Machine Design
If shearing stress in nut is half the tensile stress in a bolt, then nut length should be equal to

1.5 × diameter of bolt
1.25 × diameter of bolt
0.75 × diameter of bolt
Diameter of bolt

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Machine Design
Castle nut is a locking device in which

One smaller nut is tightened over main nut and main nut tightened against smaller one by loosening, creating friction jamming
A slot is cut partly is middle of nut and then slot reduced by tightening a screw
Through slots are made at top and a cotter pin is passed through these and a hole in the bolt, and cotter pin spitted and bent in reverse direction at other end
A hard fibre or nylon cotter is recessed in the nut and becomes threaded as the nut is screwed on the bolt causing a tight grip

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