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Which of the following examples represent periodic motion?
(a) A swimmer completing one (return) trip from one bank of
a river to other bank. (b) A freely suspended bar magnet displaced from its N-S
direction and released, (c) A hydrogen molecule rotating about its centre of
mass. (d) An arrow released from a bow.
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Which of the following examples represent (nearly) simple
harmonic motion and which represent periodic but' not simple harmonic motion?
(a) The rotation of earth about its axis. (b) Motion of an
oscillating mercury column in a U tube; (c) Motion of a ball bearing inside a
smooth curved bowl, when released from a point slightly above the lower most
position, (d) General vibrations of a polyatomic molecule about its equilibrium
position.
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Given below Fig.10 (NCT). l depicts four x-t plots for
linear motion of a particle. Which of the plots represent periodic motion? What
is the period of motion (in case of periodic motion)?
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Which of the following functions of time represent (a)
simple harmonic, (b) periodic but not simple' harmonic, and (c) non-periodic
motion ? Give period for each case of periodic motion; \[\log \left[
\frac{\left( x+\upsilon t \right)}{{{x}_{0}}} \right]\]co is any positive
constant).
(a) \[\log \left[ \frac{\left( x+\upsilon t
\right)}{{{x}_{0}}} \right]\] (b)
\[\text{m}{{\text{s}}^{\text{-1}}}\]
(c) \[\text{m}{{\text{s}}^{\text{-1}}}\] (d)
\[{{10}^{5}}\]
(e) \[{{\text{ }\!\!\upsilon\!\!\text{
}}_{\text{a}}}\text{=340m}{{\text{s}}^{\text{-1}}}\text{,}\] (f)
\[{{\upsilon }_{w}}=1486m{{s}^{-1}}\]
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A particle is in linear S.H.M. between two points A and B,
10 cm apart. Take the direction from A to B as the positive direction and give
the signs of velocity, acceleration and force on the particle when it is.
(a) at the end A, (b) at the end B,
(c) at the mid point of AB going towards A,
(d) at 2 cm away from 5 going towards A,
(e) at 3 cm away from A going towards B, a
(f) at 4 cm away from A going towards A.
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Which of the following relationships between the
acceleration, a and the displacement x of a particle involve simple harmonic
motion.
(a)\[\text{3cos}\left( \frac{\text{ }\!\!\pi\!\!\text{
}}{\text{4}}\text{-2 }\!\!\omega\!\!\text{ t} \right)\text{=3cos}\left( \text{2
}\!\!\omega\!\!\text{ t-}\frac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}} \right)\] (b)\[\left[
\because \cos \left( -\theta \right)=\cos \theta \right]\] (c)
\[a=-10x\] (d) \[\text{2 }\!\!\pi\!\!\text{ /2
}\!\!\omega\!\!\text{ }\text{.}\]
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The motion of a particle in S.H.M. is described by the
displacement function, \[x=A\cos (\omega t+\phi )\]. If the initial \[(t=0)\] position
of the particle is 1 cm and its initial velocity is \[\text{2
}\!\!\pi\!\!\text{ / }\!\!\omega\!\!\text{ }\] what are its amplitude and
initial phase angle? The angular frequency of the particle is \[{{\text{e}}^{\text{-}{{\text{
}\!\!\omega\!\!\text{ }}^{\text{2}}}{{\text{t}}^{\text{2}}}}}\]. If instead of
the cosine function, we choose the sine function to describe the SHM : \[x=B\,\sin
(\omega t+\alpha )\], what are the amplitude and initial phase of the particle
with the above initial conditions?
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A spring balance has a scale that reads from 0 to 50 kg. The
length of the scale is 20 cm. A body a suspended from this spring, when
displaced and released, oscillates with period of 0.60 s. What is the weight of
the body?
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A spring of force constant \[1200\,N{{m}^{-1}}\] is mounted
on a horizontal table as shown in Fig. 10 (NCT).3. A mass of 3.0 kg is attached
to the free end of the spring, pulled side ways to a distance of 2.0 cm and released.
Determine
(a)
the frequency of osculation of the mass.
(b)
the maximum acceleration of the mass.
(c) the maximum speed of the mass.
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In Q.9 above, let us take position of mass when the spring
is unstretched as \[x=0\], and the direction from left to right as the positive
direction of x-axis. Give x as a function of time t for the osculating mass if
at the moment we start the stop watch \[(t=0)\], the mass is (a) at the mean
position (b) at the maximum stretched position, and (c) at the maximum
compressed position.
In what way do these functions for S.H.M. differ from each
other, in frequency, in amplitude or the initial phase?
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Figures 10 (NCT). 4(a) and (b) correspond to two circular
motions. The radius of the circle, the period of revolution, the initial,
position, and the sense of revolution (i.e. clockwise or anticlockwise) are indicated
on each. Obtain the corresponding equations of simple harmonic motions of the
revolving particle P in each case.
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Plot the corresponding reference circle for each of the
following simple harmonic motions. Indicate the initial \[(t=0)\] position of
the particle, the radius of the circle, and the angular speed of the rotating particle.
For simplicity, the sense of rotation may be fixed to be anticlockwise m every
case; (x is in cm and t is in s).
(a)
\[x=-2\sin \] \[+\pi /2\]
(b)
\[x=\cos \,(\pi /6-t)\]
(c)
\[x=3\sin \,(2\pi +\pi /4)\]
(d)
\[x=2\cos \pi t\]
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Figure 10 (NCT). 6 (a) shows a spring of force constant k
clamped rigidly at one end and a mass m attached to its free end. A force F
applied at the free end stretches the spring. Figure 10 (NCT). 6 (b) shows the
same spring with both ends free and attached to a mass m at either end. Each
end of the spring in Fig. 10 (NCT). 6(b) is stretched by the same force F.
(a)
What is the maximum extension of the spring in the two cases?
(b)
If the mass in Fig. 10 (NCT). 6 (a) and the two masses in Fig. 10 (NCT). 6 (b)
are released free, what is the period of oscillation in each case?
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The piston in the cylinder head of a locomotive has a stroke
(twice the amplitude) of 1.0 m. If the piston moves with simple harmonic
motion with an angular frequency of 200 rev/min., what is its maximum speed?
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The acceleration due to gravity on the surface of the moon
is \[\text{T=2 }\!\!\pi\!\!\text{ }\sqrt{\frac{\text{m}}{\text{k}}}\]. What is
the time period of a simple pendulum on the surface of moon if its time period
on the surface of earth is 3.5 s? (g on the surface of earth \[\therefore \])
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Answer the following questions:
(a)
Time period of a particle in S.H.M. depends on the force constant k and mass m
of the particle \[{{g}_{e}}=9\cdot 8m{{s}^{-2}};\] A simple pendulum executes
S.H.M. approximately. Why then is the time-period of a pendulum independent of
the mass of the pendulum?
(b)
The motion of simple pendulum is approximately simple harmonic for small angles
of osculation. For large angle of oscillation, a more involved analysis (beyond
the scope of this book) shows that T is greater than \[{{T}_{m}}=?;{{T}_{e}}=3\cdot
5{{s}^{-1}}\] Think of a qualitative argument to appreciate this result.
(c)
A man with a wrist watch on his hand falls from the top of a tower. Does the
watch give correct time during the free fall?
(d) What is the frequency of oscillation of a simple
pendulum mounted in a cabin that is freely falling under gravity?
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A simple pendulum of length/and having a bob of mass M is
suspended in a car. The car is moving on a circular track of radius R with a
uniform speed \[\theta \approx \theta \]. IS the pendulum makes small
oscillations in a radial direction about its equilibrium position, what will be
its time period?
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A cylindrical piece of cork of base area A and height h
floats in a liquid of density \[\theta \]. The cork is depressed slightly and
then released. Show that the cork oscillates up and down simple harmonically
with a period \[\theta \] where \[\upsilon \] is the density of cork. (Ignore
damping due to viscosity of the liquid).
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One end of a U-tube containing mercury is connected to a
suction pump and the other end to atmosphere. A small pressure difference is
maintained between the two columns. Show that, when the suction pump is
removed, the column of mercury in the U-tube executes simple harmonic motion.
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An air chamber of volume V has a neck area of cross section
A into which a ball of mass m just fits and can move up and down without any
friction, Fig. 10 (NCT). 7. Show that when the ball is pressed down a little
and released, it executes SHM. Obtain an expression for the time period of
oscillations assuming pressure volume variations of air to be isothermal.
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You are riding an automobile of mass 3000 kg. Assuming that
you are examining the oscillation characteristics of its suspension system. The
suspension sags 15 cm when the entire automobile is placed on it. Also, the
amplitude of oscillation decreases by 50% during one complete oscillation. Estimate
the values of (a) the spring constant k and (b) the damping constant b for the
spring and shock absorber system of one wheel, assuming that each wheel
supports 750 kg. g = 10 \[\text{k=E}{{\text{A}}^{\text{2}}}\text{/V}\].
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Show that for a particle in linear S.H.M., the average
kinetic energy over a period of oscillation equals the average potential energy
over the same period.
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A circular disc of mass 10 kg is suspended by a wire
attached to its centre. The wire is twisted by rotating the disc and released.
The period of torsional oscillations is found to be 1.5s. The radius of the
disc is 15 cm. Determine the torsional spring constant of the wire. (Torsional
spring constant \[\text{(T)}\frac{\text{1}}{\text{4}}\text{m}{{\text{a}}^{\text{2}}}{{\text{
}\!\!\omega\!\!\text{ }}^{\text{2}}}\] is defined by the relation \[J=-\alpha
\theta \], where J is the restoring couple and 9 the angle of twist).
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A body describes simple harmonic motion with an amplitude of
5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when
the displacement is (a) 5 cm, (b) 3 cm, (c) 0 cm.
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A mass attached to a spring is free to oscillate, with
angular velocity \[\text{V= }\!\!\omega\!\!\text{
}\sqrt{{{\text{r}}^{\text{2}}}\text{-}{{\text{y}}^{\text{2}}}}\], in a
horizontal plane without friction or damping. It is pulled to a distance \[0\cdot
05\] and pushed towards the centre with a velocity \[\begin{align}
& A=-{{\left( 10\pi \right)}^{2}}\times 0\cdot
05=-5{{\pi }^{2}}m/{{s}^{2}} \\
& V=10\pi \times \sqrt{{{\left( 0\cdot 05
\right)}^{2}}-{{\left( 0\cdot 05 \right)}^{2}}}=0 \\
\end{align}\] at time \[t=0\]. Determine the amplitude of
the resulting oscillations in terms of the parameters \[0\cdot 03\] and \[A=-{{\left(
10\pi \right)}^{2}}\times 0\cdot 03=-3{{\pi }^{2}}m/{{s}^{2}}\]
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question_answer26)
The displacement of
a particle is represented by the equation \[y=3\,\cos \left( \frac{\pi
}{4}\,-\,2\omega t \right)\]
The
motion of the particle is
(a)
simple harmonic with period \[2\pi /\omega \]
(b)
simple harmonic with period \[\pi /\omega \]
(c)
periodic but not simple harmonic.
(d)
non-periodic.
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question_answer27)
The
displacement of a particle is represented by the equation \[y={{\sin
}^{3}}\,\omega t\]. The motion is
(a)
non-periodic.
(b)
periodic but not simple harmonic.
(c)
simple harmonic with period \[2\pi /\omega \]
(d)
simple harmonic with period \[\pi /\omega \]
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question_answer28)
The
relation between acceleration and displacement of four particles are given
below:
(a)
\[{{a}_{x}}=+2x\] (b) \[{{a}_{x}}=+2{{x}^{2}}\]
(c)
\[{{a}_{x}}=-2{{x}^{2}}\] (d) \[{{a}_{x}}=-2x\]
Which
one of the particles is executing simple harmonic motion?
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question_answer29)
Motion
of an oscillating liquid column in a U-tube is
(a)
periodic but not simple harmonic.
(b)
non-periodic.
(c)
simple harmonic and time period independent of die density of die liquid
(d)
simple harmonic and time period directly proportional to the density c the
liquid.
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question_answer30)
A
particle is acted simultaneously by mutually perpendicular simple hormonic
motion \[x=a\,\cos \omega t\] and \[y=a\sin \,\omega t\]. The trajectory, of motion
of die particle will be
(a)
an ellipse (b) a parabola
(c)
a circle (d) a straight line
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question_answer31)
The displacement of
a particle varies with time according to the relation
\[y=a\sin
\,\omega t+b\cos \,\omega t\]
(a) The motion is
oscillatory but not S.H.M.
(b) The motion is
S.H.M. with amplitude\[a+b\].
(c) The motion is
S.H.M. with amplitude \[{{a}^{2}}+{{b}^{2}}\].
(d) The motion is
S.H.M. with amplitude \[\sqrt{{{a}^{2}}+\,{{b}^{2}}}\].
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question_answer32)
Four
pendulums A, B, C and D are suspended from the same
elastic
support as shown in fig. A and C are of the same length, while B is smaller
than A and D is larger than A. If A is given a transverse displacement.
(a) D will vibrate
with maximum amplitude.
(b) C will vibrate
with maximum amplitude.
(c) B will vibrate
with maximum amplitude.
(d) All the four
will oscillate with equal amplitude.
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question_answer33)
Figure
shows the circular motion of a particle. The radius of the circle, the period,
sense of revolution and the initial position are indicated on the figure. The
simple harmonic motion of the x-projection of the radius vector of the rotating
particle P is
(a) \[x\,(t)\,\,B\,\sin
\,\left( \frac{2\pi t}{30} \right)\]
(b) \[x(t)=B\cos
\left( \frac{\pi t}{15} \right)\]
(c) \[x(t)=B\sin
\left( \frac{\pi t}{15}+\frac{\pi }{2} \right)\]
(d) \[x(t)=B\cos
\left( \frac{\pi t}{15}+\,\frac{\pi }{2} \right)\]
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question_answer34)
The
equation of motion of particle is \[x=a\cos \,(\alpha \,{{t}^{2}})\].
The
motion is
(a) periodic but
not oscillatory.
(b) periodic and
oscillatory.
(c) oscillatory but
not periodic
(d) neither
periodic nor oscillatory.
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question_answer35)
A
particle executing S.H.M. has a maximum speed of 30 cm/s and a maximum
acceleration of \[60\text{ }cm/{{s}^{2}}\]. The period of oscillation is
(a)
\[\pi s\] (b) \[\frac{\pi }{2}s\]
(c)
\[2\pi \,s\] (d)
\[\frac{\pi }{t}\,s\]
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question_answer36)
When
a mass m is connected individually to two springs \[{{S}_{1}}\] and \[{{S}_{2}}\],
the oscillation frequencies are \[{{v}_{1}}\] and \[{{v}_{2}}\]. If the same mass
is displacement 0 1 2 3 4 5 6 7 time (s) attached to the two springs as shown
in Fig., the oscillation frequency would be
(a) \[{{v}_{1}}+{{v}_{2}}\] (b) \[\sqrt{v_{1}^{2}+v_{2}^{2}}\]
(c) \[{{\left(
\frac{1}{{{v}_{1}}}+\,\frac{1}{{{v}_{2}}} \right)}^{-1}}\] (d) \[\sqrt{v_{1}^{2}-v_{2}^{2}}\]
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question_answer37)
The
rotation of earth about its axis is
(a)
periodic motion
(b)
simple harmonic motion
(c)
periodic but not simple harmonic motions.
(d)
non-periodic motion.
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question_answer38)
Motion
of a ball bearing inside a smooth curved bowl, when released from a point
slightly above the lower point is
(a)
simple harmonic motion
(b)
non-periodic motion.
(c)
periodic motion
(d)
periodic but not S.H.M.
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question_answer39)
Displacement vs.
time curve for a particle executing S.H.M. is shown in fig. Choose the correct
statements.
(a) Phase of the
oscillator is same at \[t=0\,s\] and \[t=2\,s\].
(b) Phase of the
oscillator is same at \[t=2\,s\] and \[t=6\,s\].
(c) Phase of the
oscillator is same at \[t=1\,s\] and \[t=7\,s\].
(d) Phase of the
oscillator is same at \[t=1\,s\] and \[t=5\,s\].
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question_answer40)
Which
of the following statements is/are true for a simple harmonic oscillator?
(a)
Force acting is directly proportional to displacement from the mean position
and opposite to it.
(b)
Motion is periodic.
(c)
Acceleration of the oscillator is constant.
(d)
The velocity is periodic.
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question_answer41)
The
displacement time graph of a particle executing S.H.M. is shown in fig. Which
of the following statement is/are ture?
(a)The
force is zero at \[t=\frac{3T}{4}\]
(b)
The acceleration is maximum at \[t=\frac{4T}{4}\]
(c)
The velocity is maximum at \[t=\,\frac{T}{4}\]
(d)
The P.E. is equal to K.E. of oscillation at \[t=\,\frac{T}{2}\]
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question_answer42)
A
body is performing S.H.M. Then its
(a)
average total energy per cycle is equal to its maximum kinetic energy.
(b)
average kinetic energy per cycle is equal to half of its maximum kinetic
energy.
(c)
mean velocity over a complete cycle is equal to \[\frac{2}{\pi }\] times of its
maximum velocity.
(d)
root mean square velocity is \[\frac{1}{\sqrt{2}}\] times of its
maximum velocities.
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question_answer43)
A
particle is in linear simple harmonic motion between two points A and B, 10 cm
apart (Fig. Take the direction from A to B as the + ve direction and choose the
correct statements.
(a) The sign of
velocity, acceleration and force on the particle when it is 3 cm away from A
going towards B are positive.
(b) The sign of
velocity of the particle at C going towards O is negative.
(c) The sign of
velocity, acceleration and force on the particle when it is 4 cm away from B
going towards A are negative.
(d) The sign of
acceleration and force on the particle when it is at point B is negative.
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question_answer44)
Displacement
versus time curve for a particle executing S.H.M. is shown in Fig. Identify the
points marked at which (i) velocity of the oscillator is zero,
(ii)
speed of the oscillator is maximum.
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question_answer45)
Two
identical springs of spring constant K are attached to a block of mass m and to
fixed supports as shown in Fig. When the mass is displaced from equilibrium
position by a distance \[x\] towards right, find the restoring force
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question_answer46)
What
are die two basic characteristics of a simple harmonic motion?
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question_answer47)
When
will die motion of a simple pendulum be simple harmonic?
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question_answer48)
What
is the ratio of maximum acceleration to the maximum velocity of a simple
harmonic oscillator?
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question_answer49)
What is the ratio
between the distance travelled by the oscillator in one time period and
amplitude?
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question_answer50)
In
Fig. what will be the sign of die velocity of the point P', which is the
projection of the velocity of the reference particle P. P is moving in a circle
of radius R in anticlockwise direction.
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question_answer51)
Show
that for a particle executing S.H.M. velocity and displacement have a phase
difference of \[\pi /2\].
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question_answer52)
Draw
a graph to snow the variation or P.E., K.E. and total energy of a simple
harmonic oscillator with displacement.
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question_answer53)
The
length of a second's pendulum on the surface of Earth is 1m. What will be the
length of a second's pendulum on the moon?
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question_answer54)
Find
the time period of mass M when displaced from its equilibrium position and then
released for the system shown in Fig.
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question_answer55)
Show
that the motion of a particle represented by \[y=\sin \,\omega t-\cos \,\omega
t\]is simple harmonic with a period of \[2\pi \,l\,\omega \].
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question_answer56)
Find
the displacement of a simple harmonic oscillator at which its P.E. is half of
the maximum energy of the oscillator.
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question_answer57)
A body of mass m is
situated in a potential field \[U(x)={{U}_{0}}(1-\cos \,\alpha x)\] when \[{{U}_{0}}\]
and a are
constants. Find the time period of small oscillations.
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question_answer58)
A
mass of 2 kg is attached to the spring of spring constant \[50\,\,N{{m}^{-1}}\].
The block is pulled to a distance of 5 cm from its equilibrium position at \[x=0\]
on a
horizontal frictionless surface from rest at \[t=0\]. Write the expression for
its displacement at anytime \[t\].
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question_answer59)
Consider
a pair of identical pendulums, which oscillate with equal amplitude
independently such that when one pendulum is at its extreme position making an
angle of \[{{2}^{o}}\] to
die right with die vertical, the other pendulum makes an angle of \[{{1}^{o}}\]
to the left
of the vertical. What is the phase difference between the pendulums?
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question_answer60)
A
person normally weighing 50 kg stands on a massless platform which oscillates
up and down harmonically at a frequency of \[2.0\,{{s}^{-1}}\] and an amplitude
5.0 cm. A weighing machine on the platform gives the persons weight against
time.
(a)
Will there be any change in weight of the body, during the oscillation?
(b)
If answer to part (a) is yes, what will be the maximum and minimum reading in the
machine and at which position?
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question_answer61)
A
body of mass m is attached to one end of a massless spring which is suspended
vertically from a fixed point. The mass is held in hand so that the spring is
neither stretched nor compressed. Suddenly the support of the hand is removed.
The lowest position attained by the mass during oscillation is 4 cm below the
point, where it was held in hand.
(a)
What is die amplitude of oscillation?
(b)
Find the frequency of oscillation?
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question_answer62)
A
cylindrical log of wood of height h and area of cross-section A floats
in water. It is pressed and then released. Show that the log would execute
S.H.M. with a time period.
\[T=\,2\pi
\,\sqrt{\frac{m}{A\rho g}}\]
Where
m is mass of the body and p is density of the liquid.
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question_answer63)
One
end of a V-tube containing mercury is connected to a suction pump and the other
end to atmosphere. The two arms of the tube are inclined to horizontal at an
angle of 45° each. A small pressure difference is created between two columns
when the suction pump is removed. Will die column of mercury in V-tube execute
simple harmonic motion? Neglect capillary and viscous forces.
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question_answer64)
A
tunnel is dug through the centre of the Earth. Show that a body of mass 'm'
when dropped from rest from one end of the tunnel will execute simple harmonic
motion.
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question_answer65)
A
simple pendulum of time period 1s and length \[l\] is hung from a
fixed support at O, such that the bob is at a distance H vertically above A on
the ground Fig. The amplitude is \[{{\theta }_{0}}\]. The string snaps at \[\theta
={{\theta }_{0}}/2\]. Find the time taken by the bob to bit die ground. Also find
distance from A where bob hits the ground. Assume \[{{\theta }_{0}}\] to be small so that
\[\sin \,{{\theta }_{0}}\,;\,{{\theta }_{0}}\] and \[\cos \,{{\theta
}_{0}}\,;\,1\].
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