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Mechanics | Relativity
the speed of a ball is 2.0 m/s as it falls past a window of a tall building. How fast, in m/s, is it going as it passes another window 3.0 m below?
Mechanics | Relativity
a baseball is hit so that it travels straight upward after being struck by the bat. a fan observes that it takes 3.00 seconds to reach it's maximum height. find (A) the ball initial velocity and (B) the height it reaches.
Mechanics | Relativity
a cubical vessel of of height 1 m is full of water.the minimum work done in taking water out from the vessel will be?
Mechanics | Relativity
A golfer drives golf ball with a velocity of100 ft/sec and an angle of 45 degrees.
Find the time of travel.
Find the balls range.
Find the max height of the ball
Find the time of travel.
Find the balls range.
Find the max height of the ball
Mechanics | Relativity
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?
Mechanics | Relativity
Express the rotational kinetic energy of the earth in terms of its period of rotation. The moment
of inertia of the earth about its spin axis is 8.04 × 1037 kg m2. Calculate its rotational kinetic
energy
of inertia of the earth about its spin axis is 8.04 × 1037 kg m2. Calculate its rotational kinetic
energy
Mechanics | Relativity
(a) State against each observation below whether it is true or false. Give reasons for your answer.
i. The angular momentum of an artificial satellite rotating about the earth under its gravitation
varies with time
ii. An alpha particle scattered from an atomic nucleus moves in a plane.
iii. An artificial satellite moves at greater speed when it is nearer the earth. (6)
(b) When is the earth’s orbital motion around the sun fastest and when is it slowest? In which month
of the year is the earth closest to the sun? Explain in not more than 50 words. (4)
i. The angular momentum of an artificial satellite rotating about the earth under its gravitation
varies with time
ii. An alpha particle scattered from an atomic nucleus moves in a plane.
iii. An artificial satellite moves at greater speed when it is nearer the earth. (6)
(b) When is the earth’s orbital motion around the sun fastest and when is it slowest? In which month
of the year is the earth closest to the sun? Explain in not more than 50 words. (4)
Mechanics | Relativity
Obtain an expression for the time period of a satellite orbiting the earth. A space shuttle is in a
circular orbit at a height of 250 km from the earth’s surface, where the acceleration due to earth’s
gravity is 0.93 g. Calculate the period of its orbit. Take g = 9.8 ms−2 and the radius of the earth
R = 6.37 × 106m.
circular orbit at a height of 250 km from the earth’s surface, where the acceleration due to earth’s
gravity is 0.93 g. Calculate the period of its orbit. Take g = 9.8 ms−2 and the radius of the earth
R = 6.37 × 106m.
Mechanics | Relativity
A string of length 2l, obeying Hooke 's law, is stretched so that its extension is l. The speed of transverse waves in the string is v. If the string is further stretched so that the extension becomes 4l, what will be the speed of transverse waves in the string?
Mechanics | Relativity
Water stands at a depth H in a tank, Whose side walls are vertical. A hole is made on one of the walls at a depth h below the water surface. Find at what value of h this range is maximum?
