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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?
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