JEE Main 2015 — Waves & Oscillations

A simple harmonic oscillator of angular frequency $2\mathrm{rad}{s}^{-1}$ is acted upon by an external force $F=\mathrm{sin}⁡t N.$If the oscillator is at rest in its equilibrium position at $t=0,$ its position at later times is proportional to:

$x and y$ 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) :

For a simple pendulum, a graph is plotted between its kinetic energy (K.E.) and potential energy (P.E.) against its displacement d. which one of the following represents these correctly? (graphs are schematic and not drawn to scale)

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