Rotational Motion: every key term you need (+ practice quiz)

More AP Physics C guides

• Kinematics
• Newton's Laws of Motion
• Work, Energy and Power
• Systems of Particles and Linear Momentum
• Gravitation

Angular Position and Displacement

θ measured in radians; angular displacement Δθ describes rotation; arc length s = rθ links linear and angular measure.

Angular Velocity

ω = dθ/dt; the rate of change of angular position in rad/s; related to linear speed by v = ωr.

Angular Acceleration

α = dω/dt = d²θ/dt²; the rate of change of angular velocity; tangential acceleration a_t = αr.

Rotational Kinematics

For constant α: ω = ω₀ + αt and θ = θ₀ + ω₀t + ½αt²; direct analogs of the linear equations.

Moment of Inertia

I = Σm_i r_i² (or ∫r² dm); rotational analog of mass measuring resistance to angular acceleration; depends on the axis.

Moment of Inertia (Integral)

I = ∫r² dm; integrate the squared distance from the axis over all mass elements for a continuous body.

Common Moments of Inertia

Solid cylinder ½MR², thin hoop MR², solid sphere (2/5)MR², thin rod about center (1/12)ML², rod about end (1/3)ML².

Parallel Axis Theorem

I = I_cm + Md²; the moment of inertia about any axis equals the value about a parallel axis through the CM plus Md².

Torque

τ = rF sinθ = r × F; the rotational effect of a force; depends on the force, its distance from the axis, and its angle.

Lever Arm

The perpendicular distance from the rotation axis to the line of action of the force; torque = force × lever arm.

Newton's Second Law for Rotation

τ_net = Iα; net torque equals moment of inertia times angular acceleration; the rotational analog of F = ma.

Rotational Kinetic Energy

KE_rot = ½Iω²; the kinetic energy of a rotating body; the rotational analog of ½mv².

Angular Momentum

L = Iω for a rigid body about a fixed axis; L = r × p for a particle; a vector measured in kg·m²/s.

Torque and Angular Momentum

τ_net = dL/dt; net torque equals the rate of change of angular momentum, the rotational form of F = dp/dt.

Conservation of Angular Momentum

With no net external torque, L is constant; reducing I (pulling in arms) increases ω to keep Iω fixed.

Rolling Without Slipping

v_cm = ωR and a_cm = αR; the contact point is instantaneously at rest, so static (not kinetic) friction acts.

Energy of Rolling

KE_total = ½Mv_cm² + ½Iω²; a rolling body has both translational and rotational kinetic energy.

Rolling Down an Incline

Objects with smaller I/MR² accelerate faster; a solid sphere beats a hoop of the same mass and radius down a ramp.

Tangential vs Centripetal Acceleration

a_t = αr changes speed along the path; a_c = ω²r = v²/r changes direction toward the center.

Work and Power in Rotation

W = ∫τ dθ and P = τω; the rotational analogs of W = ∫F dx and P = Fv.

Rotational-Linear Analogs

x↔θ, v↔ω, a↔α, m↔I, F↔τ, p↔L; every translational law has a rotational counterpart.

Torque as Cross Product

τ = r × F, magnitude rF sinφ; direction by the right-hand rule; only the perpendicular component of F contributes.

Moment of Inertia (Integral)

I = ∫r² dm; depends on axis choice; for a rod about its end I = ⅓ML², about its center I = (1/12)ML².

Parallel Axis Theorem

I = I_cm + Md²; the moment of inertia is minimum about the axis through the center of mass.

Perpendicular Axis Theorem

For a planar lamina, I_z = I_x + I_y; e.g. a thin disk about a diameter has I = ¼MR².

Rotational Second Law

Στ = Iα about a fixed axis or the center of mass; α = d²θ/dt².

Rolling Without Slipping

v_cm = Rω and a_cm = Rα; static friction acts and does no work; the contact point is instantaneously at rest.

Rolling Down an Incline

a = g sinθ/(1 + I/(MR²)); objects with smaller I/(MR²) reach the bottom first regardless of mass or radius.

Friction Needed for Rolling

f = (I/(MR²)) M a; rolling fails and slipping begins when f exceeds μ_sN, i.e. tanθ > μ_s(1 + MR²/I).

Massive Pulley

Tensions differ on the two sides; T₁ − T₂ = Iα/R for a pulley with rotational inertia I.

Static Equilibrium Conditions

ΣF = 0 and Στ = 0 about any point; choose the pivot to eliminate unknown forces.

Ladder Problem

Wall normal, floor normal, floor friction, weight; take torques about the floor contact to find the wall force.

Physical Pendulum Torque

τ = −Mgd sinθ ≈ −Mgdθ for small angles, giving ω = √(Mgd/I).

Angular Kinematics

θ = θ₀ + ω₀t + ½αt² and ω² = ω₀² + 2αΔθ mirror linear kinematics for constant α.

Rotational Work and Power

W = ∫τ dθ and P = τω; the rotational analog of P = Fv.

Instantaneous Axis of Rotation

A rolling wheel can be viewed as pure rotation about the contact point with I_contact = I_cm + MR².