Oscillations Questions

Two simple harmonic motions are represented by the equations $y_1=10\sin (3\pi t+\pi /4)$ and $y_2=5[\sin (3\pi t)+\sqrt{3}\cos (3\pi t)]$. The ratio of their amplitudes is:

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, the ratio $(A_Q/A_P)$ of the amplitude $A_Q$ of mass $Q$ to the amplitude $A_P$ of mass $P$ is:

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?

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

The oscillation of a body on a smooth horizontal surface is represented by the equation $X=A\cos (\omega t)$. Which one of the following graphs shows correctly the variation of acceleration $a$ with $t$?

The damping force on an oscillator is directly proportional to the velocity. The units of the constant of proportionality are:

Two particles oscillate with same frequency and amplitude. They pass each other moving in opposite directions when displacement is half the amplitude. The phase difference is:

The period of oscillation of a mass $M$ suspended from a spring of negligible mass is $T$. If along with it another mass $M$ is also suspended, the period of oscillation will now be:

The displacement of a particle along the x-axis is given by $x=a{\sin }^2\omega t$. The motion of the particle corresponds to:

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 value $k$. The mass is released from rest with the spring initially unstretched. The maximum extension produced in the length of the spring will be:

Which one of the following equations of motion represents simple harmonic motion?

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

A point performs simple harmonic oscillation of period $T$ and the equation of motion is given by $x=a\sin (\omega t+\pi /6)$. 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 Simple Harmonic Motions of angular frequency $100{\text{rad s}}^{-1}$ and $1000{\text{rad s}}^{-1}$ have the same displacement amplitude. The ratio of their maximum accelerations is:

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

Four point $+ve$ charges of same magnitude ($Q$) are placed at four corners of a rigid square frame as shown in figure. The plane of the frame is perpendicular to $Z$ -axis. If a $-ve$ point charge is placed at a distance $z$ away from the above frame ($z\ll L$) then:

Which of the following functions represents a simple harmonic oscillation?