Rotational Motion Notes

Angular Quantities

  • Deals with rotational motion, kinematics, dynamics involving torque, rotational kinetic energy and angular momentum.

  • Rigid object rotates about a fixed axis.

  • Points in the object moves in circles, centers of these circles lie on the axis of rotation.

  • R: perpendicular distance of a point from the axis of rotation.

  • r: position of a particle with reference to the origin of some coordinate system.

  • Angle θ of a line in the object with respect to some reference line (x-axis).

  • Arc length: l = Rθ (θ in radians).

  • Radian definition: angle subtended by an arc whose length is equal to the radius.

  • 1 \text{ rad} ≈ 57.3°

  • 1 \text{ rev} = 360° = 2π \text{ rad}

  • Angular displacement: Δθ = θ2 - θ1

  • Average angular velocity: ω = Δθ/Δt

  • Instantaneous angular velocity: ω = \lim_{Δt→0} Δθ/Δt = dθ/dt

  • Angular velocity units: radians per second (rad/s).

  • Angular acceleration: α = (ω2 - ω1)/Δt = Δω/Δt

  • Instantaneous angular acceleration: α = \lim_{Δt→0} Δω/Δt = dω/dt

  • Angular acceleration units: radians per second squared (rad/s²).

  • Linear velocity related to angular velocity: v = Rω

  • Tangential linear acceleration: a_{\text{tan}} = Rα

  • Radial (centripetal) acceleration: a_R = v^2/R = ω^2R

  • Frequency: f = v/(2π). Unit: Hertz (Hz).

  • Period: T = 1/f

  • Relationship between angular velocity and frequency: ω = 2πf

Vector Nature of Angular Quantities

  • Angular velocity V and angular acceleration A can be treated as vectors along the axis of rotation.

  • Right-hand rule: fingers of the right hand are curled around the rotation axis and point in the direction of the rotation, then the thumb points in the direction of V .

  • Pseudovectors (Axial Vectors): V and A are pseudovectors because they do not behave like vectors under reflection.

    • Reflection: The reflection of v points in the same direction.

Constant Angular Acceleration

  • Kinematic equations for constant angular acceleration:

    • v = v_0 + αt

    • θ = v_0t + (1/2)αt^2

    • v^2 = v_0^2 + 2αθ

    • v = (v + v_0)/2

Torque

  • Torque: τ = R_⊥ F

  • R_⊥ is the lever arm (or moment arm).

  • τ = RF sin θ where θ is the angle between vectors R and F.

  • Torque units: m⋅N.

  • Net torque: sum of torques, with counterclockwise torques positive and clockwise torques negative.

Rotational Dynamics and Rotational Inertia

  • Newton’s second law for rotation: Στ = Iα

  • Moment of inertia: I = Σmi Ri^2

  • Rotational inertia of an object depends on the mass distribution with respect to the axis of rotation.

  • Newton’s second law for rotation about the center of mass: (Στ){\text{CM}} = I{\text{CM}} α_{\text{CM}}

Solving Problems in Rotational Dynamics

  • Usefulness and power of rough estimates.

  • Newton’s second law for translation: ΣF = ma

Determining Moments of Inertia

  • Experimentally: measure angular acceleration α about a fixed axis due to a known net torque Στ and apply Newton's second law, I = Στ/α

  • Using Calculus: I = ∫R^2 dm

  • Parallel-Axis Theorem: I = I_{\text{CM}} + Mh^2 (h: distance between the two parallel axes)

  • Perpendicular-Axis Theorem:

    • For plane objects, Iz = Ix + I_y

Rotational Kinetic Energy

  • Rotational kinetic energy: K = (1/2) Iω^2

  • Work done by a torque: W = ∫{θ1}^{θ_2} τ dθ

  • Power: P = τω

  • Work-energy principle for rotation: W = (1/2)Iω2^2 - (1/2)Iω1^2

Rotational Plus Translational Motion; Rolling

  • Rolling without slipping: v = Rω

  • Total kinetic energy: K{\text{tot}} = (1/2)I{\text{CM}} ω^2 + (1/2) Mv_{\text{CM}}^2

  • Rolling objects go slower than sliding objects.

Vector Nature of Angular Quantities

  • V and A are pseudovectors (axial vectors).

  • Angular velocity and angular acceleration are vectors.

  • For a rigid object rotating about a fixed axis, both V and A point along the rotation axis.

  • The direction of V is given by the right-hand rule.

Neglected Aspects

  • Real objects are not perfectly rigid.

  • Real objects are not perfectly rigid. Sphere rolling to the Right.

  • The normal force, F, exerts a torque that slows down the sphere.

  • The deformation of the sphere and the surface it moves on have been exaggerated for detail.