Is it possible for an aircraft propeller to become supersonic? I am not referring as for the aircraft, only the rotating blade. I know it is extremely unlikely for a propeller plane to go supersonic, but could the propeller ?
Two examples I can think of immediately: the F-84H "Thundescreech" and the Soviet/Russian "Bear" bomber. The former aircraft failed to enter service, in part because it was literally so loud as to make people working on the apron physically ill. The Bear has been in service for decades, however.
Both of these aircraft had propellers that ran supersonic at the tips -- the Bear has eight of them, though they're less strident than those of the Thunderscreech (probably because slower at the tips).
Yes the blade tips can go supersonic at redline rpm if the blades are long enough, the tip speed simply being a function of those two variables, along with the right temperature, which determines the speed of sound.
Some common airplanes have propellers that produce supersonic shock waves on takeoff, generally long two bladed ones. They produce a distinctive, very loud BLAAAAAT sort of sound as they pass by you on take off. Slowing the propeller RPM just a little makes the sound go away. The speed of sound being temperature dependent, the effect is worse when the air is cooler than warmer.
It's common on airplanes with radials and two blade propellers like the DeHavilland Beaver. The Harvard/T6 trainer, with an R-1340 radial had a particularly intense "sonic blat" that is familiar if you've ever watched one at an airshow.
Cessna 180s/185s with two blade propellers also do it at takeoff RPM. A 185 taking off from a lake on a cool morning will have that distinctive searing sound that echoes off the local hills.
If a lake or other location is unusually noise sensitive, it's a common practice to back off on the RPM a little bit as soon as airborne, about 100 RPM or so, to make the "sonic blat" go away.
On of the benefits of switching to a 3 blade prop is lower takeoff noise, because the shorter blade tips stay subsonic.
The tips of propeller blades can go supersonic, however this is very inefficient so in practice this is rarely done. One notable exception is the Russian Tu-95 turboprop bomber which cruises at 380 knots (440 mph, 710 km/h). The propeller tips of the Tu-95 move at supersonic speeds, which makes the plane extremely noisy.
Tu-95 (photo: RAF, via: Wikimedia Commons, OGL v1.0)
Yes, you can figure that out yourself with simple math. The circumference of a circle is 2 Pi r, and let's say the speed of sound is 343 meters / second -- it depends on a few parameters, but 343 m / s is what the wikipedia page uses. My plane's engine redlines the rpm at 2700 rpm, that is 2700 revolutions per minute. You can now solve for r and thus determine the prop length at which the tip goes supersonic at 2700 rpm. We have:
2 Pi r 2700/60 == 343
and you get a value for r of 1.21311 meters.
You can play with the rpm, but 2700 is a typical redline for a piston plane's engine. Of course, our props aren't 1.21 meters in radius, but this equation should give you an idea on the radius / rpm relationship.