Oscillations 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)

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:

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 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 :

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 :

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 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:

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

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:

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:

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:

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?

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'?

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:

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

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

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?

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:

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?

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?