A trolley containing a liquid slides down a smooth inclined plane of angle $\alpha$ with the horizontal. Find the angle of inclination $\theta$ of the free surface with the horizontal.

A cylinder of radius $a$ and height $H$ is filled with a liquid to an unknown height $h.$ When it is rotated at an unknown constant angular velocity $\omega ,$ the base of the cylinder gets exposed when the liquid just starts spilling out, as shown in figure.
(a) Find the height $h$ of the liquid.
(b) Find the angular speed $\omega$ of the cylinder.

A U- tube of length $L$contains liquid. It is mounted on a horizontal turn table rotating with an angular speed $\omega$ about one of its arms as shown in figure. Find the difference in heights $\left(h\right)$ between the liquid columns in two vertical arms.

A rock is found to weigh $w$ in air when suspend from a spring scale. When completely submerged in water while attached to the spring scale, it weighs $w_{\mathrm{submerge}}$ as shown in figure. Find the density of the rock ${\rho }_{\mathrm{rock}}$ in terms of the scale reading and the density of water.

A large block of ice $5\mathrm{m}$ thick has a vertical hole drilled through it and is floating in the middle of a lake. What is the minimum length of a rope required to scoop up a bucket full of water through the hole? Relative density of ice $=0.9.$