Waves & Oscillations PYQ — Page 12

A pendulum with the time period of $1s$ is losing energy due to damping. At a certain time, its energy is $45J$. If after completing $15$ oscillations its energy has become $15J$, then its damping constant (in ${s}^{-1}$) will be

$x and y$ displacements of a particle are given as $x(t)=a sin ⁡\omega t and y(t)=a sin ⁡2\omega t.$ Its trajectory will look like:

A cylindrical block of wood (density = $650\mathrm{kg}{m}^{-3}$), of base area $30 {\mathrm{cm}}^{2}$ and height $54\mathrm{cm}$, floats in a liquid of density $900\mathrm{kg}{m}^{-3}$. The block is depressed slightly and then released. The time period of the resulting oscillations of the block would be equal to that of a simple pendulum of length (nearly) :

Two bodies of masses $1\mathrm{kg}$ and $4\mathrm{kg}$ are connected to a vertical spring, as shown in the figure. The smaller mass executes simple harmonic motion of angular frequency $25\mathrm{rad}{s}^{-1}$, and amplitude $1.6\mathrm{cm}$ while the bigger mass remains stationary on the ground. The maximum force exerted by the system on the floor is (take $g=10m{s}^{-2}$). 

The amplitude of a simple pendulum, oscillating in air with a small spherical bob, decreases from 10 cm to 8 cm in 40 seconds. Assuming that Stokes law is valid, and ratio of the coefficient of viscosity of air to that of carbon dioxide is 1.3, the time in which amplitude of this pendulum will reduce from 10 cm to 5 cm in carbondioxide will be close to (ln 5 = 1.601, ln 2 = 0.693).

A body is in simple harmonic motion with time period $T=0.5 \text{s}$ and amplitude $A=1 \text{cm}$. Find the average velocity in the interval in which it moves from equilibrium position to half of its amplitude.

A particle which is simultaneously subjected to two perpendicular simple harmonic motions represented by; $x={a}_{1}\mathrm{cos}⁡\omega t$ and $y={a}_{2}\mathrm{cos}⁡2\omega t$ traces a curve given by :

A transverse wave is represented by : $\text{y} = \frac{ 1 0 }{ \pi } sin ( \frac{ 2 \pi }{ \text{T} } \text{t} - \frac{ 2 \pi }{ \lambda } \text{x} )$ For what value of the wavelength the wave velocity is twice the maximum particle velocity?

The total length of a sonometer wire fixed between two bridges is $110\mathrm{cm}$. Now, two more bridges are placed to divide the length of the wire in the ratio $6:3:2$. If the tension in the wire is $400N$ and the mass per unit length of the wire is $0.01\mathrm{kg}{m}^{-1}$, then the minimum common frequency with which all the three parts can vibrate, is

The angular frequency of the damped oscillator is given by, $\omega=\sqrt{\left(\frac{\mathrm{k}}{\mathrm{m}}-\frac{\mathrm{r}^2}{4 \mathrm{~m}^2}\right)}$ where $\mathrm{k}$ is the spring constant, $\mathrm{m}$ is the mass of the oscillator and $r$ is the damping constant. If the ratio $\frac{r^2}{\mathrm{mk}}$ is $8 \%$, the change in time period compared to the undamped oscillator is approximately as follows:

A particle moves with simple harmonic motion in a straight line. In first $\tau s$, after starting from rest it travels a distance a, and in next $\tau s$ it travels 2a, in same direction, then :

A pipe of length 85 cm is closed from one end. Find the number of possible natural oscillations of air column in the pipe whose frequencies lie below 1250 Hz. The velocity of sound in air is 340 m/s.

Which of the following expressions corresponds to simple harmonic motion along a straight line, where $\mathrm{x}$ is the displacement and $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are positive constants?

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $\text{M}$. The piston and the cylinder have equal cross-sectional area $\text{A}$. When the piston is in equilibrium, the volume of the gas is ${V}_{0}$ and its pressure is ${M}_{0}$. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency [Assume the system is in space.]

A mass $m=1.0 \mathrm{~kg}$ is put on a flat pan attached to a vertical spring fixed on the ground. The mass of the spring and the pan is negligible. When pressed slightly and released, the mass executes simple harmonic motion. The spring constant is $500 \mathrm{~N} / \mathrm{m}$. What is the amplitude A of the motion, so that the mass $m$ tends to get detached from the pan ? (Take $g=10 \mathrm{~m} / \mathrm{s}^2$ ). The spring is stiff enough so that it does not get distorted during the motion. 

A sonometer wire of length $114 \mathrm{~cm}$ 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 $1: 3: 4$ ?

A uniform cylinder of length $\mathrm{L}$ and mass $M$ having cross-sectional area $\mathrm{A}$ is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half submerged in a liquid of density $\sigma$ at equilibrium position. When the cylinder is given a downward push and released, it starts oscillating vertically with a small amplitude. The time period $\mathrm{T}$ of the oscillations of the cylinder will be :

In a transverse wave the distance between a crest and neighbouring trough at the same instant is $4.0 \mathrm{~cm}$ and the distance between a crest and trough at the same place is $1.0 \mathrm{~cm}$. The next crest appears at the same place after a time interval of $0.4 \mathrm{~s}$. The maximum speed of the vibrating particles in the medium is:

A sonometer wire of length $1.5m$ is made of steel. The tension in it produces an elastic strain of $\text{1%}$. What is the fundamental frequency of steel if density and elasticity of steel are $7.7\times {10}^{3}\mathrm{kg}{m}^{-3}$ and $2.2\times {10}^{11}N{m}^{-2}$ respectively?

Bob of a simple pendulum of length $l$ is made of iron. The pendulum is oscillating over a horizontal coil carrying direct current. If the time period of the pendulum is $\mathrm{T}$ then :

Two simple pendulums of length $1 \mathrm{~m}$ and $4 \mathrm{~m}$ respectively are both given small displacement in the same direction at the same instant. They will be again in phase after the shorter pendulum has completed number of oscillations equal to:

Two charges, each equal to $\text{q}$, are kept at $\text{x} = - \text{a}$ and $\text{x} = \text{a}$ on the x-axis. A particle of mass $\text{m}$ and charge ${\text{q}}_{0}=-\frac{\text{q}}{2}$ is placed at the origin. If charge ${\text{q}}_{0}$ is given a small displacement $\text{(y << a)}$ along the y-axis, the net force acting on the particle is proportional to :

When two sound waves travel in the same direction in a medium, the displacements of a particle located at ' $x$ ' at time ' $t$ ' is given by : $\begin{aligned} & y_1=0.05 \cos (0.50 \pi x-100 \pi t) \\ & y_2=0.05 \cos (0.46 \pi x-92 \pi t) \end{aligned}$ where $y_1, y_2$ and $x$ are in meters and $t$ in seconds. The speed of sound in the medium is :

Following are expressions for four plane simple harmonic waves (i) $\quad y_1=A \cos 2 \pi\left(n_1 t+\frac{x}{\lambda_1}\right)$ (ii) $y_2=A \cos 2 \pi\left(n_1 t+\frac{x}{\lambda_1}+\pi\right)$ (iii) $y_3=A \cos 2 \pi\left(n_2 t+\frac{x}{\lambda_2}\right)$ (iv) $y_4=A \cos 2 \pi\left(n_2 t-\frac{x}{\lambda_2}\right)$ The pairs of waves which will produce destructive interference and stationary waves respectively in a medium, are