Relation Between Torque and Moment of Inertia
Simple Explanation
Just as Newton's second law relates force to mass and linear acceleration (F = ma), its rotational equivalent relates net torque to moment of inertia and angular acceleration: τ = Iα.
Why Do We Need It?
This relation is the rotational version of Newton's second law, letting you predict how fast an object will spin up or slow down under a known torque, given its moment of inertia.
Formula
Torque and Angular Acceleration
τ = Iα
This is the rotational equivalent of Newton's second law (F = ma) — net torque produces angular acceleration, with moment of inertia playing the role mass plays in linear motion.
- τ
- — Net torque, in N·m
- I
- — Moment of inertia, in kg·m²
- α
- — Angular acceleration, in rad/s²
When to use it: Use to find the angular acceleration produced by a net torque on a rotating object, or to find the torque needed to produce a given angular acceleration.
Worked Example
Finding angular acceleration from torque
A net torque of 12 N·m acts on a wheel with moment of inertia 3 kg·m². Find its angular acceleration.
Why Does This Work?
This relation follows directly from applying Newton's second law to every small piece of a rotating object and summing the results — the total torque needed to produce a given angular acceleration scales exactly with how the object's mass is distributed (its moment of inertia), mirroring how force relates to mass and linear acceleration.
Real-Life Example
Why heavier flywheels resist speeding up
Engineers use large, heavy flywheels in engines specifically to resist rapid changes in rotational speed.
A flywheel with a large moment of inertia requires a large torque to produce even a small angular acceleration (α = τ/I) — this resistance to sudden speed changes is exactly why flywheels are used to smooth out engine power delivery.
Practice
A disc with moment of inertia 5 kg·m² needs an angular acceleration of 6 rad/s². Find the required torque.
HardCommon mistake
Confusing this rotational relation with F = ma directly — the quantities are analogous (τ↔F, I↔m, α↔a) but are NOT interchangeable; always use the correct rotational quantities together, not a mix of linear and rotational terms.
Quick Review
- τ = Iα is the rotational equivalent of F = ma.
- Net torque produces angular acceleration, resisted by moment of inertia.
- A larger moment of inertia means more torque is needed for the same angular acceleration.