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Required information

A $22-$mm$-$diameter uniform steel shaft is $500$ mm long between bearings. Take the value of acceleration due to gravity

$g$ to be $9\mathrm{.}81\frac{m}{s^2},$ the value of $y$ for carbon steel to be $76\mathrm{.}5{10}^{3}N*m^{-3},$ and the value of $E$ to be $207{10}^9\frac{N}{m^2}.$

What is the lowest critical speed of the shaft in rads$/$s$?$

The lowest critical speed of the shaft is

$ra\frac{d}{s}.!$

If the goal is to double the critical speed, what is the new diameter in mm$?$

What is the critical speed of a half$-$size model of the original shaft?

The critical speed of a half$-$size model of the original shaft is

$ra\frac{d}{s}.$