1.4 Rotation: torque, angular momentum, and statics

Linear momentum has a rotational counterpart, and Newton’s second law has a rotational form that follows from it directly. The rotational law governs spinning bodies, the stability of structures, and the lever — the oldest machine.

Torque and angular momentum

For a body at position $\mathbf r$ from a chosen origin with momentum $\mathbf p$, the angular momentum about that origin is $\mathbf L = \mathbf r\times\mathbf p$, and the torque of a force is $\boldsymbol\tau = \mathbf r\times\mathbf F$. Differentiating $\mathbf L$ and using $\mathbf F = d\mathbf p/dt$ gives the rotational second law:

(The $\mathbf v\times\mathbf p$ term vanishes because $\mathbf v$ and $\mathbf p$ are parallel.) When the net torque is zero, angular momentum is conserved — the reason a spinning skater speeds up on pulling in her arms, and a planet sweeps equal areas in equal times.

For a rigid body rotating about a fixed axis at angular velocity $\omega$, the angular momentum is $L = I\omega$ and the rotational kinetic energy is $\tfrac12 I\omega^2$, where the moment of inertia $I = \sum_i m_i r_i^2$ plays the role mass plays in linear motion — it measures the resistance to angular acceleration, weighting each mass element by the square of its distance from the axis.

Statics: the balance of force and torque

A rigid body is in static equilibrium when both the net force and the net torque about every point vanish:

The two conditions are independent: a pair of equal and opposite forces offset from each other (a couple) sums to zero force but a non-zero torque, and would spin the body even though it does not translate.

The lever

The lever is the simplest non-trivial application: two forces $F_L$ and $F_R$ acting at distances $L_L$ and $L_R$ on opposite sides of a pivot balance when their torques are equal,

A small force at a long arm matches a large force at a short arm — the mechanical advantage $L_L/L_R$. The lever trades force for distance: the long arm moves through a larger displacement, and the work $F\,\Delta x$ done at each end is equal, as energy conservation demands.

Slide the pivot and the two masses to bring the torque-balance condition into and out of balance. The mechanical advantage rises sharply as the pivot approaches the short-arm mass — the principle behind every lever, gear, and bone-and-tendon system that converts a small motion into a large force or the reverse.