Torque

Long before physicists formalized the concept, ancient engineers intuitively exploited torque every time they used a lever, turned a capstan, or balanced a beam on a fulcrum. The principle that a force applied farther from a pivot produces a greater rotational effect underpins some of the earliest machines in recorded history. Formalizing this idea required centuries of mathematical development — from Archimedes' law of the lever through Newton's framework of forces and, ultimately, Euler's rotational equations of motion. Understanding torque is essential because translational mechanics alone cannot predict the behavior of any system that rotates, from a simple door hinge to a gyroscope aboard a spacecraft.

The central question torque answers is deceptively simple: how do forces cause rotation? A force applied to a door near its hinges barely moves it, while the same force at the handle swings it wide open. Torque quantifies this dependence on both the magnitude of the force and the geometry of its application point relative to the axis of rotation. Mastering torque is a prerequisite for analyzing everything from static equilibrium of beams and bridges to the angular acceleration of flywheels and planetary motion.

Torque — often represented by the Greek letter τ (tau) — is the rotational analog of force. Just as a net force causes translational acceleration, a net torque causes angular acceleration. Torque is a vector quantity whose direction is determined by the right-hand rule and whose magnitude depends on three factors: the applied force, the distance from the axis, and the angle between the force and the position vector.

The diagram above illustrates the fundamental geometry of torque. Notice that only the component of F perpendicular to r contributes to the torque — the parallel component simply pulls along the lever arm without causing rotation. This is precisely what the cross product captures: it extracts the perpendicular contribution automatically through the sin θ factor. When θ = 90°, the entire force contributes to rotation and the torque is maximized at τ = rF. When θ = 0° or 180°, the force points along or opposite to r and produces zero torque.

The same 20 N perpendicular force at lever arms 0.20 m and 0.80 m produces 4 N·m and 16 N·m.

Physics C • beam 12 kg • load 8 kg; illustrative apparatus not to scale.

The mathematical structure of torque emerges naturally from extending Newton's second law to rotating systems. We begin with the vector definition, derive the scalar magnitude, and then connect torque to rotational dynamics through the moment of inertia.

A useful derivation connects the translational and rotational forms. Consider a single particle of mass m constrained to move in a circle of radius r. The tangential component of Newton's second law gives F_⊥ = ma_t. Multiplying both sides by r yields rF_⊥ = mra_t. Since a_t = rα, we obtain τ = mr²α = Iα, confirming the rotational second law for a point mass with I = mr².

Torque problems in AP Physics C fall into two broad categories: static equilibrium (Στ = 0) and rotational dynamics (Στ = Iα). In static equilibrium problems, you choose a convenient pivot, sum torques, and solve for unknown forces or distances. In dynamics problems, you combine torque equations with translational equations (often via a string-and-pulley constraint) to find angular or linear accelerations. The diagram below illustrates the classic Atwood machine with a massive pulley — one of the most commonly tested configurations on the AP exam.

A hanging load’s tension is less than its weight while it accelerates downward. With no slip, a = rα, so its axle torque is m(g − rα)r.

For the displayed relationship τ = 0.060α + 0.012, the slope gives I = 0.060 kg·m² and the intercept gives a constant opposing friction torque of 0.012 N·m.

For a load accelerating downward, mg − T = ma. With no slip, a = rα, so the applied axle torque is Tr = m(g − rα)r. If bearing friction opposes rotation, Tr = Iα + τfriction.Open full diagram

A uniform horizontal beam of mass M = 12 kg and length L = 4.0 m is attached to a wall by a hinge at its left end. A cable attached to the right end of the beam makes an angle of 30° with the beam and connects to the wall above the hinge. A block of mass m = 8.0 kg hangs from a point 3.0 m from the hinge. Find the tension in the cable and the force exerted by the hinge.

At the AP Physics C level, torque connects to several deeper ideas that bridge the gap to upper-division mechanics. The relationship τ = dL/dt generalizes Newton's second law for rotation, relating torque to the time rate of change of angular momentum L. When the net external torque on a system is zero, angular momentum is conserved — a principle with applications ranging from figure skaters pulling in their arms to the precession of gyroscopes.

The work–energy theorem also has a rotational analog. The work done by a torque rotating through an angle dθ is dW = τ dθ, and integrating yields W = ∫τ dθ. For a constant torque, this simplifies to W = τΔθ. The rotational kinetic energy K_rot = ½Iω² connects to work via the work–energy theorem: W_net = ΔK_rot. These relationships are frequently tested in FRQ problems that require translating between representations — for instance, deriving the angular velocity of a pulley system from energy methods rather than torque–acceleration kinematics.