Topic: Other
Concepts: Dimensional analysis, Kinetic energy

Solution:

Let $\lambda$ be the maximum kinetic energy per kilogram that can be stored in the flywheel. If we double the thickness $h$ of the flywheel, that is equivalent to stacking two flywheels on top of each other. Since each has $\lambda$ energy per kilogram, the combination also has $\lambda$ energy per kilogram. Hence, $\lambda$ is independent of $h$.

We can use dimensional analysis to find $\lambda(r,\rho,\sigma)$. First, the dimensions of each variable are:

We need 1 power of $\sigma$ to have the correct number of powers of $T$:

We need $-1$ powers of $\rho$ to cancel out the mass dependence:

This has the same dimensions as $\lambda$ so there is no $r$ dependence. Thus,

where $\alpha$ is a dimensionless constant, so the answer is E.