3.11.1.2 - Rotational Kinetic Energy

Deriving rotational kinetic energy

Every small mass element in a rotating rigid body is moving, so each has kinetic energy.

An element $m_i$ at radius $r_i$ has speed $v_i=\omega r_i$. All elements share the same angular speed $\omega$, but elements farther out have greater linear speeds. Thus

The sum is the moment of inertia about the rotation axis, giving

The formula describes rotation about the specified axis. Reversing the direction does not change the energy at a given speed because $\omega$ is squared. If the object's centre also travels through space, its translational kinetic energy must be included separately.

Stored and usable energy

A flywheel that slows down releases the difference between its initial and final rotational kinetic energies.

Convert revolutions per minute before substituting: $\omega=2\pi n/60$, where $n$ is the numerical speed in rpm. A flywheel with $I=0.45\,\mathrm{kg\,m^2}$ at $3000\,\mathrm{rpm}$ has $\omega=100\pi\,\mathrm{rad\,s^{-1}}$ and

Retain the unrounded angular speed in the calculation.

If this wheel slows to $2000\,\mathrm{rpm}$, the available energy decrease is

Do not use $\tfrac12I(\omega_{\rm high}-\omega_{\rm low})^2$. Energy below the minimum useful operating speed remains stored. If the transfer efficiency is $\eta$, the load receives $\eta\Delta E$.

Flywheel capacity and losses

A flywheel is a rotating energy store that can exchange energy with a machine as its angular speed changes.

At a fixed speed, increasing $I$ increases stored energy in direct proportion. For a fixed mass and outer radius, concentrating mass nearer the rim increases $I$. At fixed $I$, doubling the angular speed gives four times the energy. These comparisons require the other quantity to stay unchanged.

Maximum usable capacity depends on the permitted speed range, $\tfrac12I(\omega_{\max}^2-\omega_{\min}^2)$, as well as the wheel's mass and distribution. Higher speeds require larger inward forces on the material and create greater mechanical stress, so material strength limits the safe speed. A larger radius cannot be increased indefinitely at the same speed without consequences for stress and size.

Bearing friction and air resistance dissipate energy during storage and transfer. Suitable bearings, lubrication and a low-pressure enclosure can reduce these losses. Reducing friction does not increase $E_k$ at the same $I$ and $\omega$; it helps retain that energy for longer.