Torque and Rotational Motion
Torque and angular momentum relate to each other exactly the way force and linear momentum do — \(\tau=dL/dt\), the direct rotational counterpart of \(F=dp/dt\) from Unit 105. Everything from there carries over: integrating a time-varying torque gives angular momentum change, integrating torque over angle gives rotational work, and \(\tau=I\alpha\) is the same special case F=ma always was, just for fixed moment of inertia.
What you'll learn
- Differentiate a given L(t) to find torque, and integrate a given τ(t) to find angular momentum or angular velocity
- Recognize τ=Iα as the rotational analog of F=ma, valid when moment of inertia is constant
- Compute rotational work from a position-dependent (angle-dependent) torque by integrating
1. Torque as the Rate of Change of Angular Momentum
Torque is defined as \(\tau = dL/dt\), directly paralleling force as \(F=dp/dt\) from Unit 105. For fixed moment of inertia, this simplifies to \(\tau=I\alpha\) exactly the way \(F=dp/dt\) simplified to \(F=ma\) when mass was fixed — the same relationship, one level removed.
This is also where the familiar geometric definition of torque comes from, rather than being a separate fact to memorize. For a single particle, \(L = r \times p\), so \(dL/dt = v \times p + r \times F\). Since \(v\) and \(p\) point the same direction, \(v \times p = 0\), leaving \(\tau = dL/dt = r \times F\) — a cross product whose magnitude is \(\tau = rF\sin\theta\), \(\theta\) being the angle between the position vector and the force. The lever-arm picture of torque is this same equation, just written out geometrically.
2. Angular Velocity from a Time-Varying Torque
When torque depends on time, \(\alpha(t)=\tau(t)/I\) is just as genuine a function of time, and integrating it (with an initial condition) recovers \(\omega(t)\) exactly as in Unit 103's angular kinematics — the only new piece is that the angular acceleration itself now comes from a physically applied torque rather than being given directly.
3. Rotational Work from an Angle-Dependent Torque
Rotational work is \(W=\int\tau(\theta)\,d\theta\), reducing to \(\tau\Delta\theta\) only when torque is constant — exactly parallel to linear work needing an integral once force stopped being constant in position.
Key equations
- τ = dL/dt — Torque is defined as the rate of change of angular momentum — the rotational analog of F=dp/dt.
- τ = rFsinθ — The geometric (lever-arm) form of the same τ=dL/dt=r×F relationship above — this is what you compute from when a force and its point of application are given directly rather than L(t).
- τ = Iα — For fixed I, τ=dL/dt=d(Iω)/dt=I(dω/dt)=Iα — the rotational analog of F=ma.
- W = ∫τ(θ)dθ — Rotational work done by a torque that depends on angular position — reduces to τΔθ only when torque is constant.