$4\mathrm{.}1$ "High speed rotation is important if a flywheel is to be used as an energy storage

device, given that the stored energy depends on the square of the rotational speed."

Appraise this statement in terms of the features that allow modern flywheels to achieve

this speed.

$4\mathrm{.}2$ A flywheel of three uniform bars, rotating about their central points as spokes of a

wheel, is made from fibres of glass with density $=2200k\frac{g}{m^3},$ and tensile strength

$3500M\frac{N}{m^2}.$ The fibres are aligned along the bars and held together by a minimal

quantity $(10\%)$ of resin of negligible tensile strength and similar density, i$.$e$.$ the tensile

strength of the bars is $90\%$ of the strength of the fibres due to the resin glue. If the

generalised moment of inertia is $I=\frac{Km{a}^2}{2}$ and the moment of inertia for a uniform bar

is $I=\frac{1}{3}m{a}^2,$ calculate:

$($a$)$ show that the shape factor $K=\frac{2}{3}$ and hence estimate the maximum energy

density obtainable.

$($b$)$ if the radius of the uniform bar $a=1\mathrm{.}0m,$ calculate the corresponding angular

velocity.