Waves & Oscillations PYQ — Page 8

${T}_{0}$ is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to $\frac{1}{16}$ times of its initial value, the modified time period is

$Y=A\mathrm{sin}(\omega t+{\phi }_{0})$ is the time-displacement equation of a $SHM.$ At $t=0$ the displacement of the particle is $Y=\frac{A}{2}$ and it is moving along negative $x$-direction. Then the initial phase angle ${\phi }_{0}$ will be:

In the reported figure, two bodies $A\mathrm{and}B$ of masses $200g$ and $800g$ are attached with the system of springs. Springs are kept in a stretched position with some extension when the system is released. The horizontal surface is assumed to be frictionless. The angular frequency will be _______ $\mathrm{rad}{s}^{-1}$ when $k=20N{m}^{-{1}^{}}$. 

In the given figure, a mass $M$ is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is $k.$ The mass oscillates on a frictionless surface with time period $T$ and amplitude $A.$ When the mass is in equilibrium position, as shown in the figure, another mass $m$ is gently fixed upon it. The new amplitude of oscillation will be: 

In the given figure, a body of mass $M$ is held between two massless springs, on a smooth inclined plane. The free ends of the springs are attached to firm supports. If each spring has spring constant $k,$ the frequency of oscillation of given body is: 

In a simple harmonic oscillation, what fraction of total mechanical energy is in the form of kinetic energy, when the particle is midway between mean and extreme position.

If the time period of a two meter long simple pendulum is $2s$, the acceleration due to gravity at the place where pendulum is executing S.H.M. is:

Given below are two statements: Statement $I$: A second's pendulum has a time period of $1$ second. Statement $\mathrm{II}$: It takes precisely one second to move between the two extreme positions. In the light of the above statements, choose the correct answer from the options given below

For what value of displacement the kinetic energy and potential energy of a simple harmonic oscillation become equal?

For a body executing S.H.M. : (a) Potential energy is always equal to its $K.E.$ (b) Average potential and kinetic energy over any given time interval are always equal. (c) Sum of the kinetic and potential energy at any point of time is constant. (d) Average $K.E.$ in one time period is equal to average potential energy in one time period. Choose the most appropriate option from the options given below :

Consider two identical springs each of spring constant $k$ and negligible mass compared to the mass $M$ as shown. Fig. $1$ shows one of them and Fig. $2$ shows their series combination. The ratios of time period of oscillation of the two SHM is $\frac{{T}_{b}}{{T}_{a}}=\sqrt{x},$ where value of $x$ is ______. (Round off to the Nearest Integer) 

Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance $\frac{R}{2}$ from the earth's centre, where $R$ is the radius of the earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simple harmonic motion with a time period:

An object of mass $0.5\mathrm{kg}$ is executing simple harmonic motion. It amplitude is $5\mathrm{cm}$ and time period (T) is $0.2s$. What will be the potential energy of the object at an instant $t=\frac{T}{4}s$ starting from mean position. Assume that the initial phase of the oscillation is zero.

A wire having a linear mass density $9.0\times {10}^{-4}\mathrm{kg}{m}^{-1}$ is stretched between two rigid supports with a tension of $900N.$ The wire resonates at a frequency of $500\mathrm{Hz}.$ The next higher frequency at which the same wire resonates is $550\mathrm{Hz}.$ The length of the wire is ___________ $m.$

A tuning fork $A$ of unknown frequency produces $5\mathrm{beats}{s}^{-1}$ with a fork of known frequency $340\mathrm{Hz}$. When fork $A$ is filed, the beat frequency decreases to $2\mathrm{beats}{s}^{-1}$. What is the frequency of fork $A$ ?

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

A student is performing the experiment of the resonance column. The diameter of the column tube is $6\mathrm{cm}$. The frequency of the tuning fork is $504\mathrm{Hz}$. Speed of the sound at the given temperature is $336m{s}^{-1}$. The zero of the meter scale coincides with the top end of the resonance column tube. The reading of the water level in the column when the first resonance occurs is:

A sound wave of frequency $245\mathrm{Hz}$ travels with the speed of $300{ms}^{-1}$ along the positive x-axis. Each point of the wave moves to and fro through a total distance of $6\mathrm{cm}$. What will be the mathematical expression of this travelling wave?

A pendulum bob has a speed of $3m{s}^{-1}$ at its lowest position. The pendulum is $50\mathrm{cm}$ long. The speed of bob, when the length makes an angle of $60^{\circ}$ to the vertical will be $(g=10m{s}^{-2})$ ______ $m{s}^{-1}$.

A particle starts executing simple harmonic motion $(\mathrm{SHM})$ of amplitude $a$ and total energy $E.$ At any instant, its kinetic energy is $\frac{3E}{4}$, then its displacement $y$ is given by:

A particle performs simple harmonic motion with a period of $2$ second. The time taken by the particle to cover a displacement equal to half of its amplitude from the mean position is $\frac{1}{a}s.$ The value of $a$ to the nearest integer is

A particle of mass $1\mathrm{kg}$ is hanging from a spring of force constant $100N{m}^{-1}.$ The mass is pulled slightly downward and released so that it executes free simple harmonic motion with time period $T.$ The time when the kinetic energy and potential energy of the system will become equal, is $\frac{T}{n}.$ The value of $n$ is ________.

A particle is making simple harmonic motion along the $X$-axis. If at a distances ${x}_{1}$ and ${x}_{2}$ from the mean position the velocities of the particle are ${v}_{1}$ and ${v}_{2}$, respectively. The time period of its oscillation is given as:

A particle executes simple harmonic motion represented by displacement function as $x(t)=A\mathrm{sin}(\omega t+\phi )$. If the position and velocity of the particle at $t=0s$ are $2\mathrm{cm}$ and $2\omega \mathrm{cm}{s}^{-1}$ respectively, then its amplitude is $x\sqrt{2}\mathrm{cm}$ where the value of $x$ is