In the given arrangement, the balls are identical each of mass $m$ placed in gravity free space with initially springs are at their natural length. When the balls are slightly displaced in the vertical direction, then the balls are in vertical SHM. Then the possible angular frequencies for that SHM (the springs are ideal springs. Each of spring constant is $k$) (Only consider both balls oscillates with same angular frequency but they crosses the mean position their velocities direction are in the same direction (or) opposite direction)

 

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 :