What is the relationship to the energy capacity of a flywheel and its radius, mass, rotational velocity etc?
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The amount of energy a flywheel can store is equal to 1/2 * (moment of inertia) * angular velocity^2.
The moment of inertia has to do with how heavy the spinning flywheel is, and how its mass is distributed around its axis of rotation. Basically, the larger the flywheel's diameter and the more mass its has, the more energy it can store.
The limiting factor for energy storage in a flywheel is its mechanical strength, because the stresses imposed on a fast-spinning flywheel will tend to make it fly to pieces. Fancy flywheels designed for energy storage must be made of exotic materials to ensure that the flywheel does not explode.
A flywheel is designed to efficiently store rotational kinetic energy. It resists changes in rpm by virtue of its rotational moment of inertia. Its stored kinetic energy is given by
Where $I$ is its moment of inertia and $ω$ is its angular velocity.
The moment of inertia is given by
Where $m$ is the mass, $r$ is radius, and $k$ is the inertial constant which depends on the shape of the flywheel. Examples are $k=1$ where the mass is concentrated at the rim of the wheel and $k=0.606$ for a flat solid circular disk of uniform thickness.
The fact that the stored kinetic energy goes up as the square of the radius of the wheel and where the mass is concentrated at the perimeter, speaks to the ability of a flywheel to store energy.
Hope this helps.