Expression for Standing Waves

Q. The transverse displacement of a string (clamped at its both ends) is given by y(x, t) = 0.06 sin (2π/3)x ( cos (120 π t) where x and y are in m and t in s. The length of the string is 1.5 m and its mass is 3.0 ×10¯²kg. Answer the following : (a) Does the function represent a travelling wave or a stationary wave? (b) Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave ? (c) Determine the tension in the string.

Q. Given below are some functions of x and t to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (a) y = 2 cos (3x) sin (10t) (b) y = 2 x − υ t (c) y = 3 sin (5x – 0.5t) + 4 cos (5x – 0.5t) (d) y = cos x sin t + cos 2x sin 2t

Q. a particle lies in space at point (2, 3, 4).find the magnitude of its position vect

Q. A plane simple harmonic progressive wave given by the equation, $y=A\sin (\omega t-kx)$ of wavelength $120\text{cm}$ is incident normally on a plane surface which is a perfect reflector (acts as fixed end). If a stationary wave is formed, then the ratio of amplitudes of vibrations at points $10\text{cm}$ and $30\text{cm}$ from the reflector is

(Here $x$ is measured from reflector)

1. $1:2$
2. $1:3$
3. $2:1$
4. $3:1$

Q. In a standing wave on a string rigidly fixed at both ends : (1) all the particles must be at their positive extremes simultaneously once in half of the time period. (2) all the particles must be at their positive extremes simultaneously once in a time period. (3) in one time period all the particles are simultaneously at rest twice. (4) all the particle are never at rest simultaneously.

Q.

In stationary wave

[MP PET 1987; BHU 1995]

1. Strain is maximum at nodes

2. Strain is maximum at antinodes

3. Strain is minimum at nodes

4. Amplitude is zero at all the points

Q. A particle moves according to the relation x=acospit, the distance covered by it in 2.5s is

Q. A uniform solid cylinder of radius 1m is placed on the inner surface of cylinder of radius 11 m find the time period of the cylinder for small oscillations. The cylinder rolls purely

Q. Statement1:Linear momemtum of the system of particles is 0. Statement2:K.E of the system of particles is 0 .Then (a)Statement 1 implies statement 2 & statememt 2 imlies 1(b)statement 1 does not imply statement 2 and statememt 2 does not imply statement 1 (c)statement 1 implies statement 2 but statement 2 does not imply statement 1 (d)statememt 1 does not imply statement 2 but statement 2 imples statement 1??.

Q. A wave disturbance in a medium is described by $y(x,t)=0.02\cos \left(10\pi x\right)\cos (50\pi t+\frac{\pi }{2})$, where $x$ and $y$ are in $\text{meter}$ and $t$ in $\text{seconds}$. Identify the incorrect option.

1. Node occurs at $x=0.15\text{m}$
2. Antinode occurs at $x=0.25\text{m}$
3. Wavelength of the constituent waves is $0.2\text{m}$
4. Speed of the constituent waves is $5\text{m/s}$

Q. What is the smallest positive phase constant which is equivalent to 7⋅5 π?

Q. (i) For the wave on a string described in Exercise 15.11, do all the points on thebstring oscillate with the same (a) frequency, (b) phase, (c) amplitude? Explain your answers. (ii) What is the amplitude of a point 0.375 m away from one end?

Q. Equation of a stationary wave is given by $y=5\cos \left(\frac{\pi x}{25}\right)\sin \left(100\pi t\right)$. Here, $x$ is in $\text{centimeter}$ and $t$ in $\text{seconds}$. Node will not occur at distance

1. $25\text{cm}$
2. $62.5\text{cm}$
3. $12.5\text{cm}$
4. $37.5\text{cm}$

Q. The equation of a stationary wave is $y=0.8\text{cos}\left(\frac{\pi x}{20}\right)\text{sin}200\pi t$, where $x$ is in $\text{cm}$ and $t$ is in seconds. The separation between a successive node and antinode will be

1. $20\text{cm}$
2. $10\text{cm}$
3. $40\text{cm}$
4. $30\text{cm}$

Q. A stationary wave set up on a string have the equation $y=\left(2\text{mm}\right)\left[\text{cos}\right(6.28{\text{m}}^{-1}\left)\text{x cos}\right(\omega t\left)\right]$. This stationary wave is created by two identical waves, of amplitude $A$ each moving in opposite directions along the string. Then,

1. $A=2\text{mm}$
2. $A=4\text{mm}$
3. The smallest length of the string is $50\text{cm}$
4. The smallest length of the string is $25\text{cm}$

Q. Let F=F0sinwt is the external periodic force acting on the oscillator. If amplitude of the oscillator is maximum for w=w1 and the energy is maximum for w=w2, then (w0 is the natural angular frequency)the relation b/w w1, w2 and w0 is)

Q. The displacement equation for a particle in standing wave is represented as $y(x,t)=0.4\sin \left(0.5x\right)\cos \left(30t\right)$, where $x$ and $y$ are in $\text{centimeter}$ and time $t$ is in $\text{seconds}$.
Magnitude of velocity of particle at $x=2.4\text{cm}$ and $t=0.8\text{s}$ is -
$\left[\sin \right(1.2)=0.93,\text{sin}(24)=-0.9]$

1. $0.4\text{cm/s}$
2. $5.2\text{cm/s}$
3. $10.1\text{cm/s}$
4. $12\text{cm/s}$

Q. A standing wave is maintained in a homogenous string of cross-sectional area $s$ and density $\rho$. It is formed by the superposition of two waves travelling in opposite directions given by the equations $y_1=a\sin (\omega t-kx)$ and $y_2=a\sin (\omega t+kx)$. The total mechanical energy confined between the sections corresponding to adjacent nodes is

1. $\frac{9}{2}\frac{\rho {\omega }^2{a}^2\pi s}{k}$
2. $\frac{7}{2}\frac{\rho {\omega }^2{a}^2\pi s}{k}$
3. $\frac{\rho {\omega }^2{a}^2\pi s}{k}$
4. $\frac{3}{2}\frac{\rho {\omega }^2{a}^2\pi s}{k}$

Q. Two identical point sources P and Q vibrating in phase with the same amplitude generate sinusoidal waves on a water surface.The sources are 6.5 cm apart. Two nearest points (from P) of destructive interference along PQ are found to be at 0.7 cm and 2.4 cm from P. The total number of points of destructive interference on the line segment PQ is (A) 4 (B) 3 (D)5 (C) 2

Q. The equation for the vibration of a string, fixed at both ends vibrating in its third harmonic, is given by

$y=\left(0·4\mathrm{cm}\right)\sin \left[\left(0·314{\mathrm{cm}}^{-1}\right)x\right]\cos \left[\left(600\pi {\mathrm{s}}^{-1}\right)t\right]$.

(a) What is the frequency of vibration? (b) What are the positions of the nodes? (c) What is the length of the string? (d) What is the wavelength and the speed of two travelling waves that can interfere to give this vibration?

Q. Which of the following statements is/are true regarding a stationary wave?

1. All particles in a particular segment between two nodes vibrate in the same phase.
2. Particles in the neighbouring segments vibrate in opposite phases.
3. All the particles of the medium attain their maximum velocities simultaneously when they pass through their mean positions.
4. All the particles of the medium, except those at nodes, execute simple harmonic motion about their mean positions.

Q. The equation of a standing wave in a stretched string is given by $y=5\sin \left(\frac{\pi x}{3}\right)\cos \left(40\pi t\right)$, where $x$ and $y$ are in $\text{cm}$ and $t$ is in $\text{sec}$. The separation between two consecutive nodes is (in $\text{cm}$)

1. $1.5$
2. $3$
3. $6$
4. $4$

Q. what are the applications of resonance of waves?
2) how beats are formed?

Q. 67.If all the the particles of a body is situated at distanced from origin, then the distance of C.M. of the body from origin is. greater than or equal to d . Why

Q. A travelling wave pulse is given by equation,
$y=\frac{4}{3x^2+48t^2+24xt+2}$, where $x$ and $y$ are in metre and $t$ is in second. The wave velocity is

Q. Two identical transverse sinusoidal waves travel in opposite directions along a string. The speed of transverse waves in the string is $0.5\text{cm/s}$. Each has an amplitude of $3.0\text{cm}$ and wavelength of $6.0\text{cm}$. The equation for the resultant wave is

1. $y=6\sin \left(\frac{\pi t}{6}\right)\cos \left(\frac{\pi x}{3}\right)$
2. $y=6\sin \left(\frac{\pi x}{3}\right)\cos \left(\frac{\pi t}{6}\right)$
3. Both $\left(a\right)$ and $\left(b\right)$ may be correct
4. Both $\left(a\right)$ and $\left(b\right)$ are wrong

Q. The equation of a stationary wave is $y=0.8\text{cos}\left(\frac{\pi x}{20}\right)\text{sin}200\pi t$, where $x$ is in $\text{cm}$ and $t$ is in seconds. The separation between a successive node and antinode will be

1. $20\text{cm}$
2. $10\text{cm}$
3. $40\text{cm}$
4. $30\text{cm}$

Q. write down possible equation of a wave travelling towards +ve z axis with velocity and particles vibrating along x axis

Q. A standing wave is represented by $y=a\sin \left(0.05x\right)\cos \left(100t\right)$, where $t$ is in $\text{second}$ and $x,y$ are in $\text{meter}$, then the velocity of the constituent wave is

1. ${10}^2\text{m/s}$
2. ${10}^5\text{m/s}$
3. ${10}^3\text{m/s}$
4. $2\times {10}^3\text{m/s}$

Q. two particles are undergoing shm. at some instant they are at +A/2 and -A/2 and are moving (i)towards x=0 (ii)away from x=0. Find time taken by them to meet in each case