The diameters of the main rotor and tail rotor of a single-engine helicopter are 7.60 m and 1.02 m,

The diameters of the main rotor and tail rotor of a single-engine helicopter are 7.60 m and 1.02 m, respectively. The respective rotational speeds are 450 rev/min and 4138 rev/ min. Calculate the speeds of the tips of both rotors. Compare these speeds with the speed of sound, 343 m/s
Problem Overview:
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Diameter of main rotor (D₁) = 7.60 m
Diameter of tail rotor (D₂) = 1.02 m
Rotational speed of main rotor (ω₁) = 450 rev/min
Rotational speed of tail rotor (ω₂) = 4138 rev/min
Speed of sound (v_sound) = 343 m/s

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Part 1: Speed of the Tip of the Main Rotor
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1. Convert the rotational speed from rev/min to rad/s:
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ω₁ (rad/s) = ω₁ (rev/min) × (2π rad/rev) × (1 min/60 s)
ω₁ (rad/s) = 450 × (2π / 60)
ω₁ (rad/s) = 47.123 rad/s

2. Find the radius of the main rotor:
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r₁ = D₁ / 2 = 7.60 / 2 = 3.80 m

3. Calculate the speed of the tip of the main rotor:
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v₁ = r₁ × ω₁
v₁ = 3.80 × 47.123
v₁ ≈ 179.07 m/s

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Part 2: Speed of the Tip of the Tail Rotor
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1. Convert the rotational speed from rev/min to rad/s:
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ω₂ (rad/s) = ω₂ (rev/min) × (2π rad/rev) × (1 min/60 s)
ω₂ (rad/s) = 4138 × (2π / 60)
ω₂ (rad/s) ≈ 433.565 rad/s

2. Find the radius of the tail rotor:
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r₂ = D₂ / 2 = 1.02 / 2 = 0.51 m

3. Calculate the speed of the tip of the tail rotor:
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v₂ = r₂ × ω₂
v₂ = 0.51 × 433.565
v₂ ≈ 221.72 m/s

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Comparison with the Speed of Sound:
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1. Compare the speeds of the tips with the speed of sound (343 m/s):
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Speed of the main rotor tip (v₁) ≈ 179.07 m/s
Speed of the tail rotor tip (v₂) ≈ 221.72 m/s

Both speeds are less than the speed of sound (343 m/s), so neither rotor tip exceeds the speed of sound.