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Question 11 Mark

For the harmonic travelling wave $\text{y}=2\cos2\pi(10\text{t}-0.0080\text{x}+3.5)$ where x and y are in cm and t is second. What is the phase difference between the oscillatory motion at two points separated by a distance of:$\frac{\lambda}{2}$

Answer

$\text{y}=2\cos2\pi(10\text{t}-0.0080\text{x+3.5})$$\text{y}=2\cos(20\pi\text{t}-0.0016\pi\text{x}+7.0\pi)$
Wave is propagated in $+\text{x}$ direction because $\omega\text{t}$ and kx are in with opposite sign standard equation $\text{y}=\text{a}\cos(\omega\text{t}-\text{kx}+\phi)$
a = 2, $\omega=20\pi,\ \text{k}=0.016\pi$ and $\phi=7\pi$
Path difference $\text{p}=\frac{\lambda}{2}$
$\Delta\phi=\frac{2\pi}{\lambda}\text{p}=\frac{2\pi}{\lambda}\times\frac{\lambda}{2}\pi\ \text{radian}$

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Question 21 Mark

In the given progressive wave $\text{y}=5\sin(100\pi\text{t+0.4x})$ where y and x are in m, t is in s. What is the:Frequency

Answer

Standard form of progressive wave travelling in $+\text{x}$ direction (kx and $\omega\text{}t$ have opposite sign is given) Eqn. is $\text{y}=\text{a}\sin(\omega\text{t}-\text{kx}+\phi)$$\text{y}=5\sin(100\pi\text{t}-0.4\pi\text{t}+0)$
Frequency $\text{v},\omega=2\pi\text{v}\Rightarrow\text{v}=\frac{\omega}{2\pi}\because\omega=100\pi$$\therefore\text{v}=\frac{100\pi}{2\pi}=50\text{Hz}$

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Question 31 Mark

In the given progressive wave $\text{y}=5\sin(100\pi\text{t+0.4x})$ where y and x are in m, t is in s. What is the:
Wave velocity

Answer

Standard form of progressive wave travelling in $+\text{x}$ direction kx and $\omega\text{}t$ have opposite sign is given) Eqn. is $\text{y}=\text{a}\sin(\omega\text{t}-\text{kx}+\phi)$$\text{y}=5\sin(100\pi\text{t}-0.4\pi\text{t}+0)$
Wave velocity $\text{v}=\text{v}\lambda=50\times5=250\text{m/ s}$

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Question 41 Mark

Given below are some functions of x and t to represent the displacement of an elastic wave.$\text{y}=4\sin(5\text{x}-\text{t/ 2})+3\cos(5\text{x}-\text{t/ 2})$

Answer

A stationary wave of the for $\text{y}=5\cos(4\text{x})\sin20\text{t}$ is a stationary wave so (b) (i).

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Question 51 Mark

For the harmonic travelling wave $\text{y}=2\cos2\pi(10\text{t}-0.0080\text{x}+3.5)$ where x and y are in cm and t is second. What is the phase difference between the oscillatory motion at two points separated by a distance of:0.5m

Answer

$\text{y}=2\cos2\pi(10\text{t}-0.0080\text{x+3.5})$$\text{y}=2\cos(20\pi\text{t}-0.0016\pi\text{x}+7.0\pi)$
Wave is propagated in $+\text{x}$ direction because $\omega\text{t}$ and kx are in with opposite sign standard equation $\text{y}=\text{a}\cos(\omega\text{t}-\text{kx}+\phi)$
a = 2, $\omega=20\pi,\ \text{k}=0.016\pi$ and $\phi=7\pi$
Path differencee p = 0.5m = 50cm
$\Delta\phi=\text{kp}=0.016\pi\times50=0.8\pi$ red.

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Question 61 Mark

The pattern of standing waves formed on a stretched string at two instants of time are shown in The velocity of two waves superimposing to form stationary waves is $360ms^{–1}$ and their frequencies are $256Hz$

Calculate the time at which the second curve is plotted.
Mark nodes and antinodes on the curve.
Calculate the distance between A′ and C′.

Answer

Given frequency of the wave v = 256Hz$\therefore\text{T}=\frac{1}{\text{v}}=\frac{1}{256}$ second = 0.00390
$\text{T}=3.9\times10^{-3}$ seconds.
(a) In stationary wave a particle passes though it's mean position after ever $\frac{\text{T}}{4}$ time$\therefore$ in II nd curve displacement of all medium particle, are zero so
$\text{t}=\frac{\text{T}}{4}=\frac{3.9\times10^{-3}}{4}=.975\times10^{-3}\sec$
$\text{t}=9.8\times10^{-4}$ secound.
(b) Point does not vibrate i.t., their displacement is zero always so nodes A, B, C, D and E. the point A' and C' are at maximam displacement so there are anti-nodes at A' and C'. Between A' and C' $=\lambda=\frac{\text{v}}{\text{V}}=\frac{360}{256}=\frac{90}{64}=1.41\text{m}.$

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Question 71 Mark

A pipe 20cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a source of 1237.5Hz?(sound velocity in air $= 330ms^{–1}$)

Answer

Length of pipe, $\text{l}=20\text{cm}=20\times10^{-2}\text{m}$ Fundamental frequency of closed organ pipe$\text{f}_0=\frac{\text{v}}{4\text{}L}=\frac{330}{4\times20\times10^{-2}}=412.5\text{Hz}$
$\frac{\text{f given}}{\text{f}_0}=\frac{1237.5}{412.5}=3$

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Question 81 Mark

For the harmonic travelling wave $\text{y}=2\cos2\pi(10\text{t}-0.0080\text{x}+3.5)$ where x and y are in cm and t is second. What is the phase difference between the oscillatory motion at two points separated by a distance of:$\frac{3\lambda}{4}$(at a given instant of time)

Answer

$\text{y}=2\cos2\pi(10\text{t}-0.0080\text{x+3.5})$$\text{y}=2\cos(20\pi\text{t}-0.0016\pi\text{x}+7.0\pi)$
Wave is propagated in $+\text{x}$ direction because $\omega\text{t}$ and kx are in with opposite sign standard equation $\text{y}=\text{a}\cos(\omega\text{t}-\text{kx}+\phi)$
a = 2, $\omega=20\pi,\ \text{k}=0.016\pi$ and $\phi=7\pi$
$\Delta\phi=\frac{2\pi}{\lambda}\text{p}=\frac{2\pi}{\lambda}\times\frac{3\pi}{4}=\frac{3}{2}\pi\ \text{radian}$

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Question 91 Mark

In the given progressive wave $\text{y}=5\sin(100\pi\text{t+0.4x})$ where y and x are in m, t is in s. What is the:
Amplitude

Answer

Standard form of progressive wave travelling in $+\text{x}$ direction (kx and $\omega\text{}t$ have opposite sign is given) Eqn. is $\text{y}=\text{a}\sin(\omega\text{t}-\text{kx}+\phi)$$\text{y}=5\sin(100\pi\text{t}-0.4\pi\text{t}+0)$
Amplitude $\text{a}=5\text{m}$

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