A mass of $5 \mathrm{kg}$ is connected to a spring. The potential energy curve of the simple harmonic motion executed by the system is shown in the figure. A simple pendulum of length $4 \mathrm{m}$ has the same period of oscillation as the spring system. What is the value of acceleration due to gravity on the planet where these experiments are performed ?

The motion of a mass on a spring, with spring constant $K$ is as shown in figure.

The equation of motion is given by, $x\left(t\right)=A\sin \omega t+$$B\cos \omega \mathrm{t}$ with $\omega =\sqrt{\frac{K}{m}}$.
Suppose that at time $t=0,$the position of mass is $x\left(0\right)$ and velocity $v\left(0\right),$ then its displacement can also be represented as $x\left(t\right)=C\cos (\omega t-\varphi ),$ where $C$ and $\varphi$ are 

In the reported figure, two bodies $A \mathrm{and} B$ of masses $200 \mathrm{g}$ and $800 \mathrm{g}$ are attached with the system of springs. Springs are kept in a stretched position with some extension when the system is released. The horizontal surface is assumed to be frictionless. The angular frequency will be _______  $\mathrm{rad} {\mathrm{s}}^{-1}$ when $k=20 \mathrm{N} {\mathrm{m}}^{-1^{}}$.

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 $q^2=10C^2$)

Consider two identical springs each of spring constant $k$ and negligible mass compared to the mass $M$ as shown. Fig. $1$ shows one of them and Fig. $2$ shows their series combination. The ratios of time period of oscillation of the two SHM is $\frac{T_b}{T_a}=\sqrt{x},$ where value of $x$ is ______.

(Round off to the Nearest Integer)