3-9. A shaft is supported by a smooth bearing at  and has a diameter  mm. The    motor operates at an angular velocity  rad/s and delivers  kW power. What is the probability that the shaft will fail if the allowable torsion shear stress is  MPa?  Assume  and  are independent.

Solution:

Torque:

 N·m                                                                                                   (1)

N·m                                                                                                       (2)

 N·m due to unit balance.

Torsion shear stress: The polar moment of inertia is  mm⁴.

 MPa                                                                                                 (3)

 MPa                                                                                                     (4)

MPa

Probability of failure:

                                                                                       (5)

Since  kW, and  MPa are independent,  also follows a

normal distribution of .

 MPa                                                                                                  (6)

 MPa                                                                  (7)

Equation (5) can be written as

Ans.