he fan blades on a jet engine make one thousand revolutions in a time of 54.9 ms. What is the angular frequency of the blades?

Hi ! Here I will discuss about the angular frequency or what is also often called the angular velocity because it has the same unit dimensions. Angular frequency occurs, when an object vibrates (either moving harmoniously / oscillating or moving in a circle). Angular frequency can be roughly interpreted as the magnitude of the change in angle (in units of rad) per unit time. So, based on this understanding, the angular frequency can be calculated using the equation :

[tex] \boxed{\sf{\bold{\omega = \frac{\theta}{t}}}} [/tex]

With the following condition :

[tex] \sf{\omega} [/tex] = angular frequency (rad/s)
[tex] \sf{\theta} [/tex] = change of angle value (rad)
t = interval of the time (s)

Problem Solving

We know that :

[tex] \sf{\theta} [/tex] = change of angle value = 1,000 revolution = 1,000 × 2π rad = 2,000π rad/s >> Remember 1 rev = 2π rad/s.
t = interval of the time = 54.9 s.

What was asked :

[tex] \sf{\omega} [/tex] = angular frequency = ... rad/s

Step by step :

[tex] \sf{\omega = \frac{\theta}{t}} [/tex]

[tex] \sf{\omega = \frac{2,000 \pi}{54.9}} [/tex]

[tex] \boxed{\sf{\omega \approx 36.43 \pi \: rad/s}} [/tex]

Conclusion :

So, the angular frequency of the blades approximately 36.43π rad/s.