If the earth's atmosphere were everywhere of the same density $\left(=1.290 {\mathrm{kgm}}^{-3}\right)$ as at the surface of the earth, how high would the atmosphere have to extend in order to give rise to the observed atmospheric pressure of $1.013\times {10}^5 \mathrm{Pa}$. 

A hollow right circular cone of height $h$ With its base on a horizontal table. With its flat base on the table. The whole volume inside the cone is filled with water of density ρ.The circular rim of the cone's base has a watertight seal with the table's surface and the top apex of the cone has a small hole. Neglecting atmospheric pressure, the total upward force exerted by water on the cone is

A U-tube contains mercury to the same level in both of its arms.If water is poured to a height of $13.6cm$ in one arm, The rise in mercury level in the other arm is $x cm$. Then $x=?$
$\left( density of water={10}^{3}kg{m}^{-3}\right)$

A straight plank of wood of uniform cross-section floats on the surface of water. Its one end is then raised above the surface of means of string attached to that end. If the specific gravity of wood is $0.80$, find

What fraction of the length of the plank is immersed in that inclined position.

A rod of length $6$ Metre has a mass of $12 \mathrm{kg}$ It is hinged at one end at a distance of $3$ Metre below a water surface.

What weight must be attached to the other end of the rod so that $5$ Metre of rod is submerged ?