Answer:
The angular velocity is [tex]w= \sqrt[4]{\frac{a_{max}^2}{r^2} - \alpha ^2}[/tex]
Explanation:
Generally the acceleration experienced by the propeller blade's is broken down into
The Radial acceleration which is mathematically represented as
[tex]a_r = \frac{v^2}{r} = w^2r[/tex]
And the Tangential acceleration which is mathematically represented as
[tex]a_r = \alpha r[/tex]
The net acceleration is evaluated as
[tex]a = \sqrt{a_r^2 + a_t^2}[/tex]
Now since angular speed varies directly with angular acceleration so when acceleration is maximum the angular velocity is maximum also and this point if the propeller blade's tip exceeds it the blade would fracture
So at maximum angular acceleration we a have
[tex]a_{max} = \sqrt{a_r^2 + a_t^2}[/tex]
[tex]a_{max}^2 = a_r^2 + a_t^2[/tex]
[tex]a_{max}^2 = (w^2r)^2 + (\alpha r)^2[/tex]
[tex]a_{max}^2 = r^2 w^4 + r^2 \alpha ^2[/tex]
[tex]a_{max}^2 = r^2 (w^4 + \alpha^2 )[/tex]
[tex]w^4 +\alpha ^2 = \frac{a_{max}^2}{r^2}[/tex]
[tex]w^4 = \frac{a_{max}^2}{r^2} - \alpha ^2[/tex]
[tex]w= \sqrt[4]{\frac{a_{max}^2}{r^2} - \alpha ^2}[/tex]
