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The tip of the propeller will always be the first point to reach mach 1 as it has the largest speed for any given rpm.

Mach number by definition depends in the speed of sound in air which changes with altitude:

Mach number = speed / speed of sound

To get Mach 1, we need the speed = speed of sound. At sea level, the speed of sound is 330m/s.

The tangential velocity is equal to the angular velocity multiplied by the radius: v=wr

w = 330 / r.

Assuming that your rotor radius is 10m, the angular speed is 330 / 10 = 33 /s

The rotational speed is 33 rotations per second which gives 33*60 = 1980 rpm.

The speed of any point along the blade can be calculated by changing the radius in the above equation.

In conclusion, the tip of a 10m rotor will travel at Mach 1 at sea level.

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