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A smooth inclined plane ends in a vertical circular loop, as shown in the figure. A small body is released from height \( h \) as shown. If the body exerts a force of three times its weight on the plane at the highest point of circle then the height \( h = \alpha R \). The value of \( \alpha \) is:

The fifth harmonic of a closed organ pipe is found to be in unison with the first harmonic of an open pipe. The ratio of lengths of closed pipe to that of the open pipe is \( \frac{5}{x} \). The value of \( x \) is __________ .

For a certain organ pipe, the first three resonance frequencies are in the ratio of \( 1:3:5 \) respectively. If the frequency of fifth harmonic is \( 405\,\text{Hz} \) and the speed of sound in air is \( 324\,\text{m}\,\text{s}^{-1} \) the length of the organ pipe is _____ \( \text{m} \)

The velocity of sound in a gas, in which two wavelengths \( 4.08 \, \text{m} \) and \( 4.16 \, \text{m} \) produce \( 40 \) beats in \( 12 \, \text{s} \), will be:

A guitar string of length \( 90 \, \text{cm} \) vibrates with a fundamental frequency of \( 120 \, \text{Hz} \). The length of the string producing a fundamental frequency of \( 180 \, \text{Hz} \) will be __________ \( \text{cm} \).

A plane progressive wave is given by \( y=2\cos 2\pi(330t-x)\, \text{m} \). The frequency of the wave is:

The first overtone frequency of an open organ pipe is equal to the fundamental frequency of a closed organ pipe. If the length of the closed organ pipe is \( 20 \, \text{cm} \). The length of the open organ pipe is _______ \( \text{cm} \).

Two strings \( (A, B) \) having linear densities \( \mu_A = 2 \times 10^{-4} \, \text{kg} / \text{m} \) and \( \mu_B = 4 \times 10^{-4} \, \text{kg} / \text{m} \) and lengths \( L_A = 2.5 \, \text{m} \) and \( L_B = 1.5 \, \text{m} \) respectively are joined. Free ends of \( A \) and \( B \) are tied to two rigid supports \( C \) and \( D \), respectively creating a tension of \( 500 \, \text{N} \) in the wire. Two identical pulses, sent from \( C \) and \( D \) ends, take time \( t_1 \) and \( t_2 \), respectively, to reach the joint. The ratio \( \frac{t_1}{t_2} \) is:

Two travelling waves produces a standing wave represented by equation.
\( y = 1.0 \, \text{mm} \cos\!\big(1.57 \, \text{cm}^{-1} \big)\,x \sin\!\big(78.5 \, \text{s}^{-1} t\big) \). The node closest to the origin in the region \( x > 0 \) will be at \( x = \) ________ \( \text{cm} \).

A transverse wave is represented by \( y = 2\sin(\omega t - kx) \; \text{cm} \). The value of wavelength (in \( \text{cm} \)) for which the wave velocity becomes equal to the maximum particle velocity, will be:

The fundamental frequency of a closed organ pipe is equal to the first overtone frequency of an open organ pipe. If length of the open pipe is \( 60 \, \text{cm} \), the length of the closed pipe will be:

A galaxy is moving away from the earth at a speed of \( 286 \, \text{km s}^{-1} \). The shift in the wavelength of a redline at \( 630 \, \text{nm} \) is \( x \times 10^{-10} \, \text{m} \).
[Take the value of speed of light \( c \), as \( 3 \times 10^{8} \, \text{m s}^{-1} \)]

In an experiment to determine the velocity of sound in air at room temperature using a resonance tube, the first resonance is observed when the air column has a length of \( 20.0 \, \text{cm} \) for a tuning fork of frequency \( 400 \, \text{Hz} \) is used. The velocity of the sound at room temperature is \( 336 \, \text{m s}^{-1} \). The third resonance is observed when the air column has a length of _______ \( \text{cm} \).

Which of the following equations represents a travelling wave?

Consider the sound wave travelling in ideal gases of \( \mathrm{He}, \mathrm{CH}_{4}, \) and \( \mathrm{CO}_{2} \). All the gases have the same ratio \( \frac{P}{\rho} \), where \( P \) is the pressure and \( \rho \) is the density. The ratio of the speed of sound through the gases \( v_{\mathrm{He}} : v_{\mathrm{CH}_{4}} : v_{\mathrm{CO}_{2}} \) is given by:

In the resonance experiment, two air columns (closed at one end) of \( 100 \, \text{cm} \) and \( 120 \, \text{cm} \) long, give \( 15 \) beats per second when each one is sounding in the respective fundamental modes. The velocity of sound in the air column is:

The distance between two consecutive points with phase difference of \( 60^{\circ} \) in a wave of frequency \( 500 \, \text{Hz} \) is \( 6.0 \, \text{m} \). the velocity with which wave is traveling is ___________ \( \text{km/s} \).

A transverse harmonic wave on a string is given by \( y(x,t)=5\sin(6t+0.003x) \) where \( x \) and \( y \) are in \( \text{cm} \) and \( t \) in sec. The wave velocity is __________ \( \text{m s}^{-1} \).

The amplitude of wave disturbance propagating in the positive \(x\)-direction is given by \(y = \dfrac{1}{(1+x)^{2}}\) at time \(t = 0\) and \(y = \dfrac{1}{1+(x-2)^{2}}\) at \(t = 1\,\text{s}\), where \(x\) and \(y\) are in metres. The shape of wave does not change during the propagation. The velocity of the wave will be _______________ \(\text{m/s}\).

A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is \( \left( \dfrac{a-1}{a} \right) \) then the value of \( a \) is ______.

A body of mass \( m \) is suspended by two strings making angles \( \theta_{1} \) and \( \theta_{2} \) with the horizontal ceiling with tensions \( T_{1} \) and \( T_{2} \) simultaneously. \( T_{1} \) and \( T_{2} \) are related by \( T_{1} = \sqrt{3}\, T_{2} \), the angles \( \theta_{1} \) and \( \theta_{2} \) are:

A wire of length \( 30 \, \text{cm} \), stretched between rigid supports, has its \( n^{\text{th}} \) and \( (n+1)^{\text{th}} \) harmonics at \( 400 \, \text{Hz} \) and \( 450 \, \text{Hz} \), respectively. If tension in the string is \( 2700 \, \text{N} \), its linear mass density is ______ \( \text{kg/m} \).

A car P travelling at \( 20 \, \text{m s}^{-1} \) sounds its horn at a frequency of \( 400 \, \text{Hz} \). Another car Q is travelling behind the first car in the same direction with a velocity \( 40 \, \text{m s}^{-1} \). The frequency heard by the passenger of the car Q is approximately. [Take: velocity of sound \( = 360 \, \text{m s}^{-1} \) ]

A closed organ pipe of length \( L \) and an open organ pipe contain gases of densities \( \rho_{1} \) and \( \rho_{2} \) respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The length of the open pipe is \( \dfrac{x}{3} L \sqrt{\dfrac{\rho_{1}}{\rho_{2}}} \) where \( x \) is ______. (Round off to the Nearest Integer)

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 ________ \( \text{Hz} \).

A tuning fork is vibrating at \( 250 \, \text{Hz} \). The length of the shortest closed organ pipe that will resonate with the tuning fork will be _______ \( \text{cm} \). (Take speed of sound in air as \( 340 \, \text{m s}^{-1} \))

An observer is riding on a bicycle and moving towards a hill at \( 18 \, \text{km h}^{-1} \). He hears a sound from a source at some distance behind him directly as well as after its reflection from the hill. If the original frequency of the sound as emitted by source is \( 640 \, \text{Hz} \) and velocity of the sound in air is \( 320 \, \text{m /s} \) the beat frequency between the two sounds heard by observer will be _________ \( \text{Hz} \).

The frequency of echo will be _______ \( \text{Hz} \) if the train blowing a whistle of frequency \( 320 \, \text{Hz} \) is moving with a velocity of \( 36 \, \text{km/h} \) towards a hill from which an echo is heard by the train driver. Velocity of sound in air is \( 330 \, \text{m/s} \).

An observer moves towards a stationary source of sound with a velocity equal to one-fifth of the velocity of sound. The percentage change in the frequency will be:

For a solid rod, the Youngs modulus of elasticity is \( 3.2 \times 10^{11} \, \text{N m}^{-2} \) and density is \( 8 \times 10^{3} \, \text{kg m}^{-3} \). The velocity of longitudinal wave in the rod will be:

In the wave equation
\( y = 0.5 \sin \dfrac{2\pi}{\lambda}(400t-x) \, \mathrm{m} \)
The velocity of the wave will be:

A tuning fork resonates with a sonometer wire of length \( 1 \, \text{m} \) stretched with tension of \( 6 \, \text{N} \). When the tension in the wire is changed to \( 54 \, \text{N} \), the same tuning fork produces \( 12 \) beats per second with it. The frequency of the tuning fork is ______ \( \text{Hz} \).

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

A longitudinal wave is represented by \( x = 10 \sin 2\pi\!\left(nt - \frac{x}{\lambda}\right) \) \( \text{cm} \). The maximum particle velocity will be four times the wave velocity if the determined value of wavelength is equal to:

Two strings with circular cross section and made of same material, are stretched to have same amount of tension. A transverse wave is then made to pass through both the strings. The velocity of the wave in the first string having the radius of cross section \( R \) is \( v_1 \), and that in the other string having radius of cross section \( R/2 \) is \( v_2 \). Then \( \frac{v_2}{v_1} = \)

The equation of a transverse wave travelling along a string is \( y(x,t) = 4.0 \sin \big( 20 \times 10^{-3} x + 600 t \big) \, \text{mm} \), where \( x \) is in \( \text{mm} \) and \( t \) is in second. The velocity of the wave is:

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 \( m \) is attached, the string vibrates with fundamental frequency of \( 50 \, \text{Hz} \). The value of \( m \) is __________ \( \text{g} \).

Displacement of a wave is expressed as \( x(t) = 5 \cos\!\left(628 t + \frac{\pi}{2}\right) \, \text{m} \). The wavelength of the wave when its velocity is \( 300 \, \text{m/s} \) is: \( (\pi = 3.14) \)

The speed of a transverse wave passing through a string of length \( 50\,\text{cm} \) and mass \( 10\,\text{g} \) is \( 60\,\text{m s}^{-1} \). The area of cross-section of the wire is \( 2.0\,\text{mm}^2 \) and its Young’s modulus is \( 1.2\times10^{11}\,\text{N m}^{-2} \). The extension of the wire over its natural length due to its tension will be \( x\times10^{-5}\,\text{m} \). The value of \( x \) is ______.

The equation of wave is given by \( Y = 10^{-2} \sin 2\pi(160t - 0.5x + \pi/4) \) where \( x \) and \( Y \) are in \( \text{m} \) and \( t \) in \( \text{s} \). The speed of the wave is ______ \( \text{km h}^{-1} \).

The amplitude and phase of a wave that is formed by the superposition of two harmonic travelling waves, \( y_1(x, t) = 4\sin(kx - \omega t) \) and \( y_2(x, t) = 2\sin(kx - \omega t + \frac{2\pi}{3}) \), are:

A transverse wave on a string is described by \( y = 3 \sin(36t + 0.018x + \pi/4) \), where \( x, y \) are in \( \text{cm} \) and \( t \) in seconds. The least distance between the two successive crests in the wave is_____ \( \text{cm} \). (Nearest integer)
(\( \pi = 3.14 \))

The frequency of a car horn encountered a change from \(400\,\text{Hz}\) to \(500\,\text{Hz}\), when the car approaches a vertical wall. If the speed of sound is \(330\,\text{m/s}\). Then the speed of car is ____________ \(\text{km/h}\).

A sound wave of frequency \( 245\,\text{Hz} \) travels with the speed of \( 300\,\text{m}\,\text{s}^{-1} \) along the positive \( x \)-axis. Each point of the wave moves to and fro through a total distance of \( 6\,\text{cm} \). What will be the mathematical expression of this traveling wave?

The equation of a wave travelling on a string is \( y = \sin[20\pi x + 10\pi t] \), where \( x \) and \( t \) are distance and time in SI units. The minimum distance between two points having the same oscillating speed is:

Two harmonic waves moving in the same direction superimpose to form a wave \( x = a \cos(1.5\, t)\cos(50.5\, t) \) where \( t \) is in seconds. Find the period with which they beat. (close to nearest integer):

A closed organ pipe \( 150 \, \text{cm} \) long gives \( 7 \) beats per second with an open organ pipe of length \( 350 \, \text{cm} \), both vibrating in fundamental mode. The velocity of sound is __________ m/s.

A sinusoidal wave of wavelength \( 7.5 \, \text{cm} \) travels a distance of \( 1.2 \, \text{cm} \) along the \( x \)-direction in \( 0.3 \, \text{sec} \). The crest \( P \) is at \( x = 0 \) at \( t = 0 \, \text{sec} \) and maximum displacement of the wave is \( 2 \, \text{cm} \). Which equation correctly represents this wave?

Two cars are approaching each other at an equal speed of \( 7.2 \, \text{km/hr} \). When they see each other, both blow horns having frequency of \( 676 \, \text{Hz} \). The beat frequency heard by each driver will be ________ \( \text{Hz} \). \([\text{velocity of sound in air is } 340 \, \text{m/s}]\)

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : A sound wave has higher speed in solids than gases.
Reason R : Gases have higher value of Bulk modulus than solids.
In the light of the above statements, choose the correct answer from the options given below:

The cars X and Y are approaching each other with velocities \( 36 \, \text{km/h} \) and \( 72 \, \text{km/h} \) respectively. The frequency of a whistle sound as emitted by a passenger in car X, heard by the passenger in car Y is \( 1320 \, \text{Hz} \). If the velocity of sound in air is \( 340 \, \text{m/s} \), the actual frequency of the whistle sound produced is ______ \( \text{Hz} \).

The speed of sound in oxygen at S.T.P. will be approximately:
(given, \( R = 8.3 \, \text{J} \, \text{K}^{-1} ,\; \gamma = 1.4 \))

In a closed organ pipe, the frequency of fundamental note is \( 30 \, \text{Hz} \). A certain amount of water is now poured in the organ pipe so that the fundamental frequency is increased to \( 110 \, \text{Hz} \). If the organ pipe has a cross-sectional area of \( 2 \, \text{cm}^2 \), the amount of water poured in the organ tube is ______ \( \text{g} \). (Take speed of sound in air is \( 330 \, \text{m/s} \))

A steel wire with mass per unit length \( 7.0 \times 10^{-3} \ \text{kg} \ \text{m}^{-1} \) is under tension of \( 70 \ \text{N} \). The speed of transverse wave in the wire will be:

A train blowing a whistle of frequency \( 320 \, \text{Hz} \) approaches an observer standing on the platform at a speed of \( 66 \, \text{m/s} \). The frequency observed by the observer will be (given speed of sound \( = 330 \, \text{m s}^{-1} \)) ______ \( \text{Hz} \).

The fundamental frequency of vibration of a string stretched between two rigid support 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 __________ \( \text{ms}^{-1} \).

Two open organ pipes of lengths \( 60 \, \text{cm} \) and \( 90 \, \text{cm} \) resonate at \( 6^{\text{th}} \) and \( 5^{\text{th}} \) harmonics respectively. The difference of frequencies for the given modes is _______ \( \text{Hz} \).
(Velocity of sound in air \( = 333 \, \text{m/s} \))

Two waves are simultaneously passing through a string and their equations are: \( y_1 = A_1 \sin k (x - v t) \), \( y_2 = A_2 \sin k (x - v t + x_0) \). Given amplitudes \( A_1 = 12 \, \text{mm} \) and \( A_2 = 5 \, \text{mm} \), \( x_0 = 3.5 \, \text{cm} \) and wave number \( k = 6.28 \, \text{cm}^{-1} \). The amplitude of resulting wave will be __________ mm.

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 equation of a plane progressive wave is given by \( y = 5 \cos \pi \left( 200 t - \frac{x}{150} \right) \) where \( x \) and \( y \) are in \( \text{cm} \) and \( t \) is in second. The velocity of the wave is ______ \( \text{m}/\text{s} \).

A person driving car at a constant speed of \( 15 \, \text{m/s} \) is approaching a vertical wall. The person notices a change of \( 40 \, \text{Hz} \) in the frequency of his car's horn upon reflection from the wall. The frequency of horn is _________ \( \text{Hz} \). (Given: Speed of sound : \( 330 \, \text{m/s} \))

A set of \( 20 \) tuning forks is arranged in a series of increasing frequencies. If each fork gives \( 4 \) beats with respect to the preceding fork and the frequency of the last fork is twice the frequency of the first, then the frequency of last fork is _______ \( \text{Hz} \).

A point source is emitting sound waves of intensity \( 16 \times 10^{-8}\ \text{W}\ \text{m}^{-2} \) at the origin. The difference in intensity (magnitude only) at two points located at a distance of \( 2\,\text{m} \) and \( 4\,\text{m} \) from the origin respectively will be ______ \( \times 10^{-8}\ \text{W}\ \text{m}^{-2} \).

A closed organ and an open organ tube are filled by two different gases having same bulk modulus but different densities \( \rho_1 \) and \( \rho_2 \), respectively. The frequency of \( 9^{\text{th}} \) harmonic of closed tube is identical with \( 4^{\text{th}} \) harmonic of open tube. If the length of the closed tube is \( 10 \, \text{cm} \) and the density ratio of the gases is \( \rho_1 : \rho_2 = 1 : 16 \), then the length of the open tube is:

A source and a detector move away from each other in absence of wind with a speed of \( 20 \, \text{m /s}^{} \) with respect to the ground. If the detector detects a frequency of \( 1800 \, \text{Hz} \) of the sound coming from the source, then the original frequency of source considering speed of sound in air \( 340 \, \text{m /s}^{} \) will be ______ \( \text{Hz} \).

If \( \vec{L} \) and \( \vec{P} \) represent the angular momentum and linear momentum respectively of a particle of mass ‘\( m \)’ having position vector as \( \vec{r} = a \left( \hat{i} \cos \omega t + \hat{j} \sin \omega t \right) \). The direction of force is:

In an open organ pipe \( \nu_3 \) and \( \nu_6 \) are \( 3^{\text{rd}} \) and \( 6^{\text{th}} \) harmonic frequencies, respectively. If \( \nu_6 - \nu_3 = 2200 \,\text{Hz} \) then length of the pipe is ________ \( \text{mm} \). (Take velocity of sound in air is \( 330 \,\text{m/s} \).)

In young’s double slit experiment performed using a monochromatic light of wavelength \( \lambda \), when a glass plate \( ( \mu = 1.5 ) \) of thickness \( x \lambda \) is introduced in the path of the one of the interfering beams, the intensity at the position where the central maximum occurred previously remains unchanged. The value of \( x \) will be

A point source is kept at the center of a spherically enclosed detector. If the volume of the detector increased by \( 8 \) times, the intensity will:

The equations of two waves are given by: \( y_{1} = 5 \sin 2\pi(x - vt) \, \text{cm} \) ; \( y_{2} = 3 \sin 2\pi(x - vt + 1.5) \, \text{cm} \) These waves are simultaneously passing through a string. The amplitude of the resulting wave is:

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JEE Main 2023 1st Feb Morning Shift

A steel wire with mass per unit length \( 7.0 \times 10^{-3} \ \text{kg} \ \text{m}^{-1} \) is under tension of \( 70 \ \text{N} \). The speed of transverse wave in the wire will be: