Questions on how helicopters fly

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When a helicopter loses power, the blades are still rotating and air is still being forced downward, but the blades slow down. Lift is reduced accordingly. The helicopter begins to accelerate downward. The air accelerated downward by the still-spinning-but-slowing blades meets the air still there from moments ago. There is viscosity. It has to go somewhere and that takes time and crucially creates a whack of drag as it works its way out from under the blades. If you are lucky, the drag created and ground effect add up to enough force to keep the downward velocity of your vehicle down to a survivable prang speed. If the blades stop spinning, almost all the lift force is lost and you prang in at something between zero and terminal velocity depending upon how high above the deck you were.

Is this when a helicopter makes a normal controlled landing with the engine still functioning?

If the engine looses power, autorotation of the blades allows the pilot to land safely with the pilot adjusting blade pitch angle and forward air speed. Just as good as a parachute.
Descent with engine failure can be catastrophic if it occurs with no or little forward airspeed at a height particular for type of helicopter - generally speaking around 500 feet - for the air flow to be set up correctly from downwards movement to upwards movement over the blades.

 

[A.T.]

Autogyros rotate because the drag on the advancing aerofoil is less than the drag on the retreating one.

In autorotation both blades can have sections that act as turbines to support the rotation of the rotor. This works theoretically even without forward motion (just sinking), so there is no advancing & retreating blade. With forward flight the balance between turbine section and propeller section shifts and becomes asymmetrical. See video below for a visualization: