Editing Flywheel (section)

== Physics ==
[[file:Volin.jpg|thumb|A mass-produced flywheel]]

The [[kinetic energy]] (or more specifically [[rotational energy]]) stored by the flywheel's [[wikt:rotor|rotor]] can be calculated by <math display="inline">\frac{1}{2} I \omega^2</math>. ω is the [[angular velocity]], and <math> I </math> is the [[moment of inertia]] of the flywheel about its axis of symmetry. The moment of inertia is a measure of resistance to [[torque]] applied on a spinning object (i.e. the higher the moment of inertia, the slower it will accelerate when a given torque is applied). The [[moment of inertia]] can be calculated for cylindrical shapes using mass (<math display="inline">m</math>) and radius (<math>r</math>). For a solid cylinder it is <math display="inline">\frac{1}{2} mr^2</math>, for a thin-walled empty cylinder it is approximately <math display="inline">m r^2</math>, and for a thick-walled empty cylinder with constant density it is <math display="inline">\frac{1}{2} m({r_\mathrm{external}}^2 + {r_\mathrm{internal}}^2) </math>.<ref>{{cite web |url=http://www.freestudy.co.uk/dynamics/moment%20of%20inertia.pdf |title=Tutorial – Moment of Inertia |access-date=2011-12-01 |url-status=live |archive-url=https://web.archive.org/web/20120105181252/http://www.freestudy.co.uk/dynamics/moment%20of%20inertia.pdf |archive-date=2012-01-05 |page=10 |website=FreeStudy.co.uk |first=D.J. |last=Dunn}}</ref>

For a given flywheel design, the kinetic energy is proportional to the ratio of the [[hoop stress]] to the material density and to the mass. The [[specific strength|specific tensile strength]] of a flywheel can be defined as <math display="inline">\frac{\sigma_t}{\rho} </math>. The flywheel material with the highest specific tensile strength will yield the highest energy storage per unit mass. This is one reason why [[carbon fiber]] is a material of interest. For a given design the stored energy is proportional to the hoop stress and the volume.{{Citation needed|date=May 2022}}

An electric motor-powered flywheel is common in practice. The output power of the electric motor is approximately equal to the output power of the flywheel. It can be calculated by <math display="inline">(V_i)(V_t)\left ( \frac{\sin(\delta)}{X_S}\right )</math>, where <math>V_i</math> is the voltage of [[Rotor (electric)|rotor]] winding, <math>V_t</math> is [[stator]] voltage, and <math>\delta</math> is the angle between two voltages. Increasing amounts of rotation energy can be stored in the flywheel until the rotor shatters. This happens when the [[hoop stress]] within the rotor exceeds the [[ultimate tensile strength]] of the rotor material. [[Tensile stress]] can be calculated by <math> \rho r^2 \omega^2 </math>, where <math> \rho </math> is the density of the cylinder, <math> r </math> is the radius of the cylinder, and <math> \omega </math> is the [[angular velocity]] of the cylinder.