$($a$)$ Design a key for a $60$ mm diameter shaft.

${}_{\text{t}\text{h}\text{a}\text{n}\text{t}}=$ shearing stress of shaft $=70$MPa

${}_{\text{k}\text{e}\text{y}}=$ shearing stress of key $=60$MPa

${}_c,=$ bearing pressure $=85$MPa

$($b$)$ Determine the power that the key can safely transmit at $120mi{n}^{-1}.$

Assume $t=\frac{D}{6}$ and $b=\frac{D}{4}.$

$(110$; $372$mm; $2970$Nm; $37,3$ kW$)$

Why should the longest key be used?