Oscillations and Waves PYQs

For a simple pendulum having time period T, the variation of kinetic energy (K.E.) with time (t) is represented by:

The sum of kinetic energy and potential energy of a simple pendulum bob is 0.02 joule. The speed of the simple pendulum bob at equilibrium position is approximately: (Consider mass of the bob = 20 g)

For a travelling harmonic wave y(x,t)=2.0 cos 2π(10t−0.0080x+0.35), where x and y are in cm and t in s. The phase difference between oscillatory motion of two points separated by a distance of 0.5 m is:

Savitha, a XI standard student, while conducting an experiment to determine the effective length of a simple pendulum L, notes down the data of time taken to complete 30 oscillations as 60 s and hence calculates the length of the simple pendulum as: (Take π² = 9.8 and g = 9.8 m/s²)

In an oscillating spring mass system, a spring is connected to a box filled with sand. As the box oscillates, sand leaks slowly out of the box vertically so that the average frequency $\omega(t)$ and average amplitude $A(t)$ of the system change with time $t$. Which one of the following options schematically depicts these changes correctly?

Two identical point masses P and Q, suspended from two separate massless springs of spring constants $k_1$ and $k_2$, respectively, oscillate vertically. If their maximum speeds are the same, then the ratio $(A_Q/A_P)$ of the amplitude $A_Q$ of the mass Q to the amplitude $A_P$ of mass P is:

A pipe open at both ends has a fundamental frequency $f$ in air. The pipe is now dipped vertically in a water drum to half of its length. The fundamental frequency of the air column is now equal to:

If 𝑥 = 5 sin(πt + π/3) m represents the motion of a particle executing simple harmonic motion, the amplitude and time period of motion, respectively, are:

If the mass of the bob in a simple pendulum is increased to thrice its original mass and its length is made half its original length, then the new time period of oscillation is 𝑥/2 times its original time period. Then the value of 𝑥 is:

The potential energy of a long spring when stretched by 2 cm is U. If the spring is stretched by 8 cm, potential energy stored in it will be :

The ratio of frequencies of fundamental harmonic produced by an open pipe to that of closed pipe having the same length is :

The x-t graph of a particle performing simple harmonic motion is shown in the figure. The acceleration of the particle at t = 2 s is :

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

Two pendulums of length 121 cm and 100 cm start vibrating in phase. At some instant, the two are at their mean position in the same phase. The minimum number of vibrations of the shorter pendulum after which the two are again in phase at the mean position is:

A body is executing simple harmonic motion with frequency 'n', the frequency of its potential energy is

A spring is stretched by 5 cm by a force of 10 N. The time period of oscillations when a mass of 2 kg suspended by it is:

The phase difference between displacement and acceleration of a particle in a simple harmonic motion is :

In a guitar, two strings A and B made of same material are slightly out of tune and produce beats of frequency 6 Hz. When tension in B is slightly decreased, the beat frequency increases to 7 Hz. If the frequency of A is 530 Hz, the original frequency of B will be :

The distance covered by a particle undergoing SHM in one time period is (amplitude = $A$)

A mass falls from a height $h$ and its time of fall $t$ is recorded in terms of time period $T$ of a simple pendulum. On the surface of earth it is found that $t = 2T$. The entire set up is taken to the surface of another planet whose mass is half of that of earth and radius the same. Same experiment is repeated and corresponding times noted as $t'$ and $T'$. Then we can say

A tuning fork with frequency 800 Hz produces resonance in a resonance column tube with upper end open and lower end closed by water surface. Successive resonances are observed at lengths 9.75 cm, 31.25 cm and 52.75 cm. The speed of sound in air is:

A tuning fork is used to produce resonance in a glass tube. The length of the air column in this tube can be adjusted by a variable piston. At room temperature of 27°C two successive resonances are obtained at 20 cm and 73 cm column length. If the frequency of the tuning fork is 320 Hz, the velocity of sound in air at 27°C is:

A pendulum is hung from the roof of a sufficiently high building and is moving freely to and fro like a simple harmonic oscillator. The acceleration of the bob of the pendulum is $20\text{ m/s}^2$ at a distance of $5\text{ m}$ from the mean position. The time period of oscillation is:

The fundamental frequency in an open organ pipe is equal to the third harmonic of a closed organ pipe. If the length of the closed organ pipe is 20 cm, the length of the open organ pipe is:

A spring of force constant $k$ is cut into lengths of ratio $1:2:3$. They are connected in series and the new force constant is $k'$. Then they are connected in parallel and force constant is $k''$. Then $k' : k''$ is

The two nearest harmonics of a tube closed at one end and open at the other end are $220\,\text{Hz}$ and $260\,\text{Hz}$. What is the fundamental frequency of the system?

A particle executes linear simple harmonic motion with amplitude $3\,\text{cm}$. When the particle is at a distance $2\,\text{cm}$ from the mean position, the magnitude of its velocity is equal to that of its acceleration. The time period of the motion is:

Two cars moving in opposite directions approach each other with speeds of $22\,\text{m/s}$ and $16.5\,\text{m/s}$ respectively. The driver of the first car blows a horn having frequency $400\,\text{Hz}$. The frequency heard by the driver of the second car is: (Velocity of sound $= 340\,\text{m/s}$)

A body of mass $m$ is attached to the lower end of a spring whose upper end is fixed. The spring has negligible mass. When the mass is slightly pulled down and released, it oscillates with a time period of $3\ \text{s}$. When the mass is increased by $1\ \text{kg}$, the time period of oscillations becomes $5\ \text{s}$. The value of $m$ in kg is:

The second overtone of an open organ pipe has the same frequency as the first overtone of a closed pipe of length $L$. The length of the open pipe will be:

Three sound waves of equal amplitudes have frequencies $(n-1)$, $n$, and $(n+1)$. They superimpose to give beats. The number of beats produced per second will be:

A particle is executing a simple harmonic motion. Its maximum acceleration is $\alpha$ and maximum velocity is $\beta$. Then, its time period of vibration will be:

A source of sound $S$ emitting waves of frequency $100\,\text{Hz}$ and an observer $O$ are located at some distance from each other. The source is moving with a speed of $19.4\,\text{m s}^{-1}$ at an angle of $60^{\circ}$ with the source-observer line as shown in the figure. The observer is at rest. The apparent frequency observed by the observer (velocity of sound in air $330\,\text{ms}^{-1}$) is:

The position vector of a particle $\vec{R}$ as a function of time is given by: $$\vec{R}=4\sin(2\pi t)\,\hat{i}+4\cos(2\pi t)\,\hat{j}$$ where R is in metres, t is in seconds and î and ĵ denote unit vectors along the x and y directions respectively. Which one of the following statements is wrong for the motion of the particle?

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

The oscillation of a body on a smooth horizontal surface is represented by the equation: $$x = A\cos(\omega t)$$ where: $$x = \text{displacement at time } t$$ $$\omega = \text{angular frequency of oscillation}$$ Which one of the following graphs shows correctly the variation of acceleration 'a' with time 't'?

If $n_1$, $n_2$ and $n_3$ 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 number of possible natural oscillations of an air column in a pipe closed at one end of length $85\,\text{cm}$ whose frequencies lie below $1250\,\text{Hz}$ are: (velocity of sound = 340 ms⁻¹)

A speeding motorcyclist sees traffic jam ahead of him. He slows down to 36 km/hour. He finds that traffic has eased and a car moving ahead of him at 18 km/hour is honking at a frequency of 1392 Hz. If the speed of sound is 343 m/s, the frequency of the horn as heard by him will be:

A wave travelling in the positive x-direction having displacement along y-direction as $1\,m$, wavelength $2\pi\,m$ and frequency $\dfrac{1}{\pi}\,Hz$ is represented by

If we study the vibration of a pipe open at both ends, then the following statement is not true

A source of unknown frequency gives $4$ beats/s, when sounded with a source of known frequency $250\,\text{Hz}$. The second harmonic of the source of unknown frequency gives five beats per second, when sounded with a source of frequency $513\,\text{Hz}$. The unknown frequency is

A train moving at a speed of 220 ${m s}^{-1}$ towards a stationary object emits a sound of frequency $1000\,\text{Hz}$. Some of the sound reaching the object gets reflected back to the train as echo. The frequency of the echo as detected by the driver of the train is: (Speed of sound in air is 330 ${m s}^{-1}$)

The equation of simple harmonic wave is given by: $$y=3\sin\left[\frac{\pi}{2}(50t-x)\right]$$ where x and y are in metres and t is in seconds. The ratio of maximum particle velocity to the wave velocity is:

Two identical piano wires kept under the same tension T have a fundamental frequency of 600 Hz. The fractional increase in the tension of one of the wires which would lead to occurrence of 6 beats/second when both the wires oscillate together would be:

Two particles are oscillating along two close parallel straight lines side by side, with the same frequency and amplitudes. They pass each other, moving in opposite directions when their displacement is half of the amplitude. The mean positions of the two particles lie on a straight line perpendicular to the paths of the two particles. The phase difference is:

A block of mass $M$ is attached to the lower end of a vertical spring. The spring is hung from a ceiling and has force constant $k$. The mass is released from rest with the spring initially unstretched. The maximum extension produced in the length of the spring will be:

The driver of a car traveling with speed $30\,\mathrm{m/s}$ towards a hill sounds a horn of frequency $600\,\mathrm{Hz}$. If the speed of sound in air is $330\,\mathrm{m/s}$, the frequency of reflected sound heard by the driver is:

A simple pendulum performs simple harmonic motion about $x=0$ with an amplitude $a$ and time period $T$. The speed of the pendulum at $x=\dfrac{a}{2}$ will be:

Which one of the following equations of motion represents simple harmonic motion? Where k, $k_0$, $k_1$ and a are all positive

A wave in a string has amplitude $2\,cm$. The wave travels in the positive $x$-direction with speed $128\,m/s$ and 5 complete waves are present in a length of $4\,m$. The equation describing the wave is:

Each of the two strings of lengths $51.6\,cm$ and $49.1\,cm$ are tensioned separately by $20\,N$. Mass per unit length of both strings is $1\,g/m$. When both strings vibrate simultaneously, the number of beats produced is:

Two Simple Harmonic Motions of angular frequency 100 and 1000 rad s⁻¹ have the same displacement amplitude. The ratio of their maximum accelerations is

The wave described by y = 0.25 sin(10πx − 2πt) where x and y are in meters and t in seconds, is a wave travelling along the

A point performs simple harmonic oscillation of period T and the equation of motion is given by x = a sin(ωt + π/6). After the elapse of what fraction of the time period will the velocity of the point be equal to half of its maximum velocity?

Two points are located at a distance of 10 m and 15 m from the source of oscillation. The period of oscillation is 0.05 s and the velocity of the wave is 300 m/s. What is the phase difference between the oscillations of the two points?

Two periodic waves of intensities I₁ and I₂ pass through a region at the same time in the same direction. The sum of the maximum and minimum intensities is

Equation for two waves is given as y₁ = a sin(ωt + φ₁), y₂ = a sin(ωt + φ₂). If amplitude and time period of resultant wave do not change, then calculate (φ₁ − φ₂).

Two stretched wires each produce a frequency of 500 Hz. By what percentage should the tension in one wire be increased so that 5 beats per second are heard?

A string having tension 360 N and mass per unit length 4 × 10⁻³ kg/m produces two consecutive resonant frequencies with a tuning fork, which are 375 Hz and 450 Hz. Find the mass of the string.

A rectangular block of mass m and area of cross-section A floats in a liquid of density ρ. If it is given a small vertical displacement from equilibrium it undergoes oscillation with a time period T. Then:

Two sound waves with wavelengths 5.0 m and 5.5 m respectively each propagate in a gas with velocity 330 m/s. We expect the following number of beats per second:

A transverse wave propagating along the x-axis is represented by y(x,t) = 8.0 sin(0.5πx − 4πt − π/4), where x is in metres and t is in seconds. The speed of the wave is:

The time of reverberation of a room A is one second. What will be the time (in seconds) of reverberation of a room having all the dimensions double of those of room A?

Which one of the following statements is true?

A point source emits sound equally in all directions in a non-absorbing medium. Two points $P$ and $Q$ are at distance $2\,m$ and $3\,m$ respectively from the source. The ratio of the intensities of the waves at $P$ and $Q$ is:

The circular motion of a particle with constant speed is:

A particle executing simple harmonic motion of amplitude $5\,\text{cm}$ has maximum speed of $31.4\,\text{cm/s}$. The frequency of its oscillation is:

The displacement $x$ of a particle varies with time $t$ as $$x = ae^{-\alpha t} + be^{\beta t}$$ where a, b, α and β are positive constants. The velocity of the particle will :

A stone tied to the end of a string 1 m long is whirled in a horizontal circle with constant speed. If the stone makes 22 revolutions in 44 s, what is the magnitude and direction of acceleration of the stone?

Two vibrating tuning forks produce progressive waves given by y₁ = 4 sin 500πt and y₂ = 2 sin 506πt. Number of beats produced per minute is: