Waves Test

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Waves Test - 71

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Weekly Quiz Competition

Question 1
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A wave of frequency 500 Hz has a phase velocity of 360 m/s. The phase difference between the two displacements at a certain point in a time interval of 10$$^{-3}$$ seconds will be how much?

A
$$\displaystyle \frac{\pi}{2}$$ radian

B
$$\pi$$ radian

C
$$\displaystyle \frac{\pi}{4}$$ radian

D
$$\displaystyle \frac{\pi}{8}$$ radian

Solution

According to the concept of simple harmonic wave,

$$y = a sin \dfrac{2 \pi }{ \lambda } (vt - \phi )$$

Now, phase angle at point "$$x$$" for time"$$t_1$$"

$$ t_{1} =\dfrac{2 \pi }{ \lambda } (vt -x)$$

now for time "$$t_2$$"

$$ t_{2} =\dfrac{2 \pi }{ \lambda } (vt -x)$$

phase difference ,$$( \phi _{2} - \phi _{1} )=\dfrac{2 \pi }{ \lambda } (v t_{2} -x) -\dfrac{2 \pi }{ \lambda } (v t_{1} -x) $$

$$=\dfrac{2 \pi }{ \lambda }v (t_{2} - t_{1}) $$

$$= \dfrac{2 \pi n \lambda}{ \lambda } (t_{2}-t_{1}) $$

$$=2 \pi n(t_{2}-t_{1}) $$

$$=2 \pi \times 500 \times 10^{-3} $$

$$= \pi radian$$
Question 2
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Travelling wave travels in medium '1' and enters into another medium '2' in which it's speed gets decreased to $$25\%$$. Then magnitude of ratio of amplitude of transmitted to reflected wave is

A
$$\dfrac{6}{5}$$

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

C
$$\dfrac{1}{7}$$

D
$$\dfrac{5}{9}$$

Solution

$${V}_{2}=d\frac { { V }_{ 1 } }{ 4 } \\ \dfrac { { A }_{ t } }{ { A }_{ r } } =\dfrac { (\dfrac { 2{ V }_{ 2 } }{ { V }_{ 2 }+{ V }_{ 1 } } ) }{ (\dfrac { { V }_{ 2 }-{ V }_{ 1 } }{ { V }_{ 2 }+{ V }_{ 1 } } ) } =(\dfrac { 2{ V }_{ 2 } }{ { V }_{ 2 }-{ V }_{ 1 } } )\\ (\dfrac { 2\dfrac { { V }_{ 1 } }{ 4 } }{ \dfrac { { V }_{ 1 } }{ 4 } -{ V }_{ 1 } } )=\dfrac { 2\dfrac { { V }_{ 1 } }{ 4 } }{ 3\dfrac { { V }_{ 1 } }{ 4 } } =\dfrac { 2 }{ 3 } $$
Question 3
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The frequency of fork is 512 Hz and the sound produced by it travels 42 metres as the tuning fork completes 64 vibrations. Find the velocity of sound :

A
336 m/sec

B
320 m/sec

C
340 m/sec

D
350 m/sec

Solution

The distance traveled for one vibration is equal to the wavelength. So $$64\lambda= 42$$

$$ \therefore \lambda = \dfrac{42}{64} = 0.656$$

Velocity of sound $$v=\lambda \nu=0.656 \times 512=336\: m/s $$
Question 4
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A certain transverse sinusoidal wave of wavelength $$20 cm$$ is moving in the positive $$x$$ direction. The transverse velocity of the particle at $$x = 0$$ as a function of time is shown. The amplitude of the motion is :

A
$$\dfrac{5}{\pi} cm$$

B
$$\dfrac{\pi}{2}cm$$

C
$$\dfrac{10}{\pi}cm$$

D
$$2\pi cm$$

Solution

$$ V_{max}=A\omega =5cm/s;$$
$$T=4 sec \Rightarrow \omega = \dfrac{2\pi}{4}=\dfrac{\pi}{2} $$

$$ \Rightarrow A =\dfrac{5}{\pi /2} = \dfrac{10}{\pi }$$cm
Question 5
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Light travels in the form of

A
Waves

B
Packets

C
Straight Lines

D
None of these

Solution
Question 6
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The effects are produced at a given point in space by two waves described by the equations $$\displaystyle y_{1}=y_{m}\sin \omega t\: \: $$and$$\: \: y_{2}=y_{m}\sin \left ( \omega t+\phi \right ) $$ where $$\displaystyle y_{m}$$ is the same for both the waves and $$\displaystyle \phi $$ is a phase angle. Tick the correct statement among the following

A
the maximum intensity that can be achieved at a point is twice the intensity of either wave and occurs if $$\displaystyle \phi=0 $$

B
the maximum intensity that can be achieved at a point is four times the intensity of either wave and occurs if $$\displaystyle \phi=0 $$

C
the maximum amplitude that can be achieved at the point its twice the amplitude of either wave and occurs at $$\displaystyle \phi=0 $$

D
When the intensity is zero the net amplitude is zero and at this point $$\displaystyle \phi=\pi/4 $$

Solution

$${ y }_{ 1 }={ y }_{ m }\sin { wt } \\ { y }_{ 2 }={ y }_{ m }\sin { \left( wt+\phi \right) } \\ y={ y }_{ 1 }+{ y }_{ 2 }\\ y={ y }_{ m }\sin { wt } +{ y }_{ m }\sin { \left( wt+\phi \right) } \\ y={ y }_{ m }2\left[ \sin { \dfrac { (wt+wt+\phi ) }{ 2 } \cos { \dfrac { \left( wt-wt-\phi \right) }{ 2 } } } \right] \\ y=2{ y }_{ m }\cos { \left( \dfrac { \phi }{ 2 } \right) } \sin { \left( \dfrac { 2wt+\phi }{ 2 } \right) } \\ \left[ y=2{ y }_{ m }\cos { \left( \dfrac { \phi }{ 2 } \right) } \sin { \left( wt+\dfrac { \phi }{ 2 } \right) } \right] \longrightarrow $$Equation of resultant wave
If $$\phi =0\\ \cos { \left( \dfrac { \phi }{ 2 } \right) } =1\\ \therefore y=2{ y }_{ m }\sin { \left( wt+\dfrac { \phi }{ 2 } \right) } $$
Where maximum intensity=$$2{ y }_{ m }$$
Hence option (A) is correct
Question 7
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The motion of the particle in simple harmonic motion is given by $$x=a\sin \omega t$$
If its speed is u, when the displacement is $$x_{1}$$ and speed is v, when the displacement is $$x_{2}$$, show that the amplitude of the motion is
.

A
$$\displaystyle a=\left [ \frac{v^{3}x_{1}^{3}-u^{2}x_{2}^{2}}{v^{2}-u^{2}} \right ]^{1/2}$$

B
$$\displaystyle a=\left [ \frac{v^{2}x_{1}^{2}-u^{2}x_{2}^{2}}{v^{2}-u^{2}} \right ]^{1/2}$$

C
$$\displaystyle a=\left [ \frac{v^{3}x_{1}^{2}-u^{3}x_{2}^{2}}{v^{3}-u^{3}} \right ]^{1/2}$$

D
$$\displaystyle a=\left [ \frac{v^{4}x_{1}^{2}-u^{4}x_{2}^{2}}{v^{4}-u^{4}} \right ]^{1/2}$$

Solution
Question 8
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The following figure depicts a wave travelling in a medium. Which pair of particles are in phase

A
A and D

B
B and F

C
C and E

D
B and G

Solution

Hint: The article which has the same phase are in a similar position on the wave.
Step1: Finding the pair of particles of the same phase.
The particles which have identical positions and have distance between them equal to the wavelength are the points of the same phase.

Hence, B and H are in phase.
The correct option is (D)
Question 9
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Directions For Questions

Dolphins emit high frequency sound waves typically $$10^5\space Hz$$ and use the echos for guidance and for hunting. The corresponding wave length in water is $$1.48\space cm$$ with this sonar system they can sense objects as small as (roughly) the wavelength. Ultrasonic imaging is a medical technique that uses exactly the same physical principle, sound waves of very short wavelength (or high frequency) called ultrasound are scanned over the human body the echos from the interior organs are used to create image. Ultrasound is more sensitive than x-rays in distinguishing various kinds of tissues and does not have the radiation hazards associated with the x-rays. Ultra sound is used for the study of heart valve action, detection of tumors and prenatal tests.

...view full instructions

Find the size of object which can be featured with $$5\space MHz$$ in water.

A
0.148 mm

B
0.3 mm

C
0.5 mm

D
0.1 mm

Solution

$$\quad v = 1.48\times10^5\times10^{-2} = 1480\space ms^{-1}$$

$$\quad \lambda = \displaystyle\frac{v}{f} = \displaystyle\frac{1480}{5\times10^6} = 0.3\space mm$$
Question 10
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Waves transfer

A
Matter

B
Particles

C
Energy

D
Water

Solution