Question

When the tips of the rotating blade of an airplane propeller have a linear speed greater than the speed of sound, the propeller makes a lot of noise, which is undesirable. If the tip-to-tip length of the blade is \(2.601 \mathrm{~m}\), what is the maximum angular frequency (in revolutions per minute) with which it can rotate? Assume that the speed of sound is \(343.0 \mathrm{~m} / \mathrm{s}\) and that the blade rotates about its center.

Short answer

Solution: 1. Calculate the radius of the circle traced by the propeller tips: Radius (r) = 2.601 m / 2 2. Convert the speed of sound to revolutions per second: ω = (v / r) × (1 rev / 2π rad) 3. Calculate the maximum angular speed in revolutions per minute: Max angular speed (ω) = (343.0 m/s / (2.601 m / 2)) × (1 rev / 2π rad) × (60 s / min) Answer: Plug the numbers into the equation to find the maximum angular frequency of the airplane propeller.

Step-by-step solution

Calculate the radius of the circle traced by the propeller tips

Given the tip-to-tip length of the propeller is \(2.601 \mathrm{~m}\), we can determine the radius of the circle traced by the propeller tips by dividing the length by 2. Radius (r) = \(\frac{2.601}{2}\)

Convert the speed of sound to revolutions per second

The speed of sound (v) is given in meters per second (\(343.0 \mathrm{~m} / \mathrm{s}\)). To convert it to revolutions per second (ω), we can use the relationship between linear speed, angular speed, and radius: Linear speed (v) = ω × r The result will be in radians per second, and we can then convert it to revolutions per second by using the conversion factor: 1 revolution = \(2\pi\) radians ω = \(\frac{v}{r}\) × \(\frac{1 \mathrm{~rev.}}{2\pi \mathrm{~rad.}}\)

Calculate the maximum angular speed in revolutions per minute

Since the magnitude of the angular speed in radians per second must not exceed the speed of sound, we insert the speed of sound and the radius into our previous equation to find the maximum angular speed in revolutions per second, and then multiply by 60 to convert it to revolutions per minute: Max angular speed (ω) = \(\frac{343.0 \mathrm{~m} / \mathrm{s}}{\frac{2.601 \mathrm{~m}}{2}}\) × \(\frac{1 \mathrm{~rev.}}{2\pi \mathrm{~rad.}}\) × \(60 \mathrm{~s} / \mathrm{min}\) Perform the calculations, and we will get the answer.