Waves Test

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Waves Test - 67

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Weekly Quiz Competition

Question 1
1 / -0

Two sine waves travel in the same direction in a medium. The amplitude of each wave is A and the phase difference between the two waves is $120^{\circ}$ . The resultant amplitude will be

A
$A$

B
$2A$

C
$4A$

D
$\sqrt{2 A}$

Solution
Question 2
1 / -0

Two point lie on a ray are emerging from a source of simple harmonic wave having period $0.045$ . The wave speed is $300\ m/s$ and points are at $10\ m$ and $16\ m$ from the source. They differ in phase by :

A
$\pi$

B
$\pi/2$

C
$0$ or $2\pi$

D
$none\ of\ these$

Solution
Question 3
1 / -0

A simple harmonic wavetrain of amplitude $5\ cm$ and frequency $100\ Hz$ is travelling in the positive $x-$ direction with a velocity of $30\ m/s$ . The displacement velocity and acceleration at $t=3s$ of a particle of the medium situated $100\ cm$ from the origin are respectively.

A
$-3.44\ cm+1750\ cm/s, 17\times cm/s^{2}$

B
$4.33\ cm+1570\ cm/s,71\times 10^{4}\ cm/s^{2}$

C
$-4.33\ cm,+1570\ cm/s,171\times 10^{4}\ cm/s^{2}$

D
$-3.44\ cm,-1750\ cm/s,171\times 10^{4}\ cm/s^{2}$

Solution

$y=A\sin (\dfrac{2\pi \mu}{v}(vt-x))$

$y=5\sin (\dfrac{2\pi 100}{3000}(3000\times 3-100))=-4.45cm$

velocity= $\dfrac{dy}{dx}=A\times 2\pi\mu\cos(\dfrac{2\pi\mu}{v}(vt-x))=5\times 2\pi\times 100\cos(\dfrac{2\pi\times 100}{3000}(3000\times 3-180))=1571cm/s$

accleration= $\dfrac{d^2y}{dx^2}=-A\times 4\pi^2\mu^2sin(\dfrac{2\pi\mu}{v}(vt-x))=-5\times 4\pi^2\times 10000\sin(\dfrac{2\pi\times 100}{3000}(3000\times 3-180))=171\times 10^{4}cm/s^2$
Question 4
1 / -0

Two identical pulses move in opposite directions with same uniform speeds on a stretched string. The width and kinetic energy of each pulse is $L$ and $k$ respectively. At the instant they completely overlap, the kinetic energy of the width $L$ of the string where they overlap is:

A
$k$

B
$2k$

C
$4k$

D
$8k$

Solution

$y_1=a\sin(2\pi x /L)$
When they get combined: $y_2=2asin(2\pi x/L)$
We know particle velocity= wave velocity x dy/dx
As in both the cases wave velocities are the same
$\dfrac{v_1}{v_2}=\dfrac{\dfrac{dy_1}{dx}}{\dfrac{dy_2}{dx}}=\dfrac{1}{2}\Rightarrow v_2=2v_1$

$\dfrac{KE_1}{KE_2}=\dfrac{{v_1}^2}{{v_2}^2}=\dfrac{1}{4}\Rightarrow KE_2=4\times KE_1=4k$
Question 5
1 / -0

A wave motion has the function $Y=a_{0}\sin (\omega t-kx)$ . The graph in the figure shows how the displacement $y$ at a fixed point varies with time $t$ . Which one of the labeled points shows a displacement equal to that at the position $x=\pi/2k$ at time $t=0$

A
$P$

B
$Q$

C
$R$

D
$S$

Solution

Given ,
$Y=a_{0}\sin (\omega t-kx)$ .
Now, at $x=\dfrac{\pi}{2k} \ and\ t=0$
we get,
$Y=a_{0}\sin (\omega (0)-k(\dfrac{\pi }{2k})$ .

$Y=-a_0$
from the figure point (Q) shows a displacement equal to (-a_0).
hence, option B is correct
Question 6
1 / -0

A simple harmonic plane wave propagates along x-axis in a medium. The displacement of the particles as a function of time is shown in figure, for $x = 0$ (curve 1) and $x = 7$ (curve 2).
The two particles are within a span of one wavelength.
The speed of the wave is

A
$12 \,m/s$

B
$24 \,m/s$

C
$8 \,m/s$

D
$16 \,m/s$

Solution

Let general wave equation is $y = A \,sin (\omega t - kx + \phi)$
$v = \dfrac {dy}{dt} = A \omega cos (\omega t - kx + \phi)$
for curve $(1 ) , x = 0$
at $t = 0, x = 0 ,$ we have $y = 0$
$\Rightarrow 0 = A \,sin[\phi] \Rightarrow sin \phi = 0$
$\Rightarrow \phi = 0 \,or \,\pi$
here $\phi = \pi$ (because velocity is negative)
for curve $(2), x = 7 \,cm$
at $t = 0, x = 7 \,cm, y = -1$
$-1 = sin(-k \times 7 + \pi)$
$\Rightarrow sin(-7k + \pi) = -1/2$
$\Rightarrow -7k + \pi = 2n\pi + \dfrac {7\pi}{6}$ or $2n\pi + \dfrac {11\pi}{6}$
here $\Rightarrow -7k + \pi = 2n\pi + \dfrac {11\pi}{6}$
(because at t = 0, velocity is positive)
$\Rightarrow -7 \left ( \dfrac {2\pi}{\lambda} \right ) = 2n\pi + \dfrac {5\pi}{6}$
$\Rightarrow \lambda = \dfrac {-14\pi}{5\pi/6 + 2n\pi}$
$\Rightarrow \lambda = \dfrac {-84}{12n + 5}$
for $n = -1, \lambda = 12 \,cm$
for $n = -2, \lambda = \dfrac {84}{19}cm$ (not possible)
because $\lambda > 7 \,cm$
$v = f\lambda = 100 \times \dfrac {12}{100} = 12 \,m/s$
Question 7
1 / -0

Two separated sources emit sinusoidal travelling waves but have the same wavelength $\lambda$ and are in phase at their respective sources. One travels a distance $l_{1}$ to get to the observation point while the other travels a distance $l_{2}$ . The amplitude is minimum at the observation point, if $l_{1}-l_{2}$ is an

A
odd integral multiple of $\lambda$

B
even integral multiple of $\lambda$

C
odd integral multiple of $\lambda / 2$

D
odd integral multiple of $\lambda / 4$

Solution
Question 8
1 / -0

Two vibration strings of the same material but lengths L and 2L have radii 2r and r, respectively. The are stretched under the same tension. Both the strings vibrate in their fundamental modes, the one of length L with frequency $n_1$ and the other with frequency $n_2$ . The ratio $n_1/n_2$ is given by

A
2

B
4

C
8

D
1

Solution

$n_{1}=\frac{1}{2l}\sqrt{\left [ \frac{T}{4\pi r^{2}\rho } \right ]}$

and $n_{2}=\frac{1}{4l}\sqrt{\left [ \frac{T}{\pi r^{2}\rho } \right ]}$

$\therefore \frac{n_{1}}{n_{2}}=2\times \frac{1}{2}=1$
Question 9
1 / -0

Figure 5.55 shows a student setting up wave on a long stretched string. The student's hand makes one complete up and down movement in $0.4 \,s$ and in each up and down movement the hand moves by a height of $0.3 \,m.$ The wavelength of the waves on the string is $0.8 \,m.$
The amplitude of the wave is

A
$0.15 \,m$

B
$0.3 \,m$

C
$0.075 \,m$

D
cannot be predicted

Solution

$Frequency = \dfrac {1}{Time \,Period} = \dfrac {1}{0.4} = 2.5 \,Hz$
$Amplitude = \dfrac {1}{2} \times 0.3 \,m = 0.15 \,m$
Question 10
1 / -0

Four pieces of string each of length L are joined end to end to make a long string of length $4L.$ The linear mass density of the strings are $\mu, 4\mu, 9\mu$ and $16\mu,$ respectively. One end of the combined string is tied to a fixed support and a transverse wave has been generated at the other end having frequency $f$ (ignore any reflection and absorptions). String has been stretched under a tension $F.$
Find the ratio of wavelengths of the waves on four strings, starting from right hand side.

A
$12 : 6 : 4 : 3$

B
$4 : 3 : 2 : 1$

C
$3 : 4 : 6 : 12$

D
$1 : 2 : 3 : 4$

Solution

Frequency of wave is same on all the four strings. So,
$\lambda_1 = \dfrac {v_1}{f},$ $\lambda_2 = \dfrac {v_2}{f},$
$\lambda_3 = \dfrac {v_3}{f},$ $\lambda_4 = \dfrac {v_4}{f},$
$\lambda_4 : \lambda_3 : \lambda_2 : \lambda_1 = \dfrac {1}{4} : \dfrac {1}{3} : \dfrac {1}{2} : 1 = 3 : 4 : 6 : 12$

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Re-Attempt Weekly Quiz Competition

Waves Test - 67

Result

Waves Test - 67

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SHARING IS CARING

Question 1

AAA

2A2A2A

4A4A4A

2A\sqrt{2 A}2A​

Solution

Question 2

π\piπ

π/2\pi/2π/2

000 or 2π2\pi2π

none of thesenone\ of\ thesenone of these

Solution

Question 3

−3.44 cm+1750 cm/s,17×cm/s2-3.44\ cm+1750\ cm/s, 17\times cm/s^{2}−3.44 cm+1750 cm/s,17×cm/s2

4.33 cm+1570 cm/s,71×104 cm/s24.33\ cm+1570\ cm/s,71\times 10^{4}\ cm/s^{2}4.33 cm+1570 cm/s,71×104 cm/s2

−4.33 cm,+1570 cm/s,171×104 cm/s2-4.33\ cm,+1570\ cm/s,171\times 10^{4}\ cm/s^{2}−4.33 cm,+1570 cm/s,171×104 cm/s2

−3.44 cm,−1750 cm/s,171×104 cm/s2-3.44\ cm,-1750\ cm/s,171\times 10^{4}\ cm/s^{2}−3.44 cm,−1750 cm/s,171×104 cm/s2

Solution

Question 4

kkk

2k2k2k

4k4k4k

8k8k8k

Solution

Question 5

PPP

QQQ

RRR

SSS

Solution

Question 6

12 m/s12 \,m/s12m/s

24 m/s24 \,m/s24m/s

8 m/s8 \,m/s8m/s

16 m/s16 \,m/s16m/s

Solution

Question 7

odd integral multiple of λ \lambda λ

even integral multiple of λ \lambda λ

odd integral multiple of λ/2 \lambda / 2 λ/2

odd integral multiple of λ/4 \lambda / 4 λ/4

Solution

Question 8

2

4

8

1

Solution

Question 9

0.15 m0.15 \,m0.15m

0.3 m0.3 \,m0.3m

0.075 m0.075 \,m0.075m

cannot be predicted

Solution

Question 10

12:6:4:312 : 6 : 4 : 312:6:4:3

4:3:2:14 : 3 : 2 : 14:3:2:1

3:4:6:123 : 4 : 6 : 123:4:6:12

1:2:3:41 : 2 : 3 : 41:2:3:4

Solution

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$A$

$2A$

$4A$

$\sqrt{2 A}$

$\pi$

$\pi/2$

$0$ or $2\pi$

$none\ of\ these$

$-3.44\ cm+1750\ cm/s, 17\times cm/s^{2}$

$4.33\ cm+1570\ cm/s,71\times 10^{4}\ cm/s^{2}$

$-4.33\ cm,+1570\ cm/s,171\times 10^{4}\ cm/s^{2}$

$-3.44\ cm,-1750\ cm/s,171\times 10^{4}\ cm/s^{2}$

$k$

$2k$

$4k$

$8k$

$P$

$Q$

$R$

$S$

$12 \,m/s$

$24 \,m/s$

$8 \,m/s$

$16 \,m/s$

odd integral multiple of $\lambda$

even integral multiple of $\lambda$

odd integral multiple of $\lambda / 2$

odd integral multiple of $\lambda / 4$

$0.15 \,m$

$0.3 \,m$

$0.075 \,m$

$12 : 6 : 4 : 3$

$4 : 3 : 2 : 1$

$3 : 4 : 6 : 12$

$1 : 2 : 3 : 4$