Newton’s Laws for Rotation: Equilibrium and Dynamics (AP Physics 1 Unit 5)

Rotational equilibrium

A condition in which the net external torque about a chosen axis is zero, so angular velocity does not change (no angular acceleration).

Translational equilibrium

A condition in which the net external force is zero, so velocity does not change (no linear acceleration).

Newton’s First Law (rotational form)

If the net external torque about an axis is zero, the object’s angular velocity about that axis remains constant (could be zero or nonzero).

Static equilibrium

Equilibrium where the object is not translating and not rotating (both linear and angular velocities are zero).

Dynamic equilibrium

Equilibrium where the object may translate at constant velocity and/or rotate at constant angular velocity (no acceleration, but not necessarily at rest).

Torque (τ)

The rotational effect of a force that tends to change an object’s rotation about an axis; depends on force, lever arm distance, and angle.

Torque magnitude formula (angle form)

For a force applied a distance r from the axis with angle θ between r and F: τ = rF sin(θ).

Perpendicular lever arm (r⊥)

The shortest distance from the axis to the force’s line of action; used to compute torque via τ = F r⊥.

Torque magnitude formula (lever arm form)

Torque can be computed as τ = F r⊥, emphasizing that only the moment arm (perpendicular distance) matters.

Units of torque

Newton-meters (N·m).

Torque sign convention (2D)

A chosen rule for positive/negative torque, commonly counterclockwise positive and clockwise negative, based on the rotation tendency about the axis.

Rotational equilibrium condition

Στ = 0, which implies α = 0 (no angular acceleration and thus constant angular velocity).

Full static equilibrium conditions (common in AP Physics 1)

To have no sliding and no rotation: ΣF = 0 and Στ = 0.

Axis choice for torque calculations

You may compute torques about any chosen axis (point) as long as you are consistent; in static equilibrium, net torque is zero about any point.

Convenient pivot (best axis) strategy

Choose an axis that makes unknown forces produce zero torque (e.g., pick the hinge so hinge forces don’t appear in the torque equation).

Zero-torque line-of-action rule

If a force’s line of action passes through the chosen axis, its perpendicular lever arm is zero and it produces zero torque about that axis.

Uniform beam center of mass location

For a uniform beam, weight acts at the center, located at L/2 from either end.

Couple (pure rotation possibility)

A situation where forces can sum to zero (ΣF = 0) but still create a nonzero net torque (Στ ≠ 0), causing rotation without translation.

Moment of inertia (I)

A measure of resistance to changes in rotational motion; rotational analog of mass, dependent on mass distribution relative to the axis.

Moment of inertia for point masses

Defined by I = Σ m r², where r is each mass’s distance from the axis.

Units of moment of inertia

Kilogram–meter squared (kg·m²).

Axis dependence of moment of inertia

The same object can have different I values depending on the chosen axis; I is not an intrinsic constant independent of axis.

Newton’s Second Law (rotational form)

Net external torque about an axis equals moment of inertia times angular acceleration: Στ = Iα.

No-slip (string/pulley) kinematic link

If a string does not slip on a pulley/disk, the linear acceleration of the string relates to angular acceleration by a = αr.

Massive pulley tension difference idea

For a pulley with rotational inertia, tensions on the two sides can differ; the difference in tensions provides the net torque that produces angular acceleration.