Problem Statement: A gear shaft for a ships reduction gear box is to be evaluated for deflection and critical speed. The

shaft is made of carbon steel $(E=30\frac{M}{P}SI),$ is $20$ inches long, and has a diameter of $8\mathrm{.}0$ inches. It weight $284$ lbs that can

be modeled as a uniformly distrubuted load accross its length. The overall gear reaction is $21,000$ lbs and since the

facewidth of the gear is the same length of the shaft, can also be modeled as a uniformly distributed load accross the

length. Assume the weight and gear reaction act in the same plane. For critical speed calculation assume we have a shaft

of constant cross section with no attached masses $($the large gear acts as if it were part of the shaft$).$ Utilize table A$-9$ for

deflection equations and Schaums Ch $8$ Handout for critical speed equations.

Calculate:

a$)$ the Deflection of the shaft $($in inches$)$

b$)$ the Critical Speed of the shaft $($in RPM$)$