Unit 5 Rotation: Understanding Torque, Rotational Dynamics, and Rolling (AP Physics C: Mechanics)

Torque (τ)

The rotational “push” that tends to change an object’s rotational motion; depends on how a force is applied relative to an axis of rotation.

Torque magnitude (τ = rF sinθ)

Magnitude of torque from a force F applied at position vector r: τ = rF sinθ, where θ is the angle between r and F (perpendicular component produces torque).

Lever arm / Moment arm (r⊥)

The perpendicular distance from the axis to the line of action of the force; lets you compute torque quickly via τ = r⊥F.

Line of action (of a force)

The infinite line extending along the direction of the applied force through its point of application; r⊥ is measured from the axis to this line.

Net torque (Στ)

The sum of all individual torques (with sign) about a chosen axis: Στ = τ1 + τ2 + …

Torque sign convention (CCW vs CW)

Torques are treated as positive or negative depending on whether they tend to cause counterclockwise (CCW) or clockwise (CW) rotation about the chosen axis; must be consistent (right-hand rule).

Gravitational torque about a pivot

In uniform gravity, weight (mg) can be treated as acting at the center of mass (COM), so gravitational torque is found by applying mg at the COM relative to the pivot.

Moment of inertia / Rotational inertia (I)

Measures an object’s resistance to changes in rotational motion about a particular axis; depends on both total mass and how that mass is distributed relative to the axis.

Point-mass moment of inertia (I = Σ mᵢ rᵢ²)

For discrete masses mᵢ at perpendicular distances rᵢ from the axis: I = Σ mᵢ rᵢ².

Continuous moment of inertia (I = ∫ r² dm)

For a continuous mass distribution: I = ∫ r² dm, where r is the perpendicular distance from the axis to the mass element dm.

Moment of inertia units

SI units of moment of inertia are kg·m².

Thin hoop/ring moment of inertia (about center axis)

For a thin hoop (mass M, radius R) about an axis through the center perpendicular to the plane: I = MR².

Solid disk/cylinder moment of inertia (about center axis)

For a solid disk or cylinder (mass M, radius R) about its symmetry axis: I = (1/2)MR².

Thin rod moment of inertia (about center)

For a thin rod (mass M, length L) about an axis through its center perpendicular to the rod: I = (1/12)ML².

Thin rod moment of inertia (about one end)

For a thin rod (mass M, length L) about an axis through one end perpendicular to the rod: I = (1/3)ML².

Solid sphere moment of inertia (about a diameter)

For a solid sphere (mass M, radius R) about a diameter: I = (2/5)MR².

Thin spherical shell moment of inertia (about a diameter)

For a thin spherical shell (mass M, radius R) about a diameter: I = (2/3)MR².

Parallel-axis theorem

Relates moment of inertia about any axis parallel to the COM axis: I = I_cm + Md², where d is the perpendicular distance between axes.

Perpendicular-axis theorem

For a flat (planar) object in the xy-plane: Iz = Ix + I_y for perpendicular axes through the same point.

Rotational kinetic energy (K_rot)

Energy of rotation: K_rot = (1/2)Iω², where ω is angular speed.

Newton’s second law for rotation (Στ = Iα)

For rotation about a fixed axis with constant I: the net torque equals moment of inertia times angular acceleration: Στ = Iα.

Torque–angular momentum relation (Στ = dL/dt)

The general rotational dynamics statement: net external torque equals the time rate of change of angular momentum L; reduces to Στ = Iα when I is constant and L = Iω.

Rolling without slipping speed constraint (v_cm = ωR)

Pure rolling condition: the center-of-mass speed equals angular speed times radius, because the contact point is instantaneously at rest relative to the ground.

Rolling without slipping acceleration constraint (a_cm = αR)

For pure rolling along a surface: the center-of-mass tangential acceleration relates to angular acceleration by a_cm = αR.

Static friction in pure rolling

In rolling without slipping, friction is typically static; it can be nonzero to provide the needed torque, and in the ideal model it often does no mechanical work because the contact point is instantaneously at rest.

Rolling with slipping

If static friction is insufficient, the object slips: vcm ≠ ωR, friction becomes kinetic (fk = μ_k N), kinetic friction does negative work, and mechanical energy is not conserved.