Wave on a String

Q. A transverse wave is travelling on stretched string with velocity $u$. Tension in the string is $T$ and mass density of string is $\mu$. 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. 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 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. 
   $-250\text{N}$
2. $+250\text{N}$
3. $+239\text{N}$
4. $-239\text{N}$

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.

Vishanth took a rope of linear mass density of $0.04kg{m}^{-1}$ and a length of $3m$. He tied the rope tightly between the two walls and disturbed it to vibrate. He noticed that transverse wave is progressing in rope at a speed $xm{s}^{-1}$. If tension in the string is $1N,$ then $x=\_\_\_\_\_\_\_\_\_.$

1. $5cm{s}^{-1}$

2. $10m{s}^{-1}$

3. $5m{s}^{-1}$

4. $10cm{s}^{-1}$

Q. If the individual atoms and molecules in the medium move with simple harmonic motion, the resulting periodic wave has a sinusoidal form.

1. True
2. False

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

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

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