Theory of Machine
The natural frequency of free transverse vibrations due to uniformly distributed load acting over a simply supported shaft is (where δS = Static deflection of simply supported shaft due to uniformly distributed load)

0.5615/ √δS
0.4985/ √δS
0.571/ √δS
0.6253/ √δS

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Theory of Machine
In gears, interference takes place when

Gears do not move smoothly in the absence of lubrication
Pitch of the gears is not same
Gear teeth are undercut
The tip of a tooth of a mating gear digs into the portion between base and root circles

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Theory of Machine
The velocity of piston in a reciprocating steam engine is given by (where ω = Angular velocity of crank, r = Radius of crank pin circle, θ = Angle turned by crank from inner dead center, and n = Ratio of length of connecting rod to the radius of crank)

ω²r [cos θ + (cos 2θ/n)]
ωr [cos θ + (cos 2θ/n)]
ωr [sin θ + (sin 2θ/n)]
ω²r [sin θ + (sin 2θ/n)]

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