it related to wavelength

? (3+3+4)

e) A 125 cm long string has a mass 2.0 g and it is str

etched with a tension of 7.0 N between

fixed supports. What is the speed of the transverse wave

on the string?

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