$A$ long cylindrical vessel is half filled with a liquid. When the vessel is rotated about its own vertical axis,the liquid rises up near the wall. If the radius of the vessel is $5 \, cm$ and its rotational speed is $2$ rotations per second,then the difference in the heights between the centre and the sides,in $cm$,will be

$A$ cylindrical vessel partially filled with water is rotated about its vertical central axis. Its surface will

$A$ cylindrical vessel of height $500 \ mm$ has an orifice (small hole) at its bottom. The orifice is initially closed and water is filled in it up to height $H$. Now the top is completely sealed with a cap and the orifice at the bottom is opened. Some water comes out from the orifice and the water level in the vessel becomes steady with the height of the water column being $200 \ mm$. Find the fall in height (in $mm$) of the water level due to the opening of the orifice.
[Take atmospheric pressure $= 1.0 \times 10^5 \ N/m^2$,density of water $= 1000 \ kg/m^3$ and $g = 10 \ m/s^2$. Neglect any effect of surface tension.]

When an air bubble rises from the bottom of a lake to the surface,its radius is doubled. The atmospheric pressure is equal to that of a column of water of height $H$. The depth of the lake is:

$A$ pump motor is used to deliver water at a certain rate from a given pipe. To obtain twice as much water from the same pipe in the same time, the power of the motor has to be increased to: (in $\times$)

$A$ cylindrical container is filled with liquid. When the container is rotated about its axis,the liquid rises along its sides. If the radius of the container is $r$ and the angular velocity of the container is $\omega$ rad/s,what will be the difference in the height of the liquid at the center and at the sides?