Wave on a String

Q. If the initial tension on a stretched string is doubled, then the ratio of the initial and final speeds of a transverse wave along the string is:

1. $1:2$
2. $\sqrt{2}:1$
3. $1:1$
4. $1:\sqrt{2}$

Q. A wave travelling along a string is described by $y(x,t)=0.008\sin (85x-4t)$ in $\text{SI}$ units, where $x$ is in metres and $t$ is in seconds . Calculate the wavelength and frequency of the wave.

1. $8.4\text{cm},1\text{Hz}$
2. $14.8\text{cm},0.64\text{Hz}$
3. $7.4\text{cm},1.28\text{Hz}$
4. $7.4\text{cm},0.64\text{Hz}$

Q. Wave pulse on a string shown in figure is moving to the right without changing shape. Consider two particles at positions $x_1=1.5mand{x}_2=2.5m$. Their transverse velocities at the moment shown in figure are along directions.

1. positive y-axis and positive y-axis, respectively
2. negative y-axis and y-axis, respectively
3. negative y-axis and negative y-axis, respectively
4. positive y-axis and negative y-axis, respectively

Q.

Two waves, each having a frequency of 100 Hz and a wavelength of 2.0 cm, are travelling in the same direction on a string. What is the phase difference between the waves (a) if the second wave was produced 0.015 s later than the first one at the same place, (b) if the two waves were produced at the same instant but the first one was produced a distance 4.0 cm behind the second one ? (c) If each of the waves has an amplitude of 2.0 mm, what would be the amplitudes of the resultant waves in part (a) and (b) ?

Q. A $200\text{Hz}$ sinusoidal wave is travelling in the positive x-direction along a string with a linear mass density of $4\times {10}^{-3}\text{kg/m}$ and a tension of $40\text{N}$. At time $t=0$, the point $x=0$ has maximum displacement in the positive y-direction. Next when this point has zero displacement, the slope of the string is $\pi /20$. Which of the following expressions represents the displacement of string as a function of $x$ (in metres) and $t$ (in seconds)?

1. $y=0.025\cos (400\pi t-2\pi x)$
2. $y=0.0125\cos (400\pi t-4\pi x)$
3. $y=0.025\cos (400\pi t-4\pi x)$
4. $y=0.0125\cos (200\pi t-4\pi x)$

Q. A non uniform string of mass $45\text{kg}$ and length $1.5\text{m}$ has a variable linear mass density given by $\mu =Kx$, where $x$ is the distance from one end of the string and $K$ is a constant. Tension in the string is $15\text{N}$ (uniform). Find the time (in seconds) required for a pulse generated at one end of the string to travel to the other end.

Q. A transverse wave is travelling on stretched string with velocity $u$. Tension in the string is $T$ and mass density of string is $r$. Find the velocity of cart $V$ such that waves seem stationary w.r.t cart.

1. $\sqrt{\frac{T}{r}}$
2. $2\sqrt{\frac{T}{r}}$
3. $3\sqrt{\frac{T}{r}}$
4. None of these

Q. The displacement wave in a string is $y=\left(3\text{cm}\right)\sin 6.28(0.5x-50t)$ where $x$ is in centimeters and $t$ in seconds. The wavelength and velocity of the wave is

1. $2\text{cm},100{\text{cms}}^{-1}$
2. $10\text{cm},50{\text{cm}}^{-1}$
3. $20\text{cm},2{\text{ms}}^{-1}$
4. $2\text{m},100{\text{ms}}^{-1}$

Q. A figure shows a snap photograph of a vibrating string at $t=0$. The particle $P$ is observed moving up with velocity $20\sqrt{3}\text{cm/s}$. The tangent at $P$ makes an angle ${60}^{\circ }$ with $x$-axis. Then choose the correct option(s):

1. Wave is moving along $(+)ve$ $x$-axis.
2. Wave is moving along $(-)ve$ $x$-axis.
3. Equation of wave is $y=\left(0.4\text{cm}\right)\sin \left[\right(10\pi {\text{s}}^{-1})t+(\frac{\pi }{2}{\text{cm}}^{-1}x+\frac{\pi }{4}]$.
4. Equation of wave is $y=\left(0.2\text{cm}\right)\sin \left[\right(10\pi {\text{s}}^{-1})t+(\frac{\pi }{2}{\text{cm}}^{-1})x-\frac{\pi }{2}]$.

Q. A wave moves with speed $400\text{m/s}$ on a wire , which is under a tension of $900\text{N}$. If the wave moves with $450\text{m/s}$, then the change in tension of the wire is

1. $+239\text{N}$
2. $-239\text{N}$
3. $+250\text{N}$
4. 
   $-250\text{N}$

Q. A figure shows a snap photograph of a vibrating string at $t=0$. The particle $P$ is observed moving up with velocity $20\sqrt{3}\text{cm/s}$. The tangent at $P$ makes an angle ${60}^{\circ }$ with $x$-axis. Then choose the correct option(s):

1. Wave is moving along $(+)ve$ $x$-axis.
2. Wave is moving along $(-)ve$ $x$-axis.
3. Equation of wave is $y=\left(0.4\text{cm}\right)\sin \left[\right(10\pi {\text{s}}^{-1})t+(\frac{\pi }{2}{\text{cm}}^{-1}x+\frac{\pi }{4}]$.
4. Equation of wave is $y=\left(0.2\text{cm}\right)\sin \left[\right(10\pi {\text{s}}^{-1})t+(\frac{\pi }{2}{\text{cm}}^{-1})x-\frac{\pi }{2}]$.

Q. A wave travelling along a string is described by $y(x,t)=0.008\sin (85x-4t)$ in $\text{SI}$ units, where $x$ is in metres and $t$ is in seconds . Calculate the wavelength and frequency of the wave.

1. $8.4\text{cm},1\text{Hz}$
2. $14.8\text{cm},0.64\text{Hz}$
3. $7.4\text{cm},1.28\text{Hz}$
4. $7.4\text{cm},0.64\text{Hz}$

Q. A wave travelling in the $+ve$ x-direction having displacement along y-direction as 1 m, wavelength $2\pi m$ and frequency of $\frac{1}{\pi }Hz$ is represented by

1. $y=\sin (x-2t)$
2. $y=\sin (2\pi x-2\pi t)$
3. $y=\sin (10\pi x-20\pi t)$
4. $y=\sin (2\pi x+2\pi t)$

Q. The equation of a standing wave produced on a string fixed at both the ends is $y=0.4\sin \left(0.314x\right)\cos \left(600\pi t\right)$, where $x$ is in $\text{cm}$, $t$ is in seconds. The smallest possible length of the string would be

1. $20\text{cm}$
2. $40\text{cm}$
3. $10\text{cm}$
4. None of these

Q. A string of length 60 cm fixed at both end is vibrating in its 2nd overtone. If mass per unit length of string is 0.2 kg/m, tension in string is 80 N and maximum amplitude of standing wave is 0.5 cm, then maximum speed of any point of string is

1. $50\pi cm/s$
2. $40cm/s$
3. $20\pi cm/s$
4. $60\pi cm/s$

Q. Wave pulse on a string shown in figure is moving to the right without changing shape. Consider two particles at positions $x_1=1.5mand{x}_2=2.5m$. Their transverse velocities at the moment shown in figure are along directions.

1. positive y-axis and positive y-axis, respectively
2. negative y-axis and positive y-axis, respectively
3. positive y-axis and negative y-axis, respectively
4. negative y-axis and negative y-axis, respectively

Q. A $200\text{Hz}$ sinusoidal wave is travelling in the positive $x$-direction along a string with a linear mass density of $4\times {10}^{-3}\text{kg/m}$ and a tension of $40\text{N}$. At time $t=0$, the point $x=0$ has maximum displacement in the positive $y$-direction. Next when this point has zero displacement, the slope of the string is $\frac{\pi }{20}$. Which of the following expressions respresent the displacement of string as a function of $x$ (in metre) and $t$ (in seconds).

1. $y=0.025\cos (400\pi t+2\pi x)$
2. $y=0.025\cos (4\pi x-400\pi t)$
3. $y=0.0125\cos (4\pi x-400\pi t)$
4. $y=0.0125\cos (4\pi x+400\pi t)$

Q. A wave travelling along a string is described by $y(x,t)=0.008\sin (85x-4t)$ in $\text{SI}$ units, where $x$ is in metres and $t$ is in seconds . Calculate the wavelength and frequency of the wave.

1. $8.4\text{cm},1\text{Hz}$
2. $14.8\text{cm},0.64\text{Hz}$
3. $7.4\text{cm},1.28\text{Hz}$
4. $7.4\text{cm},0.64\text{Hz}$

Q. A $5.5m$ length string has a mass of $0.035kg$. If the tension in the string is $77N.$ The speed of a transverse wave on the string is

1. $110m{s}^{-1}$
2. $165m{s}^{-1}$
3. $102m{s}^{-1}$
4. $77m{s}^{-1}$

Q. y - x curve at an instant for a wave travelling along x axis on a string is shown. Slope at the point A on the curve, as shown , is $\tan {53}^{\circ }$

1. Transverse velocity of the particle at point A is positive if the wave is travelling along positive x axis.
2. Transverse velocity of the particle at point A is positive if the wave is travelling along negative x axis.
3. Magnitude of transverse velocity of the particle at point A is greater than wave speed.
4. Magnitude of transverse velocity of the particle at point A is lesser than wave speed.

Q. Two long strings $\text{A}$ and $\text{B}$ each having linear mass density $1.2\times {10}^{-2}\text{kg/m}$ are stretched by different tensions $4.8\text{N}$ and $7.5\text{N}$ respectively and are kept parallel to each other with their left ends at $x=0$. Wave pulses are produced on the string at the left ends at $t=0$ on string$\text{A}$ and $t=20\text{ms}$ on string $\text{B}$. When will the pulse on $\text{B}$ overtake that on $\text{A}$ for the first time?

1. $0.1\text{s}$
2. $0.4\text{s}$
3. $0.5\text{s}$
4. $0.25\text{s}$

Q. A transverse pulse generated at the bottom of a uniform rope of length L, travels in upward direction. The time taken by it to travel the full length of rope will be

1. $\sqrt{\frac{L}{2g}}$
2. $\sqrt{\frac{L}{g}}$
3. $\sqrt{\frac{4L}{g}}$
4. $\sqrt{\frac{2L}{g}}$

Q. A metallic wire of 1 m length has a mass of $10\times {10}^{-2}$ kg. If a tension of 100 N is applied to a wire, what is the speed of transverse wave?

1. 31.6 $m{s}^{-1}$
2. 10 $m{s}^{-1}$
3. 0.1 $m{s}^{-1}$
4. 200 $m{s}^{-1}$

Q. The electric field part of an electromagnetic wave in an medium is represented by $E_x$ = 0,
$E_y=2.5\frac{N}{C}\cos [(2\pi \times {10}^6\frac{rad}{s})t-(\pi \times {10}^{-2}\frac{rad}{m})x]$
$E_z$ = 0. The wave is

Q. Two block of masses $40kg$ and $20kg$ are connected by a wire that has a linear mass density of $1g/m$. These blocks are being pulled across horizontal frictionless floor by horizontal force $F$ that is applied to $20kg$ block. A transverse wave travel on the wave between the blocks with a speed of $400m/s$ (relative to the wire). The mass of the wire is negligible compared to the mass of the blocks The magnitude of $F$ is

1. $160N$
2. $320N$
3. $240N$
4. $400N$

Q. Which of the following is not true for this progressive wave $y=4\sin 2\pi (\frac{t}{0.02}-\frac{x}{100})$ where y and x are in cm & t in sec. [CMPT 2003]

1. Its amplitude is 4 cm
2. Its wavelength is 100 cm
3. Its frequency is 50 cycles/sec
4. Its propagation velocity is $50\times {10}^3$ cm/sec

Q. The amplitude of a wave disturbance propagating in the positive $x$-directions is given by
$y=\frac{1}{1+x^2}$ at time $t=0$ and by
$y=\frac{1}{[1+(x-1{)}^2]}$ at $t=2$ seconds
where $x$ and $y$ are in metres. The shape of the wave disturbance does not change curing the propagation, then the velocity of the wave is

1. $0.5m{s}^{-1}$
2. $2.0m{s}^{-1}$
3. none of the above
4. $1.0m{s}^{-1}$

Q. The electric field of an electromagnetic wave traveling through a vacuum is given by equation $E=E_{0}sin(kx-t)$. The quantity that is independent of wavelength is?

1. $k$
2. $\frac{k}{\omega }$
3. $k^2$
4. $k^3$

Q. The electric field associated with an e.m. wave in vacum is given by $\overrightarrow{E}=\hat{i}40cos(kz-6\times {10}^{8}t)$, where E, z and t are in volt/m, meter and seconds respectively.The value of wave vector k is:

1. $6m^{-1}$
2. ${3m}^{-1}$
3. ${2m}^{-1}$
4. $0.5m^{-1}$

Q. Find the displacement of a point on the string as a function of position and time.