Oscillations Test

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Oscillations Test - 71

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Question 1
1 / -0

A block of mass $$M$$ is performing $$SHM$$ with amplitude $$A$$ on a smooth horizontal surface$$.$$ At the extreme position a small block of mass $$m$$ falls vertically and sticks to$$M.$$ then$$,$$ amplitude of oscillation will be

A
$$A$$

B
$$A\sqrt {\dfrac{M}{{M + m}}} $$

C
$$A\left( {\dfrac{M}{{M + m}}} \right)$$

D
$$A\left( {\dfrac{{M + m}}{m}} \right)$$

Solution
Question 2
1 / -0

A particle of mass $$m$$ is allowed to oscillate on a smooth parabola: $${ x }^{ 2 }=4ay,a>1$$ as shown in the figure. The angular frequency $$\left( \omega \right) $$ of small oscillations is

A
$$\omega =\sqrt { \dfrac { g }{ 4a } } $$

B
$$\omega =\sqrt { \dfrac { g }{ 2a } } $$

C
$$\omega =\sqrt { \dfrac { 2g }{ a } } $$

D
$$\omega =\sqrt { \dfrac { g }{ a } } $$

Solution

Ref.Image.
$$ x^{2} = 4ay$$
$$ 2x = 4a \dfrac{dy}{dx}$$
$$tan\theta = \dfrac{dy}{dx} = \dfrac{2x}{4a}$$
fresing = -mg $$sin\theta $$
$$ tan\theta -sin\theta \simeq \theta .tan\theta $$ is very small
$$ tan\theta = \dfrac{x}{20}$$
$$ F=-mg \dfrac{X}{20}$$
$$ F = -k/l$$
$$ K= \dfrac{mg}{2a} K = mw^{2}$$
$$ \dfrac{mg}{2a} = mw^{2}$$
$$ w2 \sqrt{\dfrac{y}{2a}}$$
Question 3
1 / -0

As shown in figure a horizontal platform with a mass $$m$$ placed on it is executing SHM along y-axis. If the amplitude of oscillation is $$2.5cm$$, the minimum period of the motion for the mass not to be detached from the platform is
($$g=10m/{sec}^{2}={\pi}^{2}$$)

A
$$\cfrac{10}{\pi}s$$

B
$$\cfrac{\pi}{10}s$$

C
$$\cfrac{\pi}{\sqrt{10}}s$$

D
$$\cfrac{1}{\sqrt{10}}s$$

Solution
Question 4
1 / -0

A horizontal plank has a rectangular block placed on it. The plank starts oscillating vertically and simple harmonically with an amplitude of $$40\ cm$$. The block just loses contact with the plank when the latter is at momentary rest. Then

A
the period of oscillation is $$\left( \dfrac { 2\pi }{ 5 } \right) $$

B
the block weight double its weight, when the plank is at one of the position of momentary rest

C
the block weighs 0.5 times its weight on the plank halfway up

D
the block weighs 1.5 times its weight on the plank halfway down.

Solution
Question 5
1 / -0

$$ABC$$ is an equilateral triangle structure up of a light rigid material. Find the frequency of small vertical oscillations of mass $$m$$ along $$AG$$. Conisider $${k}_{1}={k}_{2}={k}_{3}={k}_{4}=k$$

A
$$\sqrt{\cfrac{5k}{2m}}$$

B
$$\sqrt{\cfrac{4k}{m}}$$

C
$$\sqrt{\cfrac{3k}{5m}}$$

D
$$\sqrt{\cfrac{10k}{9m}}$$
Question 6
1 / -0

The length of simple pendulum is about 100 cm known to have an accuracy of 1 mm. Its period of oscillation is 2 s determined by measuring the time for 100 oscillations using a block of 0.1 s resolution. What is the accuracy in the determined value g?

A
0.2%

B
0.5%

C
0.1%

D
2%

Solution
Question 7
1 / -0

A spring is placed in vertical position by suspending it from a hook at its top. A similar hook on the bottom of the spring is at $$11\ cm$$ above a table top. A mass of $$75\ g$$ and of negligible size is then suspended from the bottom hook, which is measured to be $$4.5\ cm$$ above the table top. The mass is then pulled down a distance of $$4\ cm$$ and released. Find the approximate position of the bottom hook after $$s$$?
Take $$g=10m/{s}^{2}$$ and hooks mass to be negligible.

A
$$5cm$$ above the table top

B
$$4.5cm$$ above the table top

C
$$9cm$$ above the table top

D
$$0.5cm$$ above the table top

Solution
Question 8
1 / -0

A particle executing S.H.M. given by equation $$ y= 8 sin 6 \pi t $$ is sending out waves in a continuous medium traveling at 200 cm/s. the resultant displacement of the particle 150 cm, from B and one second after commencement of vibration of B is:

A
4 cm

B
8 cm

C
-8 cm

D
-3 cm
Question 9
1 / -0

A particle executes simple harmonic motion with an amplitude of 5 cm. When the particle is at 4 cm from the mean position, the magnitude of its velocity in SI units is equal to that of its acceleration. Then, its periodic time in seconds is:

A
$$\frac{8\pi }{3}$$

B
$$\frac{3}{8}\pi $$

C
$$\frac{4\pi }{3}$$

D
$$\frac{7}{3}\pi $$

Solution
Question 10
1 / -0

A simple pendulum of length 1 m is oscillating with an angular frequency 10 rad/s. The support of the pendulum starts oscillating up and down with a small angular frequency off 1 rad/s and an amplitude of $${ 10 }^{ -2 }.$$ The relative change in the angular frequency of the pendulum is best given by:

A
$${ 10 }^{ -5 }rad/s$$

B
$${ 10 }^{ -1 }rad/s$$

C
$${ 10 }^{ }rad/s$$

D
$${ 10 }^{ -3 }rad/s$$