The diameters of the main rotor and tail rotor of a single-engine helicopter are 7.59 m and 0.98 m, respectively. The respective rotational speeds are 456 rev/min and 4143 rev/min. Calculate the speeds of the tips of both rotors.
main rotor:
tail rotor:
Compare these speeds with the speed of sound, 343 m/s.
speedtip= w*r where w=2PI*rev/min*1min/60sec
To calculate the speeds of the tips of both rotors, we need to first find the circumferences of both rotors. The circumference of a circle can be calculated using the formula:
Circumference = 2 * π * radius
In this case, we are given the diameters of both rotors. The radius can be calculated by dividing the diameter by 2:
Main rotor radius = 7.59 m / 2 = 3.795 m
Tail rotor radius = 0.98 m / 2 = 0.49 m
Now, we can calculate the circumferences of both rotors:
Main rotor circumference = 2 * π * 3.795 m
Tail rotor circumference = 2 * π * 0.49 m
Next, we need to convert the given rotational speeds from revolutions per minute (rev/min) to radians per second (rad/s). We can do this by multiplying by 2π/60, since there are 2π radians in a full circle and 60 minutes in an hour:
Main rotor speed = 456 rev/min * (2π/60) rad/s
Tail rotor speed = 4143 rev/min * (2π/60) rad/s
Now we have the respective speeds of the main and tail rotor in radians per second.
Finally, to calculate the speeds of the tips of both rotors, we multiply the respective circumferences by their respective speeds:
Main rotor tip speed = Main rotor circumference * Main rotor speed
Tail rotor tip speed = Tail rotor circumference * Tail rotor speed
Comparing these speeds with the speed of sound:
Main rotor tip speed / Speed of sound
Tail rotor tip speed / Speed of sound
Let's now substitute the given values and perform the calculations:
Main rotor circumference = 2 * π * 3.795 m = 23.84 m
Tail rotor circumference = 2 * π * 0.49 m = 3.08 m
Main rotor speed = 456 rev/min * (2π/60) rad/s ≈ 47.91 rad/s
Tail rotor speed = 4143 rev/min * (2π/60) rad/s ≈ 432.91 rad/s
Main rotor tip speed = 23.84 m * 47.91 rad/s ≈ 1140.51 m/s
Tail rotor tip speed = 3.08 m * 432.91 rad/s ≈ 1332.71 m/s
Comparing these speeds with the speed of sound:
Main rotor tip speed / Speed of sound ≈ 1140.51 m/s / 343 m/s ≈ 3.32 ≈ Mach 3.32
Tail rotor tip speed / Speed of sound ≈ 1332.71 m/s / 343 m/s ≈ 3.88 ≈ Mach 3.88
Therefore, the speed of the tips of the main rotor is approximately Mach 3.32, and the speed of the tips of the tail rotor is approximately Mach 3.88.