The value of universal gravitational constant $G=6.67\times {10}^{-11} \mathrm{N}\mathrm{ }{\mathrm{m}}^2\mathrm{ }\mathrm{k}{\mathrm{g}}^{-2}$. The value of $G$ in units of ${\mathrm{g}}^{-1} \mathrm{c}{\mathrm{m}}^3\mathrm{ }{\mathrm{s}}^{-2}$ is $6.67\times {10}^{-n}$ . The value of $n$ is .

Six objects are placed at the vertices of $\mathrm{a}$ regular hexagon. The geometric centre of the hexagon is at the origin with objects $1$ and $4$ on the $X$-axis (see figure). The mass of the $\mathrm{k}$ th object is ${\mathrm{m}}_{\mathrm{k}}={\mathrm{k}}^{\mathrm{i}}\mathrm{M}\left|{\mathrm{cos\theta }}_{\mathrm{k}}\right|,$ where $\mathrm{i}$ is an integer, $\mathrm{M}$ is $\mathrm{a}$ constant with dimension of mass and ${\mathrm{\theta }}_{\mathrm{k}}$ is the angular position of the $\mathrm{k}$ th vertex measured from the positive $\mathrm{x}$ -axis in the counter-clockwise sense. If the net gravitational force on $\mathrm{a}$ body at the centroid vanishes, the value of $\mathrm{i}$ is

Four identical particles of mass $\mathrm{M}$ are located at the corners of a square of side $‘a’$ . What should be their speed if each of them revolves under the influence of other’s gravitational field in a circular orbit circumscribing the square?

A solid sphere of radius $R$ carries a charge $\left(Q+q\right)$ distributed uniformly over its volume. A very small point-like piece of it of mass $m$ gets detached from the bottom of the sphere and falls down vertically under gravity. This piece carries charge $q$. If it acquires a speed $v$ when it has fallen through a vertical height $y$ (see figure), then (assume the remaining portion to be spherical)

A person whose mass is $100 \mathrm{kg}$ travels from Earth to Mars in a spaceship. Neglect all other objects in sky and take acceleration due to gravity on the
surface of the Earth and Mars as $10 \mathrm{m} {\mathrm{s}}^{-2}$ and $4 \mathrm{m} {\mathrm{s}}^{-2}$, respectively. Identify from the below figures, the curve that fits best for the weight of the passenger as a function of time.