Required information

Problem $06\mathrm{.}035-$ Unloaded helicopter

An unloaded helicopter of mass $\backslash (12,700\backslash$mathrm$\{$~kg$\}\backslash )$ hovers at sea level while it is being loaded. In the unloaded hover mode, the blades rotate at $550$ rpm $.$ The horizontal blades above the helicopter cause a $18-$m$-$diameter air mass to move downward at an average speed proportional to the overhead blade rotational speed $($rpm$).$ A load of $\backslash (13,600\backslash$mathrm$\{$~kg$\}\backslash )$ is loaded onto the helicopter, and the helicopter slowly rises.

Take the density of atmospheric air to be $\backslash (1\mathrm{.}18\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ).$ Assume air approaches the blades from the top through a large area with negligible speed and air is forced by the blades to move down with a uniform speed through an imaginary cylinder whose base is the blade span area.

Problem $06\mathrm{.}035.$b $-$ Number of revolutions per minute taken by the helicopter blades to hover

Determine the number of revolutions per minute $($rpm$)$ taken by the helicopter blades to hover with the $13,600-$kg load and the required power input.

The number of revolutions per minute of the loaded helicopter is rpm$.$

The required power output is kW $.$