Waves Test

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Waves Test - 45

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Weekly Quiz Competition

Question 1
1 / -0

Two particles execute simple harmonic motions of same amplitude and frequency along the same straight line. They cross one another when going opposite directions. The phase difference between them when their displacements are one half of their amplitudes is then

A
$$60^0$$

B
$$30^0$$

C
$$120^0$$

D
$$150^0$$

Solution

The equation for SHM is, $$y = A sin(\omega + \phi)$$
As the displacement is half of the amplitudes $$(y = \dfrac{A}{2})$$, or $$\dfrac{A}{2} = A sin (\omega t + \phi) or sin (\omega t + \phi) = \dfrac{1}{2}$$
$$\therefore \omega t + \phi = 30^0 or 150^0$$.
Since the two particles are going in opposite directions, the phase of one is $$30^0$$ and that of the other $$150^{0}$$. Hence the phase difference between the two particles = $$150^0 - 30^0 = 120^0$$.
Question 2
1 / -0

Equation of plane progressive wave is given by $$y = 0.6 sin 2\pi\left(t - \dfrac{x}{2}\right)$$. On reflection from a denser medium its amplitude becomes 2/3 of the amplitude of the incident wave. The equation of the reflected wave is

A
$$y = 0.6 sin 2\pi\left(t + \dfrac{x}{2}\right)$$

B
$$y =- 0.4 sin 2\pi\left(t + \dfrac{x}{2}\right)$$

C
$$y = 0.4 sin 2\pi\left(t + \dfrac{x}{2}\right)$$

D
$$y =- 0.4 sin 2\pi\left(t - \dfrac{x}{2}\right)$$

Solution

Amplitude of reflected wave $$= \dfrac{2}{3} \times 0.6 = 0.4$$
On reflection from a denser medium there is phase change of $$\pi$$.
$$\therefore$$ Equation of reflected wave is
$$y = 0.4 sin 2\pi\left(t + \dfrac{x}{2} + \pi\right) = - 0.4 sin 2\pi\left(t + \dfrac{x}{2}\right)$$
Question 3
1 / -0

In the standing waves that form as a result of reflection of waves from an obstacle, the ratio of the amplitude at antinode to the amplitude at a node is s. The fraction of the energy that passes past the obstacle is:

A
$$\dfrac{4}{s^2}$$

B
$$\dfrac{s + 1}{s - 1}$$

C
$$(\dfrac{s + 1}{s - 1})^2$$

D
non of the above.

Solution

Let $$A_0$$ be amplitux of original wave
$$A_r$$ be amplitux of reflected wave
$$s/a$$
$$\dfrac{A_0+A_r}{A_0-A_r}=S$$
Applying comp. and dive.
$$\dfrac{A_0+A_r+A_0-A_r}{A_0+A_r-A_0+A_r}=\dfrac{s+1}{s-1}$$
$$\Rightarrow A_r=\dfrac{s-1}{s+1}A_0$$
Now
$$4\propto A^2$$
$$\therefore$$ Req ratio
$$\dfrac{A_0^2-A_r^2}{A_0^2}=1-\left(\dfrac{s-1}{s+1}\right)^2$$
$$=\dfrac{2s}{(s+1)^2}$$
Remark: Non of option is correct
Question 4
1 / -0

The equation of the stationary wave is
$$y = 2A sin(\dfrac{2 \pi c t}{\lambda}) cos (\dfrac{2 \pi x}{\lambda})$$
Which of the following statements is wrong?

A
The unit of ct is same as that of $$\lambda$$.

B
The unit of x. is same as that of $$\lambda$$

C
The unit of ct is same as that of $$x$$

D
The unit of ct, $$\lambda$$ and $$x$$ are same

Solution

$$y=2A\sin\left( \dfrac { 2\Pi t }{ \lambda } \right) \cos\left( \dfrac { 2\Pi x }{ \lambda } \right) $$
Arguments of $$sin$$ and $$cos$$ are unit less.
$$\Rightarrow \dfrac { Ct }{ \lambda } ,\dfrac { x }{ \lambda } $$ have dimension $${ M }^{ 0 }{ L }^{ 0 }{ T }^{ 0 }$$.
$$\Rightarrow$$ $$Ct,x$$ and $$\lambda $$ have same dimensions.
Question 5
1 / -0

Assertion - On reflection from a rigid boundary there takes place a complete reversal of phase.
Reason - On reflection from a denser medium, both the particle velocity and wave velocity are reversed in sign.

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

C
Assertion is correct but Reason is incorrect

D
Both Assertion and Reason are incorrect

Solution

The reflection from a rigid boundary is same as the reflection from a medium of high density or a denser medium. When the reflection takes place, there is a reversal in the sign of the wave velocity as well as the particle velocity i.e. the complete reversal of the phase.

Thus, option $$A$$ is correct.
Question 6
1 / -0

The maximum velocity of a body undergoing S.H.M. is $$0.2 m/s$$ and its acceleration at $$0.1 m$$ from the mean position is $$0.4 m/s^{-2}$$ The amplitude of the S.H.M. is:

A
$$0.25 m$$

B
$$0.3 m$$

C
$$0.1 m$$

D
$$1.05 m$$

Solution

$${V}_{max}= \omega A= 0.2 m/s$$
$$a--{ \omega }^{ 2 }x=0.4 m/{s}^{2}$$
$${ \omega }^{ 2 }=4$$ $$\Rightarrow $$ $$(\omega =2)$$
$$\omega A=0.2$$
$$2 \times A=0.2$$
$$(A=0.1m)$$
Question 7
1 / -0

At a particular position the velocity of a particle in SHM with amplitude a is $$\dfrac{{\sqrt 3 }}{2}$$ that at its mean position. In this position, its displacement is:

A
$$\dfrac{a}{2}$$

B
$$\sqrt 3 \dfrac{a}{2}$$

C
$$a\sqrt 2 $$

D
$$\sqrt {2a} $$

Solution

Let the equation of SHM is $$x=a sin(\omega t)$$,
$$\therefore v=a\omega cos(\omega t) $$ $$\Rightarrow $$ Velocity at mean position $$v= a\omega$$,
As given in question, $$\dfrac{\sqrt 3}{2}a\omega= a\omega cos (\omega t)$$ $$\Rightarrow cos(\omega t) = \dfrac{\sqrt 3}{2}$$ $$\Rightarrow \omega t= \dfrac{\pi}{6}$$
So Displacement $$x= a sin(\dfrac{\pi}{6})=\dfrac{a}{2}$$
Question 8
1 / -0

The amplitude of an electromagnetic wave in vacuum is doubled with no other changes made to the wave. As a result of this doubling of amplitude, which of the following statement is correct?

A
The speed of the wave propagation changes only

B
The frequency of the wave changes only

C
The wavelength of the waves changes only

D
None of these

Solution

Velocity of EM wave is given by,
$$v =\dfrac{1}{\sqrt{\epsilon_0 \mu_0}}$$

This speed of the wave is the same as the speed of light in vacuum.
$$v=3*10^8m/s$$

This velocity of the electromagnetic wave is independent of the amplitude of the wave and also independent of frequency.
Question 9
1 / -0

A particle moves according to $$x= a cos \Big(\frac{\pi}{2}t\Big)$$. The distance covered by it in time interval between t=0 to t=2 s is

A
2a

B
3a

C
4a

D
a

Solution

$$\dfrac{\pi}{2}=\frac{2\pi}{T} \Rightarrow T= 4 $$sec

$$S_{0-2}= 2a$$
Question 10
1 / -0

Given that $$y = A \sin \left[\left(\dfrac{2 \pi}{\lambda}\right)(ct - x)\right]$$, where $$y$$ and $$x$$ are measured in the unit of length. Which of the following statements is true?

A
The unit of $$\lambda$$ is same that of $$x$$ and $$A$$.

B
The unit of $$\lambda$$ is same that of $$x$$ but may not be same as that of $$A$$.

C
The unit of $$c$$ is same as that of $$2 \pi/ \lambda$$

D
The unit of $$(ct - x)$$is same as that of $$2 \pi/ \lambda$$

Solution

$$y=A\sin\left( \dfrac { 2\Pi }{ \lambda } \left( ct-x \right) \right) $$
$$\dfrac { ct-x }{ \lambda } $$ is written $$\Rightarrow$$ $$x$$ and $$\lambda$$ have same unit.
$$\sin\left( \theta \right) $$ is unitless $$\Rightarrow$$ $$y$$ and $$A$$ have same unit.

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Re-Attempt Weekly Quiz Competition

Waves Test - 45

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Waves Test - 45

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SHARING IS CARING

Question 1

$$60^0$$

$$30^0$$

$$120^0$$

$$150^0$$

Solution

Question 2

$$y = 0.6 sin 2\pi\left(t + \dfrac{x}{2}\right)$$

$$y =- 0.4 sin 2\pi\left(t + \dfrac{x}{2}\right)$$

$$y = 0.4 sin 2\pi\left(t + \dfrac{x}{2}\right)$$

$$y =- 0.4 sin 2\pi\left(t - \dfrac{x}{2}\right)$$

Solution

Question 3

$$\dfrac{4}{s^2}$$

$$\dfrac{s + 1}{s - 1}$$

$$(\dfrac{s + 1}{s - 1})^2$$

non of the above.

Solution

Question 4

The unit of ct is same as that of $$\lambda$$.

The unit of x. is same as that of $$\lambda$$

The unit of ct is same as that of $$x$$

The unit of ct, $$\lambda$$ and $$x$$ are same

Solution

Question 5

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Assertion is correct but Reason is incorrect

Both Assertion and Reason are incorrect

Solution

Question 6

$$0.25 m$$

$$0.3 m$$

$$0.1 m$$

$$1.05 m$$

Solution

Question 7

$$\dfrac{a}{2}$$

$$\sqrt 3 \dfrac{a}{2}$$

$$a\sqrt 2 $$

$$\sqrt {2a} $$

Solution

Question 8

The speed of the wave propagation changes only

The frequency of the wave changes only

The wavelength of the waves changes only

None of these

Solution

Question 9

2a

3a

4a

a

Solution

Question 10

The unit of $$\lambda$$ is same that of $$x$$ and $$A$$.

The unit of $$\lambda$$ is same that of $$x$$ but may not be same as that of $$A$$.

The unit of $$c$$ is same as that of $$2 \pi/ \lambda$$

The unit of $$(ct - x)$$is same as that of $$2 \pi/ \lambda$$

Solution

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