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: $Δθ = θ<em>2 - θ</em>1$

• 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: $α = (ω<em>2 - ω</em>1)/Δt = Δω/Δt$

• 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: $I = Σm<em>i R</em>i^2$

• 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: $(Στ)<em>{\text{CM}} = I</em>{\text{CM}} α_{\text{CM}}$

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, $I<em>z = I</em>x + I_y$

Rotational Kinetic Energy

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

• Work done by a torque: $W = ∫<em>{θ</em>1}^{θ_2} τ dθ$

• Power: $P = τω$

• Work-energy principle for rotation: $W = (1/2)Iω<em>2^2 - (1/2)Iω</em>1^2$

Rotational Plus Translational Motion; Rolling

• Rolling without slipping: $v = Rω$

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

• 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.