A key of width and height 6 mm fixes a gear to a shaft of diameter 20 mm. Power is 10 kW at 600 rpm; allowable key shear stress is 80 N/mm². Neglect compressive stress. Find minimum key length in mm to two decimals.
Angular velocity is $2\pi(600)/60=20\pi$ rad/s. Torque is $10000/(20\pi)=159.1549$ N·m = 159154.9 N·mm. Tangential force on the key is $F=T/(d/2)=15915.49$ N. Shear area is bL, so $L=F/(b\tau)=15915.49/(6\times80)=\mathbf{33.16}$ mm.
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