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Propellers and rotors

Air and fluids · level 6, lesson 6 of 6

A question to start

Ingenuity, the helicopter that flew on Mars, spun its rotor blades at about 2,400 turns a minute. How fast were the tips of its blades moving through the air?

The idea

A propeller or helicopter rotor is a set of spinning wings. Each blade is tilted so that, as it sweeps round, it turns air downwards (or backwards, for a propeller), and the air pushes the blade up (or forwards). This push is the thrust.

Each point on a blade moves in a circle. In one turn, a point at radius \(r\) travels \(2\pi r\). At \(n\) turns each second, its speed is

\[v = 2\pi r n.\]

The tips move fastest, and the tips do most of the work. But a tip must not get too close to the speed of sound, or shock waves form and the blade loses lift and shakes.

tip: fastestblade speed grows with radius

Ingenuity had a mass of 1.8 kg and two rotors, each 1.2 m from tip to tip, spinning in opposite directions at about 2,400 turns a minute.

Blade tips at 151 m/s Mars Earth
Speed of sound 234 m/s (carbon dioxide, 217 K) about 340 m/s (air, 15 °C)
Tip speed as a share of it 65% 44%

The two rotors spin in opposite directions. A single rotor would push the air round one way, and by Newton's third law the body would spin the other way. Two rotors cancel this out.

Worked examples

1. Turns each second. \(2400/60 = 40\) turns a second.

2. Tip speed. \(r = 0.6\) m: \(v = 2\pi \times 0.6 \times 40 = 151\) m/s, about 540 km/h.

3. Compared with sound. \(151/234 = 0.65\): the tips moved at about 65% of the speed of sound in Mars air.

Try it

  1. Why do a rotor's tips move faster than the parts near the middle?
  2. A fan blade 0.2 m long spins at 1,200 turns a minute. Find its tip speed.
  3. At what spin rate would Ingenuity's tips reach the speed of sound in Mars air?
  4. Why did Ingenuity have two rotors spinning in opposite directions?
Answers
  1. They go round a bigger circle in the same time.
  2. \(2\pi \times 0.2 \times 20 = 25\) m/s.
  3. \(n = 234/(2\pi \times 0.6) = 62\) turns a second, about 3,700 turns a minute.
  4. So the body would not spin the opposite way to a single rotor.

Common mistake

Using the diameter as the radius. Ingenuity's rotors are 1.2 m across, so the radius is 0.6 m.

Where it leads