I think the basis of the question is that intuitively, it doesn’t seem to take a lot of power to spin a propeller. But obviously from a physics standpoint, a great deal of thrust is required to accelerate a heavy airplane and power its ascent. So the size and pitch of the propeller blades and their rotational speed must obviously be such as to be able to convert a great deal of power into thrust by accelerating a large volume of air to a very high speed. So IOW, the simple answer to the question is that the intuitive perception that it doesn’t take a lot of power to do this is wrong.
This. Moving something through the air produces resistance, commonly called drag in the aviation biz. While it seems “intuitive” that spinning a propeller through “thin air” is “easy”, in fact it ain’t, not when that propeller is pulling its way forward through that air and pulling a whole airplane along behind it.
What the engine and propeller are doing is throwing air backwards. Air has mass and resistance. So the more powerful the engine, the more air you can throw backwards per second, which means the plane goes forward faster.
I think it’s instructive to consider big outboard motors on boats, and the dinky little propeller at the bottom. It boggles my mind to consider an engine putting several hundred horsepower into that little blade. If I had a loose propeller in my hand and held it in water, I could obviously turn it. But propellers bog down terribly when you try to spin them faster, in water or in air.
I assume you’ve, at some point in your life, pushed a car down the road. Now try getting it up to highway speeds.
Also, it’s not just about RPM, it’s also about torque. A CD spins at, what 200 RPMs, but I can stop it with my hand. OTOH, if I put a CD sized piece of metal on a bigger motor and spin it that fast and grab it, it’ll probably take the skin off my hand.
Even if an airplane engine and a lawnmower engine spin at the same speed (and I doubt they do), the lawnmower engine doesn’t have the strength to pull the airplane through the air.
Pretty good. Aerodynamics gets weird, because you need to to take into account the viscosity of the air/fluid, and various conservation laws (which gets you to Navier-Stokes), as well, but the basics are there.
Where I think most people go wrong in thinking about aerodynamics is that there is little intuitive appreciation of how much air weighs. As a good approximation air at sea level has one thousandth the density of water. Which doesn’t sound like much, but consider that - say my lounge/dining room is about 1052.4 metres. Thus it contains about 120kg of air.
A small aircraft - say a Cessna 172 has a propeller very close to 2metres in diameter. So a disk area close to 3 metres[sup]2[/sup]. It can generate (depending on model) about 750 pounds = 3,300 Newtons of static force with the 160hp engine at full power. A cubic metre of air weighs about 1kg, so you can see that this isn’t unreasonable.
When you get to jet aircraft, especially the high bypass fanjets, they are very pleased to tell you how many tons of air they shift per second.
In addition to air resistance, you also have to consider the inertia of the propeller itself. Some examples were pretty freaking big (compare the propeller on the F4U Corsair to the guy standing next to the plane). As others have said, you’re going to need to impart a shit-ton* of torque on that propeller to get it spinning and keep it that way, especially if you need to change pitch or RPM rapidly as a fighter pilot might find reason to do in combat (or in just trying to put the thing down on an aircraft carrier without crashing spectacularly into the sea).
As an interesting aside, one post-war development with the advent of the Jet engine was the Turboprop: A jet engine that used the energy produced to spin a propeller, rather than trying to propel the aircraft with raw jet thrust. It is, I’m told, a very efficient design, more fuel efficient than a jet engine (those things drink fuel like sailors consume coffee and alcohol) and easier to maintain than a Piston engine (jets have far fewer moving parts than piston engines do, largely because they don’t have any pistons). Examples of modern-day Turboprop planes include the Lockheed C-130J Super Hercules and the Lockheed P-3 Orion. Downsides include not being able to impart as much raw thrust as a turbojet or turbofan engine can, and also being very noisy thanks to the propeller blades chopping through the air. They make a sort of whining drone sound, compare to the “Buzzzzzzz” you associate with smaller propeller airplanes.
*Please pardon the highly technical engineering jargon. It’s an SI unit, I assure you.
You are generally on more or less the right path but we can tidy up quite a few things in here. It would take zero power to keep either of your propellers spinning in a perfect vacuum (ignoring some very small effects like power loss through your framework for vibration). However, it would take some energy to bring the propeller up to speed, so the lawnmower might take a while longer to accelerate it. With less power, the lawnmower would require more time to produce that much energy. But, the propeller would give you back that energy while it is slowing down.
It doesn’t matter much that the air is viscous. Viscosity tells you how quickly a fluid converts mechanical energy into heat energy by friction. The other effect is how dense the air is (how much inertia a volume of it has at a given velocity), which converts propeller mechanical energy into kinetic energy in the air (which is what you’re actually trying to do). The ratio of the inertial effect to the viscous effect is called the Reynolds number, and it’s high for airplane propellers. It’s equal to the size times the speed times the density divided by the viscosity, and should be a few million for an airplane propeller. We call this situation “inviscid” because it’s practically without viscous effect.
You say " for every unit of m accelerated by the propeller, there is an equivalent unit of thrust". You have the dimensions inconsistent here, because mass accelerated by the propeller happens over some time, whereas thrust is instantaneous. Thrust is force, which is instantaneous. What you are missing is the idea of “impulse”, which is the product of force and time. To get a certain amount of thrust, you have to be accelerating some quantity of mass per time at some level of acceleration.
You also say “to move a mass of an aircraft M at acceleration a, it requires a displacement of air of mass M”. However, you could accelerate less mass at a greater acceleration, or vice versa, to do the same job of accelerating the aircraft. There’s no particular need to make the masses the same.
So, you were sort of on the right track, but this is a bit more accurate.