Torque and rotational inertia
Also known as: moment of force
Torque is the rotational effect of a force, equal to τ = rF sin θ — the force times its lever arm about a pivot. Rotational inertia (moment of inertia) measures an object's resistance to changes in its rotation.
Torque is the rotational analog of force: it measures how effectively a force twists an object about an axis. Its magnitude is τ = rF sin θ, where r is the distance from the pivot to where the force is applied, F is the force, and θ is the angle between them. Only the force component perpendicular to the lever arm creates torque — pushing straight along a wrench handle toward the bolt turns nothing, while pushing perpendicular at the handle's end turns it best. Torque is measured in newton-meters (N·m) and has a direction, conventionally counterclockwise positive.
Rotational inertia (moment of inertia, I) is the rotational analog of mass: it measures resistance to changes in rotational motion. Unlike mass, it depends on how mass is distributed — the farther the mass sits from the axis, the larger the rotational inertia. For a point mass, I = mr². That is why a figure skater spins faster after pulling in her arms: she reduces her rotational inertia, and angular momentum conservation speeds her rotation.
The two quantities combine in Newton's second law for rotation, τ = Iα: net torque produces angular acceleration, just as net force produces linear acceleration. When the net torque on a system is zero, it is in rotational equilibrium — the principle behind balanced seesaws, ladders leaning against walls, and levers. In human movement, muscles generate torque about joints, with tendon insertion points setting the lever arms.
Torque and rotational inertia are core mechanics topics on the AP Physics 1 exam and the MCAT's physics section, both of which test equilibrium and rotational dynamics problems. The CSCS exam applies the same concepts to biomechanics, where joint torques and body lever systems explain human strength and power.
Key takeaways
- Torque is τ = rF sin θ — the product of the lever arm and the perpendicular force component.
- Rotational inertia measures resistance to rotational change and grows as mass moves farther from the axis (I = mr² for a point mass).
- Newton's second law for rotation is τ = Iα.
- Zero net torque means rotational equilibrium — the basis of lever and balance problems.
- AP Physics 1, the MCAT, and the CSCS all test torque, from physics problems to joint biomechanics.