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A flywheel can rotate in order to store kinetic energy. The flywheel is a uniform disk made of a material with a density $\rho $ and tensile strength $\sigma $ (measured in Pascals), a radius $r$ , and a thickness $h$ . The flywheel is rotating at the maximum possible angular velocity so that it does not break. Which of the following expression correctly gives the maximum kinetic energy per kilogram that can be stored in the flywheel ? Assume that $\alpha $ is a dimensionless constant
$\alpha \sqrt {\frac{{\rho \sigma }}{r}} $
$\alpha h\sqrt {\frac{{\rho \sigma }}{r}} $
$\alpha \left( {\frac{h}{{{r^2}}}} \right)\left( {\frac{\sigma }{\rho }} \right)$
$\frac{{\alpha \sigma }}{\rho }$
Solution
Dimensions of kinetic energy per unit mass $=L^{2} T^{-2}$
$\left[\frac{\alpha \sigma}{\rho}\right]=\frac{M L^{-1} T^{-2}}{M L^{-3}}=L^{2} T^{-2}$
