You can sometimes detect earth rotation coriolis effects on falling objects (decreasing the radius of spin) such as dense objects dropped from a helicopter or airplane.
However, a helicopter "floating" in air is also restricted in how fast the air can go by.
At this latitude, the spin of the earth rotates relatively fast: Cos(35 degrees)*25,000 miles/24 hours = 853 Miles/hour.
So for your helicopter to hover with respect to the spinning earth -- It would have to be traveling at more than Mach 1 in the air. Can you relate this to the time lags perceived when traveling in a jet airliner either east to west or west to east?

Imagine that you are in a helicopter.When the rotor blades start spinning you get some lift which allows you to take off. To move around, you tilt the blades a bit which gives you some thrust in the direction you wish to go. A key point here is that you need air for the helicopter to work and any movement is with respect to the atmosphere.
Now lets say you are hoving above a point on the Earth and someone below you on the ground is watching you. If you see that person moving away from you, you might think that the Earth is rotating away from you. But what would the person on the ground think? If the Earth really was rotating away from helicopter then it must also be rotating away from the atmosphere. If that was true I think the person on the ground would experience a pretty heavy duty wind. If the Earth did rotate with the tmosphere staying "still" how fast do you think the wind would be?>