Stationary Waves & Vibrating Strings

For a stationary wave, $y = 10\cos(\pi x/15)\sin(48\pi t)$ cm, the distance between a node and the successive antinode is

The equation of a stationary wave along a stretched string is $y = 5\sin\frac{\pi x}{3}\cos 40\pi t$ where $x$ and $y$ are in cm and $t$ in second. The separation between two adjacent nodes is

For stationary wave, $y = 12\cos\left(\frac{\pi x}{10}\right)\sin(36\pi t)$ cm, the distance between a node and the successive antinode is

Two waves of same frequency ($n$) approaching each other with same velocity of $18$ m/s interfere. The distance between two consecutive antinodes is

In case of stationary wave pattern, which of the following statement is CORRECT?

In a stationary wave, all the particles

A stretched string of length '$L$' is fixed at both the ends, has '$n$' nodes ($n > 2$). The length of the string in terms of $n$ and wavelength $\lambda$ is

The fundamental frequency '$n$' of the tuning fork is 288 Hz, it will not resonate with the frequency

When a sonometer wire vibrates in the third overtone, the number of antinodes and nodes formed on the wire are respectively

A sonometer wire is vibrating in third overtone. There are

A uniform wire of length $L$, diameter $D$ and density $\rho$ is stretched under tension $T$. The relation between its fundamental frequency $f$, length $L$ and diameter $D$ is

To increase the frequency of transverse oscillations of a stretched string by 40%, the tension must be increased by

A stretched wire of length $110$ cm is divided in to 3 segments whose frequencies are in the ratio $1:2:3$. Their respective lengths must be

A spring of mass $0.2$ kg/m has length $l = 0.6$ m. It is fixed at both ends and stretched such that is has a tension $80$ N. The string vibrates in three segments with amplitude $0.5$ cm. The amplitude of transverse (particle) velocity is

In sonometer experiment, the string of length $'L'$ under tension vibrates in second overtone between the two bridges. The amplitude of vibration is maximum at length

A wire of length 'L' and linear density 'm' is stretched between two rigid supports with tension 'T'. It is observed that wire resonates in the $P^{th}$ harmonic at a frequency of 320 Hz and resonates again at next higher frequency of 400 Hz in two successive modes. The value of 'P' is

Two vibrating metal wires of the same material of length 'L' and '3L' and have radii '2r' and 'r' respectively. They are stretched under same tension. Both the wires vibrate in their fundamental modes. The ratio $\frac{n_1}{n_2}$ is [$n_1$ and $n_2$ are the frequencies of first and second wire respectively]

Two uniform strings A and B made of steel are vibrating under the same tension. If the first overtone of A is equal to the second overtone of B and if the radius of A is twice that of B, the ratio of the lengths of the strings is

A wire of length '$L$', diameter '$d$', density of material '$\rho$' is under tension '$T$' has fundamental frequency of vibration '$n_A$'. Another wire of length '$2L$', diameter '$3d$', density of material '$2\rho$' is vibrated under tension '$2T$', the fundamental frequency of vibration becomes '$n_B$'. The ratio $n_B : n_A$ is

The fundamental frequency of sonometer wire is $50$ Hz for some length and tension. If the length is increased by $25\%$ by keeping the tension same, then the frequency change of second harmonic is

Two uniform wires of same material are vibrating under the same tension. If the first overtone of the first wire is equal to the second overtone of the second wire and radius of first wire is twice the radius of second wire then the ratio of length of first to second wire is

A sonometer wire resonates with a given tuning fork, forming standing waves with five antinodes between the two bridges, when a mass of $9$ kg is suspended from the wire. When this mass is replaced by mass $M$, the wire resonates with the same tuning fork, forming three antinodes for the same positions of the bridges. The value of $M$ (in kg) is

A sonometer wire resonates with a given tuning fork forming standing wave with $5$ antinodes between two bridges when mass of $9$ kg is suspended from the wire. When mass $M$ is suspended from the wire with same fork and same length between two bridges, $3$ antinodes are formed. Mass $M$ is

In a sonometer experiment, the fundamental frequency of vibration of wire is '$n_1$' when wire is stretched by hanging a metal bob. If the bob is completely immersed in water, the frequency of vibration of wire becomes '$n_2$'. The relative density of the metal of the bob is

A sonometer wire resonates with $4$ antinodes between the two bridges for a given tuning fork when $1$ kg mass is suspended from the wire. Using same fork, when mass M is suspended, the wire resonates producing $2$ antinodes between the two bridges. (Distance between the bridges as before) The value of M is

Two waves of same frequency ( n) are approaching each other with same velocity $12\text{ m/s}$ along the same linear path and interfere. The distance between two consecutive nodes is

In Sonometer experiment, the frequency of a tuning fork used is 288 Hz. Harmonics will 'NOT' be produced at the frequency

The frequency of a tuning fork is 256 Hz. It will not resonate with the tuning fork of frequency

Fundamental frequency of sonometer wire is ' $n$ '. If the tension and length are increased 3 times and diameter is increased twice, the new frequency will be

The fundamental frequency of sonometer wire is ' $n$ '. If the tension and length are increased 3 times and diameter is increased twice, the new frequency will be

A string of mass per unit length $0.1\text{ kg m}^{-1}$ has length 0.9 m. It is fixed at both ends and stretched such that it has a tension of 40 N. The string vibrates in three segments with amplitude 0.3 cm. The amplitude (maximum) of the particle velocity is (in $\text{m/s}$)

The fundamental frequency of a sonometer wire is 50 Hz for some length and tension. If the length is increased by $25\%$ by keeping tension same then frequency change of second harmonic is

Two uniform wires of same material are vibrating under the same tension. If the $1$st overtone of $1$st wire is equal to the $2$nd overtone of $2$nd wire and radius of $1$st wire is twice the radius of $2$nd wire, the ratio of length of $1$st wire to that $2$nd wire is

A wire of length L, diameter ' d ' density of material ' e ' is under tension ' T ', having fundamental frequency of vibration $n_A$. Another wire of length 2 L, tension 2 T, density 2 e and diameter 3 d has fundamental frequency of vibration $n_B$. The ratio $n_B : n_A$ is

At poles, a stretched wire of a given length vibrates in unison with a tuning fork. At the equator, for same setting, to produce resonance with same fork, the vibrating length of wire

A stationery wave is represented by $y = 12\cos\left(\frac{\pi}{6}x\right)\sin(8\pi t)$, where $x \,\&\, y$ are in cm and $t$ in second. The distance between two successive antinodes is

When the string is stretched between two rigid supports, under certain tension and vibrated

In a vibrating string with fixed ends the waves are of type

When a sonometer wire vibrates in third overtone there are

Stationary wave is produced along the stretched string of length 80 cm. The resonant frequencies of string are $90\text{ Hz}, 150\text{ Hz}$ and 210 Hz. The speed of transverse wave in the string is

A stationary wave is formed having 3 nodes along the length of the string 90 cm. The wavelength of the wave is

A stretched string is fixed at both ends. It is made to vibrate so that the total number of nodes formed in it is ' $x$ '. The length of the string in terms of the wavelength of waves formed in it is ( $\lambda =$ wavelength $)$

When the tension in string is increased by $3\text{ kg wt}$, the frequency of the fundamental mode increases in the ratio $2:3$. The initial tension in the string is

A violin emits sound waves of frequency ' $n_1$ ' under tension T. When tension is increased by $44\%$, keeping the length and mass per unit length constant, frequency of sound waves becomes ' $n_2$ '. The ratio of frequency ' $n_2$ ' to frequency ' $n_1$ ' is

A sonometer wire of resonating length $90\text{ cm}$ has a fundamental frequency of $400\text{ Hz}$ when kept under some tension. The resonating length of the wire with fundamental frequency of $600\text{ Hz}$ under same tension _______ $\text{cm}$.

The length of a sonometer wire 'AB' is 110 cm. Where should the two bridges be placed from end ' $A$ ' to divide the wire in three segments whose fundamental frequencies are in the ratio $1:2:3$?

Two progressive waves $Y_1 = \sin 2\pi\left(\frac{t}{0.4} - \frac{x}{4}\right)$ and $Y_2 = \sin 2\pi\left(\frac{t}{0.4} + \frac{x}{4}\right)$ superpose to form a standing wave. ' $x$ ' and ' $y$ ' are in SI system. Amplitude of the particle at $x = 0.5\text{ m}$ is $\left[\sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}}\right]$

A string is under tension of 180 N and mass per unit length $2 \times 10^{-3}\text{ kg/m}$. It produces two consecutive resonant frequencies with a tuning fork, which are 375 Hz and 450 Hz. The mass of the string is

A wire of length ' $L$ ' and linear density ' $m$ ' is stretched between two rigid supports with tension ' $T$ '. It is observed that wire resonates in the $P$th harmonic at a frequency of 320 Hz and resonates again at next higher frequency of 400 Hz. The value of ' $p$ ' is

Two uniform strings A and B made of steel are made to vibrate under the same tension. If first overtone of A is equal to the second overtone of $B$ and if the radius of $A$ is twice that of $B$, the ratio of the length of string $B$ to that of $A$ is

A string A has twice the length, twice the diameter, twice the tension and twice the density of another string B. The overtone of A which will have the same fundamental frequency as that of $B$ is

At the poles of earth, a stretched wire of a given length vibrates in unison with a tuning fork. At the equator of earth, for same setting, to produce resonance with same fork, the vibrating length of wire

A sonometer wire is in unison with a tuning fork of frequency ' $n$ ' when it is stretched by a weight of specific gravity ' $d$ '. When the weight is completely immersed in water, ' $x$ ' beats are produced per second, then

A sonometer wire is stretched by hanging a metal bob, the fundamental frequency of the wire is ' $n_1$ '. When the bob is completely immersed in water, the frequency of vibration of wire becomes ' $n_2$ '. The relative density of the metal of the bob is

A sonometer wire is stretched by hanging a metal bob. The fundamental frequency of vibration of wire is ' $n_1$ '. When the bob is completely immersed in water, the frequency of vibration of wire becomes ' $n_2$ '. The relative density of the metal of the bob is

Two progressive waves are travelling towards each other with velocity $50\text{ m/s}$ and frequency $200\text{ Hz}$. The distance between the two consecutive antinodes is

A string fixed at both the ends forms standing wave with node separation of $5\text{ cm}$. If the velocity of the wave on the string is $2\text{ m/s}$, then the frequency of vibration of the string is

In case of a stationary wave pattern which of the following statements is CORRECT?

Stationary waves can be produced in

A guitar string of length 90 cm vibrates with a fundamental frequency of 120 Hz. The length of the string producing a fundamental frequency of 180 Hz will be _________ cm.

When a string of length '$l$' is divided into three segments of length $l_1, l_2$ and $l_3$. The fundamental frequencies of three segments are $n_1, n_2$ and $n_3$ respectively. The original fundamental frequency '$n$' of the string is

Sound waves of frequency $600\text{ Hz}$ fall normally on a perfectly reflecting wall. The shortest distance from the wall at which all particles will have maximum amplitude of vibration is (speed of sound $= 300\text{ m s}^{-1}$)

A string is stretched between two rigid supports separated by $75\text{ cm}$. There are no resonant frequencies between $420\text{ Hz}$ and $315\text{ Hz}$. The lowest resonant frequency for the string is

The fundamental frequency of vibration of a string stretched between two rigid supports is $50\text{ Hz}$. The mass of the string is $18\text{ g}$ and its linear mass density is $20\text{ g/m}$. The speed of the transverse waves so produced in the string is ___________ $\text{m s}^{-1}$

A uniform string is vibrating with a fundamental frequency '$n$'. If radius and length of string both are doubled keeping tension constant then the new frequency of vibration is

A sonometer wire '$A$' of diameter '$d$' under tension '$T$' having density '$\rho_1$' vibrates with fundamental frequency '$n$'. If we use another wire '$B$' which vibrates with same frequency under tension '$2T$' and diameter '$2d$' then density '$\rho_2$ ' of wire '$B$' will be

Two uniform wires of same material are vibrating under the same tension. If the first overtone of first wire is equal to the $2$nd overtone of $2$nd wire and radius of the first wire is twice the radius of the $2$nd wire then the ratio of length of first wire to $2$nd wire is

If the length of stretched string is reduced by $40\%$ and tension is increased by $44\%$ then the ratio of final to initial frequencies of stretched string is

A wire of density $8 \times 10^{3}\text{ kg/m}^3$ is stretched between two clamps $0.5\text{ m}$ apart. The extension developed in the wire is $3.2 \times 10^{-4}\text{ m}$. If $Y = 8 \times 10^{10}\text{ N/m}^2$, the fundamental frequency of vibration in the wire will be ___________ $\text{Hz}$.

In an experiment with sonometer when a mass of $180\text{ g}$ is attached to the string, it vibrates with fundamental frequency of $30\text{ Hz}$. When a mass $\text{m}$ is attached, the string vibrates with fundamental frequency of $50\text{ Hz}$. The value of $\text{m}$ is ___________ g.

A stationary wave is represented by $y = 10\sin\left(\frac{\pi x}{4}\right)\cos(20\pi t)$ where $x$ and $y$ are in $\text{cm}$ and $t$ in second. The distance between two consecutive nodes is

Two travelling waves of equal amplitudes and equal frequencies move in opposite directions along a string. They interfere to produce a stationary wave whose equation is given by $y = \left(10\cos\pi x\sin\frac{2\pi t}{T}\right)$ cm

The amplitude of the particle at $x = \frac{4}{3}$ cm will be ___________ cm.

Consider the following statements about stationary waves.

A. The distance between two adjacent nodes or antinodes is equal to $\frac{\lambda}{2}(\lambda =$ wavelength of the wave)

B. A node is always formed at the open end of the open organ pipe.

Choose the correct option from the following.

A wire of length 30 cm, stretched between rigid supports, has it's nth and (n + 1)th harmonics at 400 Hz and 450 Hz, respectively. If tension in the string is 2700 N, it's linear mass density is ____________ kg/m.

Two travelling waves produce a standing wave represented by equation,

y = 1.0 mm cos(1.57 cm${}^{-1}$) x sin(78.5 s${}^{-1}$)t.

The node closest to the origin in the region x > 0 will be at x = .............. cm.

A wire having a linear mass density 9.0 $\times$ 10$-$4 kg/m is stretched between two rigid supports with a tension of 900 N. The wire resonates at a frequency of 500 Hz. The next higher frequency at which the same wire resonates is 550 Hz. The length of the wire is ____________ m.

The length and diameter of a metal wire used in sonometer is doubled. The fundamental frequency will change from 'n' to

A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of $9\text{ kg}$ is suspended from the wire. When this mass is replaced by a mass $M$, the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. The value of '$M$' is

A sonometer wire resonates with 4 antinodes between two bridges for a given tuning fork, when 1 kg mass is suspended from the wire. Using same fork, when mass M is suspended, the wire resonates producing 2 antinodes between the two bridges (distance between two bridges is as before). The value of M is

Two wires of same material of radius 'r' and '2r' respectively are welded together end to end. The combination is then used as a sonometer wire under tension 'T'. The joint is kept midway between the two bridges. The ratio of the number of loops formed in the wires such that the joint is a node is

The length of the string of a musical instrument is 90 cm and has a fundamental frequency of 120 Hz. Where should it be pressed to produce fundamental frequency of 180 Hz?

Two identical strings X and Z made of same material have tension TX and TZ in them. If their fundamental frequencies are 450 Hz and 300 Hz, respectively, then the ratio TX/TZ is

A wire of length L and mass per unit length 6.0 × 10–3 kgm–1 is put under tension of 540 N. Two consecutive frequencies that it resonates at are: 420 Hz and 490 Hz. Then L in meters is:

Two identical strings of length $l$ and $2l$ vibrate with fundamental frequencies $N\text{ Hz}$ and $1.5N$ Hz, respectively. The ratio of tensions for smaller length to large length is

A wire of density 9 $\times$ 10–3 kg cm–3 is stretched between two clamps 1 m apart. The resulting strain in the wire is 4.9 $\times$ 10–4. The lowest frequency of the transverse vibrations in the wire is: (Young’s modulus of wire Y = 9 $\times$ 1010 Nm–2), (to the nearest integer), _________

At the poles, a stretched wire of a given length vibrates in unison with a tuning fork. At the equator, for same setting to produce resonance with same fork, the vibrating length of wire

A sonometer wire under suitable tension having specific gravity $\rho$, vibrates with frequency $n$ in air. If the load is completely immersed in water the frequency of vibration of wire will become

When tension $T$ is applied to a sonometer wire of length $l$, it vibrates with the fundamental frequency $n$. Keeping the experimental setup same, when the tension is increased by 8 N, the fundamental frequency becomes three times the earlier fundamental frequency $n$. The initial tension applied to the wire (in newton) was

Find the wrong statement from the following about the equation of stationary wave given by $Y = 0.04\cos(\pi x)\sin(50\pi t)\text{ m}$ where $t$ is in second. Then for the stationary wave.

A stretched string fixed at both ends has ' $m$ ' nodes, then the length of the string will be

A string 2.0 m long and fixed at its ends is driven by a 240 Hz vibrator. The string vibrates in its third harmonic mode. The speed of the wave and its fundamental frequency is :-

A stretched wire of length 260 cm is set into vibrations. It is divided into three segments whose frequencies are in the ratio $2:3:4$. Their lengths must be

A string is clamped at both the ends and it is vibrating in its 4th harmonic. The equation of the stationary wave is Y = 0.3 sin(0.157x) cos(200$\pi$t). The length of the string is: (All quantities are in SI units.)

A string of length 1 m and mass 5 g is fixed at both ends. The tension in the string is 8.0 N. The string is set into vibration using an external vibrator of frequency 100 Hz. The separation between successive nodes on the string is close to -

The fundamental frequency of sonometer wire increases by 9 Hz, if its tension is increased by $69\%$, keeping the length constant. The frequency of the wire is

A sonometer wire is in unison with a tuning fork, when it is stretched by weight $w$ and the corresponding resonating length is $L_1$. If the weight is reduced to $\left(\frac{w}{4}\right)$, the corresponding resonating length becomes $L_2$. The ratio $\left(\frac{L_1}{L_2}\right)$ is

A wire of length 2L, is made by joining two wires A and B of same length but different radii r and 2r and made of the same material. It is vibrating at a frequency such that the joint of the two wires forms a node. If the number of antinodes in wire A is p and that in B is q then the ratio p: q is:

A standing wave is formed by the superposition of two waves travelling in opposite directions. The transverse displacement is given by

y(x, t) = 0.5 sin $\left(\frac{5\pi}{4}x\right)$ cos(200 $\pi$t).

What is the speed of the travelling wave moving in the positive x direction?

(x and t are in meter and second, respectively.)

Two wires W1 and W2 have the same radius r and respective densities $\rho$1 and $\rho$2 such that ρ2 = 4$\rho$1. They are joined together at the point O, as shown in the figure. The combination is used as a sonometer wire and kept under tension T. The point O is midway between the two bridges. When a stationary wave is set up in the composite wire, the joint is found to be a node. The ratio of the number of antinodes formed in W1 to W2 is:

A string is stretched between fixed points separated by 75.0 cm. It is observed to have resonant frequencies of 420 Hz and 315 Hz. There are no other resonant frequencies between these two. The lowest resonant frequency for this string is

If n1, n2 and n3 are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency n of the string is given by

The total length of a sonometer wire between fixed ends is 110 cm. Two bridges are placed to divide the length of wire in ratio 6: 3: 2. The tension in the wire is 400 N and the mass per unit length is 0.01 kg/m. What is the minimum common frequency with which three parts can vibrate?

The length of the wire between two ends of a sonometer is 100 cm. What should be the positions of two bridges below the wire so that the three segments of the wire have their fundamental frequencies in the ratio 1: 3: 5.

A sonometer wire of length 114 cm is fixed at both the ends. Where should the two bridges be placed so as to divide the wire into three segments whose fundamental frequencies are in the ratio 1: 3: 4?