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f the magnitude of the acceleration of a propeller blade's tip exceeds a certain value amaxamax, the blade tip will fracture. If the propeller has radius rr, is initially at rest, and has angular acceleration of magnitude αα, at what angular speed ωω will the blade tip fracture? Express your answer in terms of the variables amaxamaxa_max, rrr, and ααalpha.

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okpalawalter8

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]        

                     

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