Oscillations Test

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Oscillations Test - 51

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Weekly Quiz Competition

Question 1
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Which of the following is simple harmonic motion ?

A
Particle moving in a circle with uniform speed

B
Wave moving through a string fixed at both ends

C
Earth spinning about its axis

D
Ball bouncing between two rigid vertical walls

Solution

Simple harmonic waves are set up in string sixed at the two ends. So, option $$B$$ is correct.
Question 2
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The function $$sin^{2}(\omega t)$$ represents :

A
a simple harmonic motion with a period $$2\pi /\omega $$.

B
a simple harmonic motion with a period $$\pi /\omega $$

C
a periodic, but not simple harmonic motion with period $$2\pi /\omega $$

D
a periodic, but not simple harmonic motion with period $$\pi /\omega $$

Solution
Question 3
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A highly rigid cubical block A of small mass M and side L is fixed rigidly onto another cubical block B of same dimensions and of low modulus of rigidly $$\eta$$ such that lower face of A completely covers the upper face of B. The lower face of B is rigidly held on a horizontal surface. A small force is applied perpendicular to one the side face of A. After the force is withdrawn, block A executes small oscillations, the time period of which is given by

A
$$2\pi\sqrt{M\eta L}$$

B
$$2\pi\sqrt{\dfrac{M}{\eta L}}$$

C
$$2\pi\sqrt{\dfrac{ML}{\eta }}$$

D
$$\sqrt{\dfrac{M\eta}{L}}$$

Solution

Modulus of rigidity, $$\eta=\dfrac{F}{A\theta}$$

Here, $$A=L^2$$ and $$\theta=\dfrac xL$$

Therefore, restoring force is,

$$F=-\eta A\theta=-\eta Lx$$

or

Acceleration, $$a=\dfrac FM=-\dfrac{\eta L}{M} x$$

Since, $$a\propto -x$$, oscillations are simple harmonic in nature, time period of which is given by,

$$T=2\pi \sqrt{|\dfrac{Displacement}{Acceleration}|}=2\pi\sqrt{|\dfrac xa|}$$

$$=2\pi\sqrt{\dfrac{M}{\eta L}}$$
Question 4
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A particle is executing shm withh amplitude a and has maximum velocity $$v_0$$ .its speed at displacement $$3a/4$$ will be?

A
$$\frac{{{v_0}\sqrt 7 }}{4}$$

B
$$\frac{{{v_0}\sqrt 4 }}{4}$$

C
$$\frac{{{v_0}\sqrt 7 }}{3}$$

D
$$\frac{{{v_0}\sqrt 9 }}{3}$$

Solution

$$\begin{array}{l} { V_{ \max } }={ V_{ 0 } }=A\omega \\ V=\omega \sqrt { { A^{ 2 } }-{ x^{ 2 } } } \\ =\omega \sqrt { { A^{ 2 } }-{ { \left( { \dfrac { { 3A } }{ 4 } } \right) }^{ 2 } } } \\ =\omega \sqrt { { A^{ 2 } }-\dfrac { { 9{ A^{ 2 } } } }{ { 16 } } } \\ =\omega \sqrt { \dfrac { { 16{ A^{ 2 } }-9{ A^{ 2 } } } }{ { 16 } } } \\ =\omega \sqrt { \dfrac { { 7{ A^{ 2 } } } }{ { 16 } } } \\ =\dfrac { { \omega A } }{ 4 } \sqrt { 7 } \\ =\dfrac { { { v_{ 0 } }\sqrt { 7 } } }{ 4 } \end{array}$$
Hence, Option $$A$$ is correct .
Question 5
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If the frequency of a wave is 20 Hz, its time period will be:

A
$$0.05\ s$$

B
$$0.5\ s$$

C
$$0.2\ s$$

D
$$2 s$$

Solution
Question 6
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A particle of mass $$0.1 kg$$ executes SHM under a force $$F = (-10 x) N$$. Speed of particle at mean position is $$6 m/s$$. Then amplitude of oscillations is

A
$$0.6 m$$

B
$$0.2 m$$

C
$$0.4 m$$

D
$$0.1 m$$

Solution

$$\begin{array}{l} M=0.1\, Kg \\ f=-10x \\ \Rightarrow k=10 \\ w=\sqrt { \frac { k }{ m } } \\ =\sqrt { \frac { { 10 } }{ { 0.1 } } } =10 \\ { V_{ \max } }=6m/s \\ \Rightarrow A=\frac { 6 }{ { 10 } } =0.6m \\ Hence,\, the\, option\, A\, is\, the\, correct\, answer. \end{array}$$
Question 7
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If the maximum velocity of a particle in SHM is $$v_0$$, then its velocity at half the amplitude from position of rest will be :

A
$$v_0/2$$

B
$$v_0$$

C
$${v_o}\sqrt 3 /2$$

D
$$v_0\sqrt{3}/4$$

Solution

Hence, the option $$C$$ is the correct answer.
Question 8
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A block of mass $$1$$ kg is connected to a spring of spring constant $$\pi^2 N/m$$ fixed at other end and kept on smooth level ground. The block is pulled by a distance of $$1$$ cm from natural length position and released. After what time does the block compress the spring by $$\frac{1}{2} cm$$.

A
$$\dfrac{2}{3}$$ sec

B
$$\dfrac{1}{3}$$ sec

C
$$\dfrac{1}{6}$$ sec

D
$$\dfrac{1}{12}$$ sec
Question 9
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wave length of wave is

A
$$0.4 m$$

B
$$0.2 m$$

C
$$0.16 m$$

D
$$0.8 m$$

Solution
Question 10
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A simple pendulum with length $$L$$ and mass $$m$$ of the bob is oscillating with an amplitude $$a$$.
Then the maximum tension in the string is :

A
$$mg$$

B
$$mg[1+\left(\dfrac{a}{L} \right)^{2}]$$

C
$$mg[1+\dfrac{a}{2L}]^{2}$$

D
$$mg \left[1+\left(\dfrac{a}{L}\right)\right]^{2}$$

Solution