Rotational Dynamics Complete Study Notes

Core Objectives

  • Distinguish between translational and rotational motion of rigid bodies.
  • Define the moment of inertia of a particle and solid bodies about an axis of rotation.
  • Determine the moment of a force about an axis of rotation.
  • Determine the moment of a couple.
  • Apply Newton's second law to a solid in rotation about a fixed axis.
  • Apply the conditions of equilibrium of a solid body.

Translational Motion vs. Rotational Motion

  • Translational Motion:

    • When a rigid body slides along a plane (such as a box sliding on a surface), all of its points cover equal distances during equal time intervals.
    • At any given instant, all points of the body possess identical velocity v⃗\vec{v} and identical acceleration a⃗\vec{a}.

  • Rotational Motion:

    • When a rigid body rotates about a fixed axis (such as a ceiling fan spinning about its vertical shaft), different points on the body do not cover equal distances during equal time intervals.
    • Points located at different distances from the axis of rotation have different linear velocities v⃗\vec{v} and different linear accelerations a⃗\vec{a}.

Translational vs Rotational Motion

Translational Inertia

  • Definition of Mass as Translational Inertia:

    • Mass is a direct measure of a particle's translational inertia (its resistance to changes in linear translational motion).

  • Mathematical Derivation & Comparison:

    • Consider two objects with different masses, mm and MM, where m<Mm < M.
    • Applying the same net force FF to both objects according to Newton's second law for translation:
    • For mass mm: F=ma1→a1=FmF = m a_1 \rightarrow a_1 = \frac{F}{m}
    • For mass MM: F=Ma2→a2=FMF = M a_2 \rightarrow a_2 = \frac{F}{M}
    • Since m<Mm < M, it follows directly that a1>a2a_1 > a_2.
    • Consequently, as the mass of an object increases, a given applied force produces a smaller linear acceleration. Therefore, mass quantifies the object's resistance to translational acceleration.

Moment of Inertia in Rotation

  • Definition:

    • Moment of inertia (II) represents the rotational analog of mass. It measures a body's resistance to angular acceleration about a specified axis of rotation (Δ\Delta).
    • SI Unit: kg⋅m2\text{kg}\cdot\text{m}^2

  • Moment of Inertia Formulas for Various Systems and Solids:

    • Single Particle of mass mm:
    • Axis placed at a distance rr from the particle:       I=mr2I = m r^2
    • Particle Moment of Inertia
    • System of Discrete Particles:
    • System composed of particles of masses mim_i placed at respective distances rir_i from axis Δ\Delta:       I=I1+I2+⋯=∑miri2I = I_1 + I_2 + \dots = \sum m_i r_i^2
    • System of Particles
    • Hoop of mass MM and radius RR (Central Axis):
    • Axis along the central symmetry axis of the hoop:       I=MR2I = M R^2
    • Hoop Central Axis
    • Hoop of mass MM and radius RR (Diameter Axis):
    • Axis along any diameter of the hoop:       I=12MR2I = \frac{1}{2} M R^2
    • Hollow Cylinder of mass MM and radius RR:
    • Axis along the central axis of the cylinder:       I=12MR2I = \frac{1}{2} M R^2
    • Hollow Cylinder
    • Solid Cylinder of mass MM and base radius RR:
    • Axis along the central axis of the cylinder:       I=12MR2I = \frac{1}{2} M R^2
    • Solid Cylinder
    • Thin Rod of mass MM and length LL (Center Axis):
    • Axis perpendicular to the rod, passing through its center of mass:       I=112ML2I = \frac{1}{12} M L^2
    • Thin Rod Center Axis
    • Thin Rod of mass MM and length LL (End Axis):
    • Axis perpendicular to the rod, passing through one of its extremities (ends):       I=13ML2I = \frac{1}{3} M L^2
    • Thin Rod End Axis
    • Solid Sphere of mass MM and radius RR:
    • Axis along any of its diameters:       I=25MR2I = \frac{2}{5} M R^2
    • Solid Sphere
    • Disk of mass MM and radius RR:
    • Axis through the central axis of the disk:       I=MR2I = M R^2

Moment of a Force About an Axis of Rotation

  • Rotational Effectiveness of Forces:

    • Only forces (or force components) that are perpendicular to the axis of rotation Δ\Delta can cause rotation of a body around that axis.
    • Forces whose line of action intersects the axis of rotation Δ\Delta or are parallel to Δ\Delta produce no rotational effect, meaning their moment about Δ\Delta is zero (M=0\mathcal{M} = 0).

  • General Mathematical Formula:   M=F×d×sin⁡(F⃗,d⃗)\mathcal{M} = F \times d \times \sin(\vec{F}, \vec{d})

    • Where:
    • FF is the magnitude of the applied force in Newtons (N\text{N}).
    • dd is the perpendicular distance from the axis of rotation Δ\Delta to the force line of action in meters (m\text{m}).
    • sin⁡(F⃗,d⃗)\sin(\vec{F}, \vec{d}) evaluates to 11 or −1-1 depending on the direction of torque relative to the chosen positive direction of rotation.
    • SI Unit: Newton-meter (N⋅m\text{N}\cdot\text{m}).

  • Illustrative Cases (Door Movable About Hinge Axis Δ\Delta):

    • Case 1: A weight force W⃗\vec{W} parallel to Δ\Delta has no rotational effect:     MW⃗/Δ=0\mathcal{M}_{\vec{W}/\Delta} = 0
    • Case 2: A force F⃗′\vec{F}' whose line of action intersects the axis Δ\Delta has no rotational effect:     MF⃗′/Δ=0\mathcal{M}_{\vec{F}'/\Delta} = 0
    • Case 3: A force F⃗\vec{F} perpendicular to Δ\Delta produces a rotational effect:     MF⃗/Δ=±F×d\mathcal{M}_{\vec{F}/\Delta} = \pm F \times d
    • M=+F×d\mathcal{M} = +F \times d if the force rotates the door in the designated positive direction.
    • M=−F×d\mathcal{M} = -F \times d if the force rotates the door in the designated negative direction.

  • System Subject to Multiple Forces:

    • For a rigid object rotating about an axis Δ\Delta under the action of three coplanar forces F⃗1\vec{F}_1, F⃗2\vec{F}_2, and F⃗3\vec{F}_3 with lever arms d1d_1, d2d_2, and d3d_3:     ∑M/Δ=MF⃗1/Δ+MF⃗2/Δ+MF⃗3/Δ=F1d1+F2d2−F3d3\sum \mathcal{M}/\Delta = \mathcal{M}_{\vec{F}_1/\Delta} + \mathcal{M}_{\vec{F}_2/\Delta} + \mathcal{M}_{\vec{F}_3/\Delta} = F_1 d_1 + F_2 d_2 - F_3 d_3
    • The net moment M\mathcal{M} can be positive or negative depending on the selected orientation of positive rotational sense.
    • Multiple Forces Moment

Moment of a Couple

  • Definition of a Couple:

    • A couple consists of two forces F⃗1\vec{F}_1 and F⃗2\vec{F}_2 that:
    • Are parallel to each other and non-collinear (do not share the same line of action).
    • Have opposite directions in translation (F⃗1=−F⃗2\vec{F}_1 = -\vec{F}_2).
    • Have identical magnitudes (F1=F2=FF_1 = F_2 = F).
    • Are symmetrically located with respect to the axis of rotation Δ\Delta.
    • Practical examples of applying a couple include turning a steering wheel or using a cross lug wrench on a car wheel.

  • Moment of a Driving Couple:

    • For two forces spaced a distance LL apart acting to rotate a body in the positive rotational direction about a central axis Δ\Delta:     Mcouple/Δ=MF⃗1/Δ+MF⃗2/Δ=F1(L2)+F2(L2)=F⋅L\mathcal{M}_{\text{couple}/\Delta} = \mathcal{M}_{\vec{F}_1/\Delta} + \mathcal{M}_{\vec{F}_2/\Delta} = F_1 \left(\frac{L}{2}\right) + F_2 \left(\frac{L}{2}\right) = F \cdot L

  • Moment of a Breaking / Opposing Couple:

    • If the two couple forces act in the direction opposing the positive rotational motion:     Mcouple/Δ=−F1(L2)−F2(L2)=−F⋅L\mathcal{M}_{\text{couple}/\Delta} = -F_1 \left(\frac{L}{2}\right) - F_2 \left(\frac{L}{2}\right) = -F \cdot L

  • Sign Conventions for Moments:

    • If the body rotates in the positive sense: M/Δ>0\mathcal{M}/\Delta > 0
    • If the body rotates in the negative sense: M/Δ<0\mathcal{M}/\Delta < 0

Newton's Second Law for Rotation

  • Derivation for a Particle in Circular Motion:

    • Consider a particle AA of mass mm moving in a circular path of radius RR about a fixed axis Δ\Delta under the action of a force F⃗\vec{F}.
    • Resolve F⃗\vec{F} into tangential component F⃗T\vec{F}_T and normal component F⃗N\vec{F}_N:     F⃗=F⃗T+F⃗N=ma⃗T+ma⃗N\vec{F} = \vec{F}_T + \vec{F}_N = m \vec{a}_T + m \vec{a}_N
    • Calculate the moment of F⃗\vec{F} about Δ\Delta:     MF⃗/Δ=MF⃗T/Δ+MF⃗N/Δ\mathcal{M}_{\vec{F}/\Delta} = \mathcal{M}_{\vec{F}_T/\Delta} + \mathcal{M}_{\vec{F}_N/\Delta}
    • Since F⃗N\vec{F}_N intersects the axis Δ\Delta, MF⃗N/Δ=0\mathcal{M}_{\vec{F}_N/\Delta} = 0.
    • Thus:     MF⃗/Δ=FT⋅R=maTR\mathcal{M}_{\vec{F}/\Delta} = F_T \cdot R = m a_T R
    • Substituting linear tangential acceleration aT=Rθ′′a_T = R \theta'' (where θ′′\theta'' is the angular acceleration):     MF⃗/Δ=m(Rθ′′)R=(mR2)θ′′=Iθ′′\mathcal{M}_{\vec{F}/\Delta} = m (R \theta'') R = (m R^2) \theta'' = I \theta''

  • General Formulation of Newton's Second Law for Rotation:   ∑M=Iθ′′\sum \mathcal{M} = I \theta''

  • Comparison of Translation and Rotation Equations & Units:

    • Translation: ∑F⃗=ma⃗\sum \vec{F} = m \vec{a}
    • Net Force ∑F⃗\sum \vec{F} in Newtons (N\text{N})
    • Mass mm in kilograms (kg\text{kg})
    • Linear Acceleration aa in meters per second squared (m/s2\text{m/s}^2)
    • Rotation: ∑M=Iθ′′\sum \mathcal{M} = I \theta''
    • Net Moment ∑M\sum \mathcal{M} in Newton-meters (N⋅m\text{N}\cdot\text{m})
    • Moment of Inertia II in kilogram-meter squared (kg⋅m2\text{kg}\cdot\text{m}^2)
    • Angular Acceleration θ′′\theta'' in radians per second squared (rad/s2\text{rad/s}^2)

Equilibrium Conditions for a Solid Body

  • Definition of Equilibrium:

    • A system is in mechanical equilibrium if it does not undergo any translational or rotational acceleration (remains at rest or moves with constant velocity / angular velocity).

  • Conditions for Complete Equilibrium:

    1. Translational Equilibrium (Newton's 1st law in translation):      ∑F⃗=0\sum \vec{F} = 0
    2. Rotational Equilibrium (Newton's 1st law in rotation):      ∑M=0\sum \mathcal{M} = 0

  • Examples of Static Equilibrium:

    • Chandelier at Rest:
    • Suspended by tension T⃗\vec{T} balancing weight W⃗\vec{W}.
    • ∑F⃗=T⃗+W⃗=0\sum \vec{F} = \vec{T} + \vec{W} = 0
    • ∑M=0\sum \mathcal{M} = 0
    • Chandelier Equilibrium
    • Ceiling Fan at Rest:
    • Suspended vertically without rotation.
    • ∑F⃗=T⃗+W⃗=0\sum \vec{F} = \vec{T} + \vec{W} = 0
    • ∑M=0\sum \mathcal{M} = 0
    • Ceiling Fan Equilibrium

Summary of Kinematic and Dynamic Analogies

  • Translational vs. Rotational Quantities:
Physical ElementTranslational MotionRotational Motion
Accelerationaaθ′′\theta''
InertiaMass mmMoment of Inertia II
Cause of AccelerationNet Force ∑F⃗\sum \vec{F}Net Moment ∑M\sum \mathcal{M}
Newton's 2nd Law∑F⃗ext=ma⃗\sum \vec{F}_{\text{ext}} = m \vec{a}∑M/Δ=Iθ′′\sum \mathcal{M}/\Delta = I \theta''

  • Time Equations of Motion:

  • Uniformly Varied Motion (UVM / UVRM):

    • Translation:
    • Position: x=12at2+v0t+x0x = \frac{1}{2} a t^2 + v_0 t + x_0
    • Velocity: v=at+v0v = a t + v_0
    • Acceleration: a=constanta = \text{constant}
    • Timeless relation: v2−v02=2a(x−x0)v^2 - v_0^2 = 2 a (x - x_0)
    • Rotation:
    • Angular Position: θ=12θ′′t2+θ0′t+θ0\theta = \frac{1}{2} \theta'' t^2 + \theta'_0 t + \theta_0
    • Angular Velocity: θ′=θ′′t+θ0′\theta' = \theta'' t + \theta'_0
    • Angular Acceleration: θ′′=constant\theta'' = \text{constant}
    • Timeless relation: (θ′)2−(θ0′)2=2θ′′(θ−θ0)(\theta')^2 - (\theta'_0)^2 = 2 \theta'' (\theta - \theta_0)

  • Uniform Motion (UM / URM):

    • Translation:
    • Position: x=vt+x0x = v t + x_0
    • Velocity: v=constantv = \text{constant}
    • Acceleration: a=0a = 0
    • Rotation:
    • Angular Position: θ=θ′t+θ0\theta = \theta' t + \theta_0
    • Angular Velocity: θ′=constant\theta' = \text{constant}
    • Angular Acceleration: θ′′=0\theta'' = 0