Rotational Energy and Angular Momentum

Energy of rotation

A spinning flywheel is clearly doing something energetic even though its center goes nowhere. Its energy is rotational kinetic energy, and it is the perfect echo of \tfrac12 mv^2 with I replacing m and \omega replacing v.

K_{\text{rot}} = \tfrac{1}{2}I\omega^{2} \qquad\longleftrightarrow\qquad K_{\text{trans}} = \tfrac{1}{2}mv^{2}

Rotational kinetic energy uses moment of inertia and angular velocity where linear KE uses mass and speed.

An object that both moves and spins — a rolling ball — carries both kinds at once. Its total kinetic energy is simply the sum:

K_{\text{total}} = \tfrac{1}{2}mv^{2} + \tfrac{1}{2}I\omega^{2}

The kinetic energy of a rolling body splits into translation of its center of mass plus rotation about it.

Rolling without slipping

A wheel that grips the road rolls without slipping: it does not skid, so the point touching the ground is, for that instant, perfectly still. This locks the spin to the travel — one rotation moves the wheel forward exactly one circumference.

v_{\text{cm}} = R\,\omega \,,\qquad a_{\text{cm}} = R\,\alpha

The rolling-without-slipping condition ties the center's speed to the spin — the same v = rω, now for the whole wheel's advance.

Angular momentum: rotation's momentum

Just as linear momentum p = mv measures 'how hard it is to stop' a moving object, angular momentum L measures how hard it is to stop a spinning one. For a rigid body it is moment of inertia times angular velocity; for a single particle it is L = mvr about the axis.

L = I\,\omega \qquad (\text{particle: } L = m v r_{\perp}) \qquad\longleftrightarrow\qquad p = m v

Angular momentum is the rotational partner of linear momentum, with I and ω in place of m and v.

Angular momentum is a vector, and like \vec\omega it points along the axis by the right-hand rule. That vector nature is why spinning things are directionally stubborn — a moving bicycle, a thrown football, a rifle bullet all stay pointed where their spin axis points.

Conservation of angular momentum

Here is the payoff. If no external torque acts on a system, its total angular momentum cannot change — this is the conservation of angular momentum, as fundamental as conservation of energy or momentum. So if the object reshapes itself to a smaller I, its \omega must rise to keep L = I\omega fixed.

\tau_{\text{ext}} = 0 \;\Longrightarrow\; L = I\omega = \text{constant} \;\Longrightarrow\; I_{1}\omega_{1} = I_{2}\omega_{2}

With no external torque, shrinking I forces ω up — the physics of the spinning skater.

A figure skater spins slowly with arms out, then pulls them in: her I drops, so her \omega shoots up and she blurs. A collapsing star doing the same becomes a pulsar spinning hundreds of times a second. A cat rights itself in mid-air by twisting one part against another with total L staying zero.

Why it holds: torque changes angular momentum

The deep reason is the most general form of Newton's second law for rotation: torque is the rate of change of angular momentum, mirroring F = dp/dt. If the net external torque is zero, L has zero rate of change — it is conserved. Every example above is this one equation in disguise.

\vec{\tau}_{\text{net}} = \frac{d\vec{L}}{dt} \qquad\longleftrightarrow\qquad \vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}

The most general rotational Newton's second law: torque is the rate of change of angular momentum.

This vector equation also explains gyroscopes: a torque that is sideways to \vec L does not slow the spin but swings its direction, making the axis sweep in a slow cone — precession. It is why a spinning top does not fall over and how a gyrocompass finds north.