Rotational Dynamics Designing a Propeller Velocity/Acceleration

[hitspace]

Homework Statement

You are asked to design an airplane propleller to turn at 2400 rpm. The forward airspeed of the plane is to be 75.0 m/s, and the speed of the tips of the propeller blades through the air must not exceed 270 m/s. What is the maximum radius the propeller can have? With this radius, what is the acceleration of the propeller tip?

Homework Equations

v = rw
alpha = omega^2 (r)

The Attempt at a Solution

My issue with this problem is after trying to understand the explained solution. This problem is an example in my physics book.

Both the book and I convert the 2400 rpm into rad/s

(2400(2)pi)/60 = 251 rad/s

This is where I get confused. I simply divide as so, v= rw , so v/w = r

270/251 = 1.07 m

The book doesn't do this, instead it says V_tip^2 = V_plane^2 + V_tan^2 = V_plane^2 + (r^2)(w^2)
They then solve for r and get r = 1.03 vs what I got, 1.07.

What exactly are they accounting for is I think what I am trying to understand. Thanks in advance.

 

[Orodruin]

You are forgetting the velocity component in the forward direction (which is orthogonal to the tangential, hence Pythagoras).

Edit: To expand on that, the problem states that the tips have a max velocity through the air - not relative to the centre of the propeller. The entire propeller moves relative to the air.

 

[hitspace]

Thanks for the assistance. I'm still a bit confused. Are you saying that since the problem defined the 270 m/s velocity limit through the air as both components of velocity , tangential and otherwise, I need to account for both components?

I have this inkling that it might help if you would kindly speculate on the wording the book would use if I didn't need to account for the forward velocity of the plane. As I read it now, I was under the impression that I should disregard it.

Thank you again, and I do apologize if I'm wearing on your patience.

 

[haruspex]

Are you saying that since the problem defined the 270 m/s velocity limit through the air as both components of velocity , tangential and otherwise

Not sure what you mean by that. It defines 270 m/s as the maximum speed of the tip relative to the air. Speed is the magnitude of the overall velocity. You have two components, tangential and forward, at right angles to each other, so to find the magnitude of the overall velocity vector you can use Pythagoras.

 

[Orodruin]

I have this inkling that it might help if you would kindly speculate on the wording the book would use if I didn't need to account for the forward velocity of the plane. As I read it now, I was under the impression that I should disregard it

It would say something like "tangential speed" or "rotational speed". As posed, it is rather clear that it means overall speed.

 

[hitspace]

Thanks for the insight. I appreciate you taking the time to help me out.