Question

The diameters of the main rotor and tail rotor of a single-engine helicopter are 7.62 m and 1.01 m, respectively. The respective rotational speeds are 458 rev/min and 4137 rev/min. Calculate the speeds of the tips of both rotors.

main rotor	m/s
tail rotor	m/s

Compare these speeds with the speed of sound, 343 m/s.

vmain rotor / vsound
vtail rotor / vsound

Answer 1

here,

dia of main rotor, dr = 7.62 m
dia of tail rotor, dt = 1.01 m

Velocity of main rotor, vr = 458 rev/min
wr = 458 * (2*pi/60) = 47.961 rad/s

Velocity of tail rotor, vt = 4137 rev/min
wr = 4137 * (2*pi/60) = 433.225 rad/s

as, Velocity = angular velocity * radius

for main rotor:
vr = wr * dr/2
vr = 47.961 * 7.62/2
vr = 182.731 m/s

for Tail rotor:
vt = wt * dt/2
vt = 433.225 * 1.01/2
vt = 218.778 m/s or 219 m/s(rounded off)

as Velocity of sound = 343 m/s

Case 1 :
vmain rotor / vsound = 182.731 / 343 = 0.532 , about 53 %

Case 2:
vtail rotor / vsound = 219 / 343 = 0.638 , 63.8 % or 64 %(rounded off)