In the adjacent figure, $ABCD$ are four points in a smooth horizontal plane representing the four corners of a square of diagonal $20 \mathrm{cm}$. A very small object of mass $144\mathrm{gm}$ placed at the centre of the square connected to points $A,B,C$ and $D$ by four elastic strings of same dimensions but of different moduli of elasticity as shown in the figure. Natural length and cross-sectional area of each string are $10 \mathrm{cm}$ and ${\pi }^2\times {10}^{-6}{ \mathrm{m}}^2$ respectively. The mass is slightly displaced from equilibrium and then released to perform SHM along the diagonal AC. ($Y_1=9Y,Y_2=4Y,Y_3=16Y$ and $Y_4=4Y$, where $Y=4.9\times {10}^9 \mathrm{N} {\mathrm{m}}^{-2}$). The time period of oscillation of object is $K$ millisecond. The value of $K$ is approximately.

A particle of mass $1 \mathrm{mg}$ and charge $q$ is lying at the mid-point of two stationary particles kept at a distance $2 \mathrm{m}$ when each is carrying same charge $q$. If the free charged particle is displaced from its equilibrium position through distance $x$ $\left(x<<1 \mathrm{m}\right)$. The particle executes SHM. Its angular frequency of oscillation will be _______ $\times {10}^5 \mathrm{rad} {\mathrm{s}}^{-1}$ (if $q^2=10C^2$)

A cone with half the density of water is floating in water as shown in figure. It is depressed down by a small distance $\delta \left(\ll \mathrm{H}\right)$ and released. The frequency of simple harmonic oscillations of the cone is

The position co-ordinates of a particle moving in a $3D$ coordinate system is given by

$x=a\cos \mathrm{\omega }\mathrm{t}$

$y=a\sin \omega t$

and $z=a\omega t$

The speed of the particle is:

The point $A$ moves with a uniform speed along the circumference of a circle of radius $0.36 \mathrm{m}$ and covers $30°$ in $0.1 \mathrm{s}$. The perpendicular projection $P$ from $A$ on the diameter $MN$ represents the simple harmonic motion of $P$. The restoration force per unit mass when $P$ touches $M$ will be :