The amplitude and the time period in a $S.H.M.$ are $0.5 \, cm$ and $0.4 \, s$ respectively. If the initial phase is $\pi / 2 \, rad$,then the equation of $S.H.M.$ will be

$A$ body constrained to move along the $y$-axis is subjected to a constant force $\vec{F} = -\hat{i} + 2\hat{j} + 3\hat{k} \, \text{N}$. The work done by this force in moving the body a distance of $4 \, \text{m}$ along the $y$-axis is ............... $\text{J}$.

Cyclic hydrocarbon molecule $A$ has all the carbon and hydrogen atoms in a single plane. All the carbon-carbon bonds are of the same length,which is less than $1.54 \ \mathring{A}$ but more than $1.34 \ \mathring{A}$. The $C-C$ bond angle will be

$A$ uniform rod $AB$ of length $L$ and mass $M$ is lying on a smooth table. $A$ small particle of mass $m$ strikes the rod with a velocity $v_0$ at point $C$ at a distance $x$ from the centre $O$. The particle comes to rest after the collision. The value of $x$,so that point $A$ of the rod remains stationary just after the collision,is:

Consider the following statements about Langmuir isotherm.
$I.$ The free gas and adsorbed gas are in dynamic equilibrium.
$II.$ All adsorption sites are equivalent.
$III.$ The initially adsorbed layer can act as a substrate for further adsorption.
$IV.$ The ability of a molecule to get adsorbed at a given site is independent of the occupation of neighboring sites.
The correct statements are

$A$ uniform cylinder of length $L$ and mass $M$ having cross-sectional area $A$ 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. The extension $x_0$ of the spring when it is in equilibrium is: