Waves & Oscillations PYQ — Page 10

The displacement of a damped harmonic oscillator is given by $x(t)={e}^{-0.1t}\mathrm{cos}⁡(10\pi t+\phi ).$ Here $t$ is in seconds. The time taken for its amplitude of vibration to drop to half of its initial value is close to:

The correct figure that shows, schematically, the wave pattern produced by the superposition of two waves of frequencies $9 Hz$ and$11 Hz$, is

Equation of travelling wave on a stretched string of linear density $5 \mathrm{~g} / \mathrm{m}$ is $\mathrm{y}=0.03 \sin (450 \mathrm{t}-9 \mathrm{x})$ where distance and time are measured in SI units. The tension in the string 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 ratio $p:q$ is: 

A travelling harmonic wave is represented by the equation $y(x, t)={10}^{-3}\mathrm{sin}(50t+2x)\text{,}$ where $x$ and $y$ are in meter and $t$ is in seconds. Which of the following is a correct statement about the wave?

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 $100 \mathrm{Hz}$. The separation between successive nodes on the string is close to

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

A small speaker delivers $2 W$ of audio output. At what distance from the speaker will one detect $120 dB$ intensity sound? [Given reference intensity of sound as ${10}^{-12}W/{m}^{2}$ ]

A simple pendulum oscillating in air has period $T$ . The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is $\frac{1}{16}th$ of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liquid is:

A simple pendulum of length $1 \mathrm{~m}$ is oscillating with an angular frequency $10 \mathrm{rad} / \mathrm{s}$. The support of the pendulum starts oscillating up and down with a small angular frequency of $1 \mathrm{rad} / \mathrm{s}$ and an amplitude of $10^{-2} \mathrm{~m}$. The relative change in the angular frequency of the pendulum is best given by :

A simple harmonic motion is represented by: $y=5(\mathrm{sin}3\pi t+\sqrt{3}\mathrm{cos}3\pi t)\mathrm{cm}$ The amplitude and time period of the motion are:

A rod of mass $M$ and length $2L$ is suspended at its middle by a wire. It exhibits torsional oscillations. If two masses, each of mass $m$, are attached at a distance $L/2$ from its centre on both sides, it reduces the oscillation frequency by $20%$. The value of ratio $m/M$ is close to

A resonance tube is old and has a jagged end. It is still used in the laboratory to determine the velocity of sound in air. A tuning fork of frequency $512 \mathrm{Hz}$ produces first resonance when the tube is filled with water to a mark $11 \mathrm{cm}$ below a reference mark, near the open end of the tube. The experiment is repeated with another fork of frequency $256 \mathrm{Hz}$ which produces first resonance when water reaches a mark $27 \mathrm{cm}$ below the reference mark. The velocity of sound in air, obtained in the experiment, is close to

A progressive wave travelling along the positive $x-$ direction is represented by $y(x,t)=A sin(kx-\omega t+\phi )$ . Its snapshot at $t=0$ is given in the figure.  For this wave, the phase $\phi$ is:

A pendulum is executing simple harmonic motion and its maximum kinetic energy is $\mathrm{K}_{1}$. If the length of the pendulum is doubled and it performs simple harmonic motion with the same amplitude as in the first case, its maximum kinetic energy is $\mathrm{K}_{2}$

A particle undergoing simple harmonic motion has time dependent displacement given by $x(t)=\mathrm{A} \sin \frac{\pi t}{90}$. The ratio of kinetic to potential energy of this particle at $\mathrm{t}=210 \mathrm{~s}$ will be

A particle is executing simple harmonic motion $(SHM)$ of amplitude $A,$ along the $x$ -axis, about $x=0.$ When its potential Energy $(PE)$ equal kinetic energy $(KE),$ the position of the particle will be:

A particle executes simple harmonic motion with an amplitude of $5 cm$ . When the particle is at $4 cm$ from the mean position, the magnitude of its velocity in $SI$ units is equal to that of its acceleration. Then, its periodic time in seconds is:

A hoop and a solid cylinder of same mass and radius are made of a permanent magnetic material with their respective axes. But the magnetic moment of hoop is twice of solid cylinder. They are placed in a uniform magnetic field in such a manner that their magnetic moments make a small angle with the field. If the oscillation periods of hoop and cylinder are ${T}_{h}$ and ${T}_{c}$ respectively, then:

A damped harmonic oscillator has a frequency of $5$ oscillations per second. The amplitude drops to half its value for every 10 oscillations. The time it will take to drop to $\frac{1}{1000}$ of the original amplitude is close to:

A cylindrical plastic bottle of negligible mass is filled with $310 ml$ of water and left floating in a pond with still water. If pressed downward slightly and released, it starts performing simple harmonic motion at angular frequency $\omega .$ If the radius of the bottle is $2.5 cm$ then $\omega$ is close to: $($ density of water $={10}^{3}kg/{m}^{3})$

A closed organ pipe has a fundamental frequency of $1.5 \mathrm{kHz}$. The number of overtones that can be distinctly heard by a person with this organ pipe will be $($Assume that the highest frequency a person can hear is $20,000 \mathrm{Hz}$).

Two sitar strings, $A$ and $B$ playing the note '$Dha$' are slightly out of tune and produce beats of frequency $5\mathrm{Hz}$. The tension of the string $B$ is slightly increased and the beat frequency is found to decrease by $3\mathrm{Hz}$. If the frequency of $A$ is $425\mathrm{Hz}$. The original frequency of $B$ is